High School Physics

Why physics begins with measurement

Physics is the study of matter, motion, and energy, and more than almost any other subject it is built on measurement. A physicist does not just say that a moving car is "fast" or that a star is "far away." Instead they attach a number and a unit: the car moves at 27 metres per second, the star is 4.1 × 10³ trillion kilometres away. Turning vague words into precise numbers is what lets physics predict the future, build bridges that stand, and send spacecraft to exactly the right place after a journey of years. Every idea in this course, from acceleration to electric current, is ultimately something you can measure and put a number on.

The reason measurement matters so much is that nature does not care about our opinions. A rope either holds a climber or it snaps, and the difference comes down to real numbers: the force in the rope compared with the force it can withstand. If you say a rope is "5 long," the number 5 by itself means nothing until you add a unit. Five metres is a rope for a small climbing pitch. Five centimetres is barely a shoelace. Five kilometres is longer than most hiking trails. The number and the unit are two halves of a single idea, and neither is complete without the other.

A quantity is a number times a unit

Here is the single most important idea in this lesson. Every physical quantity is written as a number multiplied by a unit, and both parts must travel together through every step of a calculation. When you write a length as 3.5 m, you are really saying "3.5 lots of the standard length we call the metre." The 3.5 is the numerical value and the m is the unit. Physicists treat the unit almost like an algebraic symbol that rides along with the number, and this habit turns out to be one of the strongest error-checking tools you have.

Think about what this means in practice. If you measure a table and get 1.5, that answer is meaningless until you say 1.5 metres, or 1.5 yards, or 1.5 arm-spans. Two students who both "measured 1.5" but used different units did not measure the same table. Whenever you report a result in physics, a bare number with no unit is considered wrong, not merely incomplete. Teachers, examiners, and engineers all treat a missing unit as a real mistake, because in the real world it causes real accidents.

A famous example makes the point. In 1999 NASA lost the Mars Climate Orbiter, a spacecraft worth well over one hundred million dollars, because one engineering team worked in pound-force seconds while another expected newton seconds. The numbers were passed along without their units being reconciled, the navigation was off, and the spacecraft was destroyed in the Martian atmosphere. The lesson for us is humble but permanent: a number without its unit is not just untidy, it can be catastrophic.

The International System of Units (SI)

If every country and every laboratory used its own units, science would be chaos. To avoid this, scientists everywhere agree to use the International System of Units, abbreviated SI from the French Systeme International d'Unites. SI is the modern form of the metric system, and it is the official language of measurement in physics worldwide. When you read a physics paper written in Japan, Brazil, or Germany, the units are the same ones you are about to learn.

SI is built on a small set of base units, each one standing for a fundamental kind of quantity. There are seven base units in total, but for a first physics course a handful do almost all the work:

Quantity	SI base unit	Symbol
Length	metre	m
Mass	kilogram	kg
Time	second	s
Electric current	ampere	A
Temperature	kelvin	K
Amount of substance	mole	mol
Luminous intensity	candela	cd

For this course the three you will use constantly are the metre for length, the kilogram for mass, and the second for time. These three are so central that mechanics is sometimes called the study of the "MKS" system, after metre, kilogram, and second. Notice one small surprise: the base unit of mass is the kilogram, not the gram, even though the kilogram sounds like it already has the prefix "kilo" attached. This is a historical quirk, but it is worth memorizing because it trips up many students.

How the base units are defined today

You might wonder where a unit like the metre actually comes from. For most of history, units were tied to physical objects. The metre was once defined by a platinum bar kept in France, and the kilogram by a metal cylinder in a vault near Paris. The trouble with physical objects is that they can be scratched, can gain a film of dust, or can slowly change, and if the standard changes then every measurement in the world quietly changes with it.

Modern SI solves this by defining the base units using unchanging constants of nature. The second is defined by the vibrations of a cesium atom, which ticks at an extraordinarily steady rate. The metre is then defined as the distance light travels in a vacuum in a tiny, exactly specified fraction of a second, because the speed of light is a fixed constant of nature (exactly 299,792,458 metres per second). Since 2019 even the kilogram is defined using a fundamental constant called the Planck constant rather than a metal cylinder. You do not need these definitions for calculations, but they show something beautiful: our units are now anchored to the universe itself, not to any single object that could be lost or damaged.

Base units versus derived units

The base units are only the beginning. Almost every quantity you meet in physics is measured in a derived unit, which is simply a base unit or a combination of base units multiplied and divided together. A base unit is defined on its own; a derived unit is built out of base units.

• Speed is length divided by time, so its unit is the metre per second, written m/s.
• Area is length times length, so its unit is the square metre, written m².
• Volume is length cubed, so its unit is the cubic metre, m³.
• Density is mass divided by volume, so its unit is the kilogram per cubic metre, kg/m³.

Some derived units are used so often that they are given their own names. The unit of force is the newton (N), but a newton is not truly a new fundamental unit. One newton is defined as exactly one kilogram-metre per second squared (kg·m/s²), because force equals mass times acceleration, and acceleration is measured in m/s². In the same way the joule (J) of energy is a newton-metre, and the watt (W) of power is a joule per second. When you learn these formulas later, you will see that their units always trace back to metres, kilograms, and seconds. This is one of the deep satisfactions of physics: the whole toolbox is built from a few simple pieces.

Metric prefixes: scaling by powers of ten

One of the great strengths of SI is that it scales by powers of ten using prefixes. A prefix is a word part like kilo or milli that multiplies a unit by a fixed factor. Because the factors are all powers of ten, you never have to memorize awkward conversions like twelve inches in a foot or 5280 feet in a mile. You just shift a decimal point. The common prefixes worth memorizing are:

Prefix	Symbol	Factor	Example
giga	G	1,000,000,000	1 GW = 1,000,000,000 W
mega	M	1,000,000	1 MJ = 1,000,000 J
kilo	k	1000	1 km = 1000 m
(base unit)	-	1	1 m = 1 m
centi	c	1/100	1 cm = 0.01 m
milli	m	1/1000	1 mm = 0.001 m
micro	µ	1/1,000,000	1 µm = 0.000001 m
nano	n	1/1,000,000,000	1 nm = 0.000000001 m

Notice that the symbol for milli is a lowercase m, the very same letter used for the metre. Context always makes the meaning clear: "mm" means millimetre, where the first m is the prefix and the second is the unit. The symbol for micro is the Greek letter mu, written µ. These prefixes appear everywhere in daily life once you notice them: a kilometre on a road sign, a megabyte or gigabyte of computer storage, a milligram of a vitamin, a nanometre in the size of a virus.

Worked example 1: reading a prefix

Question: How many metres is 3.5 km?

Solution: The prefix kilo means 1000, so you multiply by 1000. 3.5 km = 3.5 × 1000 m = 3500 m. Going the other way, how many metres is 250 cm? The prefix centi means one hundredth, so 250 cm = 250 × 0.01 m = 2.5 m. A useful sanity check runs through both answers: when you move to a smaller unit the number gets bigger, and when you move to a larger unit the number gets smaller. There are many small centimetres in a length but only a few large kilometres, so 3.5 km became a big 3500 m, while 250 cm became a small 2.5 m.

Worked example 2: chaining prefixes

Question: A computer chip feature is 90 nm wide. Express this in metres, and in micrometres.

Solution: Nano means one billionth, so 90 nm = 90 × 0.000000001 m = 0.00000009 m = 9 × 10⁻⁸ m. To convert to micrometres, remember that one micrometre is 1000 nanometres. So 90 nm = 90 / 1000 µm = 0.09 µm. This kind of tiny length is why engineers who build computer chips work at the very edge of what is physically possible.

Worked example 3: mass with a prefix

Question: A tablet contains 500 mg of a medicine. How many grams is this, and how many kilograms?

Solution: Milli means one thousandth, so 500 mg = 500 × 0.001 g = 0.5 g. To go to kilograms, recall that 1 kg is 1000 g, so 0.5 g = 0.5 / 1000 kg = 0.0005 kg. Notice how the same physical amount of medicine can be written as 500 mg, 0.5 g, or 0.0005 kg. All three are correct; the prefix simply chooses a convenient size of number for the situation. A pharmacist naturally uses milligrams, because grams and kilograms would bury the useful digits behind zeros.

Analogy: units are like currencies

Units and prefixes work much like money. A price is a number times a currency: 5 dollars and 5 euros are different amounts, just as 5 metres and 5 feet are different lengths. Prefixes are like the relationship between cents and dollars: 100 cents make a dollar just as 100 centimetres make a metre. When you convert 350 cents to 3.5 dollars, you are doing exactly the same mental move as converting 350 cm to 3.5 m. Nobody is confused that the same amount of money can be "350 cents" or "3.5 dollars," and in the same way the same length can be "350 cm" or "3.5 m." The physics is not harder than the shopping.

Real-world applications

Getting units right is not an academic nicety; it is the backbone of engineering and medicine. A nurse who confuses milligrams with grams could give a patient a thousand times too much of a drug. A pharmacist reads doses in milligrams and micrograms every day. An electrician thinks in kilowatts and megawatts. A civil engineer designing a dam works in cubic metres of water and meganewtons of force. Even your phone quietly does unit work: it reports distances in kilometres or miles depending on your region, download sizes in megabytes and gigabytes, and screen features measured in millimetres. Once you start looking, SI units and their prefixes are the hidden grammar of the modern world.

Common misconceptions

• "The gram is the SI base unit of mass." It is not. The base unit is the kilogram. This is the single most common SI mistake, and it is worth repeating to yourself until it sticks.
• "A number alone can be a physics answer." No. A physical quantity is always a number and a unit together. Writing "the speed is 20" is wrong; it must be 20 m/s, or 20 km/h, and those are different speeds.
• "Kilo always means one thousand of anything, so a kilobyte is exactly 1000 bytes." In pure SI, kilo means 1000 exactly. In computing, the same word is sometimes used loosely for 1024, which is why a separate prefix "kibi" exists for the binary value. In physics you should always take kilo to mean exactly 1000.
• "Bigger unit, bigger number." It is the opposite. Switching to a larger unit gives a smaller number, because fewer of the larger units fit in the same amount. 2000 m becomes only 2 km.
• "Units do not really matter as long as I keep track of the numbers." The Mars Climate Orbiter shows how expensive this belief can be. Units are as much a part of the answer as the digits.

Recap

Physics is built on measurement, and every physical quantity is a number multiplied by a unit, with both parts inseparable. Scientists worldwide use the International System of Units, or SI, whose base units include the metre for length, the kilogram (not the gram) for mass, and the second for time. Every other unit is a derived unit built from these, such as m/s for speed or the newton for force, which is really kg·m/s². The metric system scales by powers of ten using prefixes: kilo means 1000, centi means one hundredth, milli one thousandth, micro one millionth, and so on. Converting between prefixes just shifts a decimal point, and a quick sanity check is that moving to a smaller unit gives a larger number. Carrying units through every calculation is one of the most reliable ways to catch mistakes before they cause harm.

Sources

1. OpenStax, "College Physics 2e," Chapter 1: The Nature of Science and Physics (Sections 1.2 and 1.3 on units and unit conversion). Available free at openstax.org.
2. National Institute of Standards and Technology (NIST), "The International System of Units (SI)," NIST Special Publication 330, and the NIST Reference on Constants, Units, and Uncertainty.
3. The Physics Classroom, "1-D Kinematics" and the tutorial on Units and Measurement, at physicsclassroom.com.
4. Bureau International des Poids et Mesures (BIPM), "The International System of Units (SI Brochure)," 9th edition.

Why we convert units at all

In the last lesson you learned that a physical quantity is a number times a unit. But the same quantity can be dressed in many different units. A car's speed might be given as 90 kilometres per hour on a speedometer, yet physics formulas almost always expect metres per second. A recipe might list an oven at 180 degrees Celsius while a physics problem wants kelvin. Being able to switch between units cleanly, without accidentally changing the actual amount, is a core skill. Get it wrong and every calculation that follows is wrong, no matter how careful your algebra. This lesson gives you a foolproof method based on conversion factors, and then teaches scientific notation, the shorthand physicists use to tame the enormous and the tiny.

Conversion factors: fractions that equal one

A conversion factor is a fraction that equals exactly 1, built from a relationship between two units. Because 1000 metres is the same length as 1 kilometre, the fraction (1000 m / 1 km) is a fraction whose top and bottom are equal amounts, so it equals 1. The upside-down version (1 km / 1000 m) also equals 1, for the same reason. This is the whole trick: multiplying any quantity by 1 never changes its value, only its appearance. When you multiply a length in kilometres by (1000 m / 1 km), you are multiplying by 1, so the length is unchanged, but the km unit cancels and you are left reading the answer in metres.

The art is choosing which version of the conversion factor to use. The rule is simple: pick the version whose bottom unit matches the unit you want to get rid of, so it cancels. If you have a length in km and want metres, you need km on the bottom, so you use (1000 m / 1 km). If you have a length in m and want km, you need m on the bottom, so you flip it to (1 km / 1000 m). Units cancel just like numbers in a fraction: a km on top and a km on the bottom cross out, leaving only the unit you wanted.

Worked example 1: a simple length conversion

Question: Convert 4.5 km to metres using a conversion factor.

Solution: You want to cancel km, so put km on the bottom of the factor. 4.5 km × (1000 m / 1 km) = 4500 m. The km on top of your starting value cancels the km on the bottom of the factor, and you are left with metres. The answer, 4500 m, is the same length as 4.5 km, just measured in smaller units, so the number is larger. That is exactly what you should expect.

Worked example 2: converting a speed, step by step

Car speeds are usually given in kilometres per hour, but nearly every physics formula uses metres per second. This conversion has two parts, because both the distance unit and the time unit must change. Convert 90 km/h to m/s, cancelling units as you go:

1. Start with the given value: 90 km/h, which means 90 kilometres every hour.
2. Change kilometres to metres using (1000 m / 1 km): 90 km/h × (1000 m / 1 km) = 90,000 m/h. The km cancels, and now the speed is in metres per hour.
3. Change hours to seconds. There are 3600 seconds in an hour (60 minutes times 60 seconds), so multiply by (1 h / 3600 s): 90,000 m/h × (1 h / 3600 s) = 25 m/s. The hour cancels, leaving metres per second.

So 90 km/h equals 25 m/s. There is a handy shortcut hiding in this arithmetic. Converting km/h to m/s always means multiplying by 1000 and dividing by 3600, which is the same as dividing by 3.6. Check: 90 / 3.6 = 25. It matches. From now on you can convert km/h to m/s in your head by dividing by 3.6, and go the other way by multiplying by 3.6.

Worked example 3: converting an area

Question: A floor is 3 m². How many square centimetres is that?

Solution: This one traps many students. It is tempting to say "100 cm in a metre, so 300 cm²," but that is wrong, because area involves length squared. Since 1 m = 100 cm, then 1 m² = (100 cm) × (100 cm) = 10,000 cm². So 3 m² = 3 × 10,000 cm² = 30,000 cm². The conversion factor for area is the square of the conversion factor for length. The same idea means a volume conversion uses the cube: 1 m³ = 1,000,000 cm³.

Scientific notation: taming very big and very small numbers

Physics constantly deals with the very large and the very small. The distance from the Earth to the Sun is about 150,000,000 metres. The diameter of a hydrogen atom is about 0.0000000001 metres. Writing out all those zeros is slow and error-prone; miscount the zeros and your answer is off by a factor of ten or more. To handle this, physicists use scientific notation: every number is written as a value between 1 and 10 multiplied by a power of ten.

• 150,000,000 m becomes 1.5 × 10⁸. The decimal point moved 8 places to the left to sit after the first digit, so the exponent is +8.
• 0.00042 s becomes 4.2 × 10⁻⁴. The decimal point moved 4 places to the right to sit after the first non-zero digit, so the exponent is -4.

The rule is precise: rewrite the number so exactly one non-zero digit sits in front of the decimal point, then count how many places you moved the point. Moving the point to the left (making the front number smaller) gives a positive exponent, because the original number was large. Moving the point to the right (for a small decimal) gives a negative exponent. The exponent is really a count of how many places the decimal shifted, with a sign that records the direction.

Reading exponents as size

A positive exponent tells you how many zeros a whole number has beyond the leading digits: 10³ is 1000, 10⁶ is a million, 10⁹ is a billion. A negative exponent tells you how far to the right of the decimal point the digits sit: 10⁻³ is 0.001, 10⁻⁶ is one millionth. This connects directly to the prefixes from the last lesson: kilo is 10³, mega is 10⁶, milli is 10⁻³, micro is 10⁻⁶. Scientific notation and metric prefixes are two faces of the same powers-of-ten idea.

Multiplying and dividing in scientific notation

Scientific notation is not only compact; it makes arithmetic with huge numbers easy. To multiply, multiply the front numbers and add the exponents.

Example: (2 × 10³) × (4 × 10²) = (2 × 4) × 10^(3+2) = 8 × 10⁵. The front numbers 2 and 4 multiply to 8, and the exponents 3 and 2 add to 5.

To divide, divide the front numbers and subtract the exponents.

Example: (8 × 10⁶) / (2 × 10²) = (8 / 2) × 10^(6-2) = 4 × 10⁴.

Sometimes the front number ends up outside the range 1 to 10, and you tidy it up. For instance (5 × 10³) × (4 × 10²) = 20 × 10⁵, but 20 is 2 × 10, so the neat form is 2 × 10⁶. This final tidying step is called normalizing, and it keeps every answer in standard form.

Order of magnitude

The order of magnitude of a quantity is just the power of ten nearest its size, ignoring the front digits. The Sun's distance, 1.5 × 10⁸ m, has order of magnitude 10⁸. An atom, about 10⁻⁶ m across, has order of magnitude 10⁻⁶. Comparing orders of magnitude is a fast way to grasp how quantities relate: the Sun's distance and an atom's size differ by about 10¹⁸, an almost unimaginable ratio, yet scientific notation lets you state it in a single line. Physicists often estimate an answer to the nearest order of magnitude first, as a sanity check, before computing the exact value.

Units as a built-in safety check

The deepest reason to carry units through every calculation is that they catch mistakes automatically. If you are solving for a time and your units come out as metres, you know something went wrong before you ever look at the number. Treating units as algebra, letting them multiply, divide, and cancel just like variables, is one of the most reliable habits in all of physics. Experienced physicists sometimes rebuild a half-remembered formula purely by asking what combination of the given quantities produces the correct units for the answer. This technique, called dimensional analysis, will serve you throughout the course.

Real-world applications

Unit conversion is everywhere outside the classroom. Pilots convert between feet, metres, and nautical miles. Chemists convert between moles, grams, and litres. Anyone traveling between countries converts kilometres and miles, litres and gallons, Celsius and Fahrenheit. Scientific notation is the native language of astronomy (distances of 10²⁰ metres), of chemistry (Avogadro's number, about 6 × 10²³ particles in a mole), and of computing (a terabyte is about 10¹² bytes). Anyone who works with the very large or very small learns to think in powers of ten, because it is the only sane way to keep the zeros straight.

Common misconceptions

• "To convert km/h to m/s I just divide by 1000." No. You must convert both the distance and the time. The correct single-step shortcut is to divide by 3.6, not by 1000.
• "1 m² is 100 cm²." Wrong. Because area is length squared, 1 m² = 10,000 cm². Always square the length conversion factor for areas and cube it for volumes.
• "A negative exponent means the number is negative." Not at all. 4.2 × 10⁻⁴ is a small positive number, 0.00042. The negative sign is about the decimal point's position, not the number's sign.
• "When multiplying powers of ten, I multiply the exponents." No, you add them: 10³ times 10² is 10⁵, not 10⁶. You only multiply exponents when raising a power to a power.
• "A conversion factor changes the amount." A properly built conversion factor equals exactly 1, so it never changes the amount, only the units it is written in.

Recap

A conversion factor is a fraction equal to 1, built from a relationship between two units, and multiplying by it changes the unit without changing the amount. Choose the version that puts the unwanted unit on the bottom so it cancels. Converting km/h to m/s takes two steps or the shortcut of dividing by 3.6. Area and volume conversions need the length factor squared or cubed. Scientific notation writes any number as a value between 1 and 10 times a power of ten, moving the decimal point until one non-zero digit sits in front; leftward moves give positive exponents, rightward moves give negative ones. To multiply, multiply the front numbers and add the exponents; to divide, divide and subtract. Order of magnitude, the nearest power of ten, gives a quick feel for size. Above all, carry units through every step, because they will warn you when something has gone wrong.

Sources

1. OpenStax, "College Physics 2e," Chapter 1, Sections 1.3 (Unit Conversion and Dimensional Analysis) and 1.4 (Approximation). Free at openstax.org.
2. The Physics Classroom, "Metric System" and "Scientific Notation" tutorials, at physicsclassroom.com.
3. HyperPhysics, "Units, Powers of 10, and Scientific Notation," hosted by Georgia State University at hyperphysics.phy-astr.gsu.edu.
4. National Institute of Standards and Technology (NIST), "Guide for the Use of the International System of Units (SI)," Special Publication 811.

Every measurement has a limit

No measuring tool is perfect, and no measurement is exact. A ruler marked in millimetres cannot tell you a length to the nearest micrometre. A kitchen scale reading in whole grams cannot detect a difference of half a gram. Every instrument has a finest division it can resolve, and any measurement it gives you carries some uncertainty in its final digit. The digits you can genuinely trust in a measurement are called its significant figures. Learning to count them, and to round answers to the right number of them, is part of the honesty of science. Reporting more digits than you actually measured is a subtle kind of dishonesty: it claims a precision you never had. This lesson shows you how to count significant figures, how to round the results of calculations correctly, and how to tell the important difference between accuracy and precision.

What a significant figure really means

Suppose you measure a pencil with a ruler marked in millimetres and read 14.3 cm. The 1 and the 4 are certain, and the 3 is your best estimate of the fraction between the marks. All three digits carry real information about the pencil, so all three are significant. If you wrote 14.30 cm, you would be claiming you could resolve hundredths of a centimetre, which the ruler cannot do, so that extra zero would be a false claim. The number of significant figures is a compact statement of how good your measuring tool was. This is why 3 m, 3.0 m, and 3.00 m are not the same statement in physics: they claim precision to the nearest metre, tenth of a metre, and hundredth of a metre respectively.

Counting significant figures: the rules

A small set of rules covers every case. Work through them slowly the first few times and they will soon become automatic.

• All non-zero digits count. The number 24.7 has three significant figures; 815 has three; 6.4832 has five.
• Zeros between non-zero digits count. These are called sandwiched or captive zeros. 2005 has four significant figures; 40.06 has four; 7.001 has four.
• Leading zeros never count. Zeros before the first non-zero digit only mark the decimal place. 0.0042 has two significant figures (just the 4 and the 2); 0.0000500 has three (the 5 and the two trailing zeros).
• Trailing zeros after a decimal point count. 3.60 has three significant figures, because the final zero was deliberately measured and written; 0.500 has three; 12.000 has five.
• Trailing zeros in a whole number with no decimal point are ambiguous. In 1500, it is unclear whether the zeros were measured. To be unambiguous, write it in scientific notation: 1.5 × 10³ has two significant figures, while 1.500 × 10³ has four.

Worked example 1: counting significant figures

Question: State the number of significant figures in each: (a) 0.00560, (b) 30.05, (c) 8000, (d) 1.020.

Solution: (a) 0.00560 has three. The three leading zeros do not count; the 5, the 6, and the trailing 0 do. (b) 30.05 has four; the two zeros are sandwiched between non-zero digits. (c) 8000 written this way is ambiguous, usually taken as just one significant figure unless more context is given; writing 8.000 × 10³ would make it four. (d) 1.020 has four; the middle zero is captive and the trailing zero after the decimal counts.

Rounding the results of calculations

When you combine measurements in a calculation, the answer cannot be more precise than the least precise measurement that went into it. A chain is only as strong as its weakest link. Two rules cover most situations:

• Multiplication and division: the answer keeps as many significant figures as the factor with the fewest significant figures.
• Addition and subtraction: the answer keeps as many decimal places as the term with the fewest decimal places.

These rules stop a calculator from inventing precision. A calculator will happily display 9.030000, but if your least precise measurement had only two significant figures, most of those digits are fiction.

How to round

To round to a chosen number of significant figures, look at the first digit you are about to drop. If it is 5 or more, round the last kept digit up; if it is 4 or less, leave the last kept digit unchanged. For example, rounding 12.746 to three significant figures: keep 1, 2, 7, and look at the next digit, 4, which is less than 5, so you round down and get 12.7. Rounding 12.751 to three significant figures: keep 1, 2, 7, look at the next digit, 5, which means round up, giving 12.8.

Worked example 2: significant figures in a product

Question: A rectangle is measured as 4.2 m by 2.15 m. What is its area, reported correctly?

Solution: On the calculator, 4.2 × 2.15 = 9.03 m². Now apply the multiplication rule. The factor 4.2 has two significant figures, and 2.15 has three; the smaller is two. So the answer must be rounded to two significant figures: 9.0 m². Reporting 9.03 would falsely claim you had measured to the nearest hundredth of a square metre, when your least precise ruler reading was only good to two figures. Notice that we keep the trailing zero in 9.0 to show that both figures are significant.

Worked example 3: significant figures in a sum

Question: Add the measured lengths 12.11 m, 8.4 m, and 1.075 m.

Solution: The raw sum is 12.11 + 8.4 + 1.075 = 21.585 m. For addition, the rule is about decimal places, not significant figures. The term 8.4 has only one decimal place, the fewest, so the answer is rounded to one decimal place: 21.6 m. The single loosely-measured length limits the whole sum.

Accuracy versus precision

These two words are used loosely in everyday speech, but in physics they mean genuinely different things, and confusing them causes real errors in reasoning.

• Accuracy is how close a measurement is to the true value. An accurate scale reads close to the real mass.
• Precision is how close repeated measurements are to one another, regardless of whether they are correct. A precise scale gives nearly the same reading every time.

The two are independent. A scale that always reads 2.000 kg for an object whose true mass is 1.5 kg is highly precise (its readings agree with each other perfectly) but not accurate (they are all wrong by the same amount). Such a consistent error is called a systematic error, and precision can never reveal it, because the readings agree with each other while all being wrong. On the other hand, readings that scatter widely around the true value are not precise, even if their average happens to be accurate; that scatter is called random error.

The dartboard picture

A dartboard makes the distinction vivid. Imagine the bullseye is the true value and each dart is a measurement. A tight cluster of darts far from the bullseye is precise but not accurate: the throws agree with one another but all miss the same way, like a scale with a systematic error. Darts scattered widely but centred on the bullseye are accurate on average but not precise: the individual throws disagree, showing random error, even though they average out near the truth. The ideal, of course, is a tight cluster right on the bullseye: both accurate and precise.

Uncertainty and the last digit

A careful scientist reports not just a value but a range of uncertainty, often written with a plus-or-minus. A length recorded as 14.3 plus or minus 0.1 cm says the true value almost certainly lies between 14.2 and 14.4 cm. The significant-figure rules are really a quick, informal way of tracking this uncertainty: the last significant figure is understood to be the uncertain one. When you write 9.0 m², you are quietly saying the true area is close to 9.0, with the tenths digit being the doubtful one. This is why keeping the correct number of significant figures is not fussy bookkeeping; it is a compressed honesty about how much you really know.

Real-world applications

Significant figures and the accuracy-precision distinction matter far beyond the classroom. A machinist cutting an engine part to 50.00 mm is working to a very different tolerance than one cutting to 50 mm; the extra figures cost more and mean more. A laboratory calibrating a thermometer cares whether its readings are accurate (close to a known standard) as well as precise (repeatable). Drug manufacturing, aircraft engineering, and satellite navigation all live or die by understanding the uncertainty in every measured quantity. Even a fitness tracker that always reads your heart rate a few beats too high is precise but inaccurate, and knowing that difference helps you interpret the number.

Common misconceptions

• "More digits always means a better answer." No. Digits beyond your measurement's precision are noise dressed up as information. A good answer has exactly the right number of significant figures, no more.
• "Leading zeros count." They never do. 0.0042 has two significant figures. Leading zeros only locate the decimal point.
• "Accurate and precise mean the same thing." They are independent. You can be precise but wrong (systematic error) or accurate on average but scattered (random error).
• "Rounding should be done at every step." Rounding repeatedly during a long calculation can accumulate error. Keep extra digits through the working and round only the final answer.
• "The trailing zero in 3.60 is pointless." On the contrary, it is meaningful: it states that the hundredths place was measured and found to be zero, giving three significant figures.

Recap

Every measurement is limited by its instrument, and its trustworthy digits are its significant figures. Non-zero digits always count, sandwiched zeros count, leading zeros never count, and trailing zeros after a decimal point count. When multiplying or dividing, the answer keeps as many significant figures as the least precise factor; when adding or subtracting, it keeps as many decimal places as the least precise term. Round by looking at the first dropped digit: 5 or more rounds up. Accuracy is closeness to the true value, while precision is agreement among repeated measurements, and the two are independent, as the dartboard picture shows. The last significant figure is understood to be uncertain, which is why reporting the right number of figures is really a statement of how much you know.

Sources

1. OpenStax, "College Physics 2e," Chapter 1, Section 1.3 (Accuracy, Precision, and Significant Figures). Free at openstax.org.
2. The Physics Classroom, "Measurement and Uncertainty" resources, at physicsclassroom.com.
3. National Institute of Standards and Technology (NIST), "Guidelines for Evaluating and Expressing the Uncertainty of Measurement Results," Technical Note 1297.
4. HyperPhysics, "Significant Figures," Georgia State University, at hyperphysics.phy-astr.gsu.edu.

Understanding uncertainty keeps you honest. In physics, a well-reported answer says both what you measured and how sure you are of the last digit.

The starting point of all motion

The branch of physics that describes motion without yet asking what causes it is called kinematics. Before you can explain why a ball speeds up or a car turns, you first need a precise vocabulary for describing where things are and how they move. This lesson builds the four foundation words of that vocabulary: distance, displacement, speed, and velocity. They sound like everyday synonyms, and in casual conversation people use them interchangeably. In physics they are not the same, and the differences matter enormously. Getting them straight now will make every later topic, from acceleration to momentum, far easier, because those topics are all built on top of these four ideas.

The single organizing idea behind this lesson is the split between quantities that have only a size and quantities that have a size and a direction. Two of our four words (distance and speed) care only about size. The other two (displacement and velocity) carry a direction as well. That one distinction explains almost every "surprising" result in this lesson, including the fact that you can travel a long way and yet have moved nowhere.

Scalars and vectors

A scalar is a quantity described completely by a single number and a unit. Temperature is a scalar: 21 degrees Celsius tells you everything, and there is no "direction" to a temperature. Mass, time, and energy are also scalars. When you add scalars you just add the numbers: 3 kg of flour plus 2 kg of flour is 5 kg of flour, no direction required.

A vector is a quantity that needs both a size (called its magnitude) and a direction to be fully described. "Walk 5 metres" is not a complete instruction, because you could walk 5 metres in any direction and end up somewhere different each time. "Walk 5 metres east" is complete. Force, displacement, and velocity are all vectors. Because vectors carry a direction, adding them is not as simple as adding the numbers; direction has to be taken into account, and two vectors of the same size pointing in opposite directions can cancel out entirely.

Keep this scalar-versus-vector idea in the front of your mind for the whole lesson. Distance and speed are the scalars. Displacement and velocity are the vectors. Every worked example below is really just an exercise in remembering which is which.

Distance: how far you traveled

Distance is the total length of the path an object actually travels. It is a scalar, so it has no direction, and it can only ever increase or stay the same as motion continues. Distance does not care where you started or where you finished, only about the total ground covered along the way. If you walk around every aisle of a supermarket and come back to the entrance, the distance you walked might be several hundred metres even though you finished exactly where you began.

Think of distance as what a car's odometer measures, or what a fitness tracker counts as your steps. Those devices tick upward with every metre you cover and never tick back down, no matter which way you turn. That "only ever grows" behavior is the signature of a scalar path length.

Displacement: how far and in what direction your position changed

Displacement is the straight-line change in position from the starting point to the finishing point, together with the direction from start to finish. It is a vector. Displacement does not care about the winding path you took in between; it draws a single straight arrow from where you began to where you ended and reports the length and direction of that arrow.

Because displacement only looks at start and end, it can be smaller than the distance, and it can even be zero. If you jog one full lap of a 400 m running track and stop at the starting line, your distance is 400 m but your displacement is zero, because your final position is the same as your starting position: the arrow from start to finish has no length at all. If instead you jog exactly half a lap, your distance is 200 m along the curved track, but your displacement is the straight line across the field to the opposite side, which is shorter than 200 m and points in a definite direction. Distance follows the track; displacement cuts straight across.

A useful rule to remember: displacement can never be larger than distance, because the straight line between two points is always the shortest path between them. They are equal only when the motion is in a single straight line without doubling back. The moment the path curves or reverses, the distance pulls ahead of the size of the displacement.

Worked example 1: comparing distance and displacement

Question: A hiker walks 3 km east, then turns and walks 4 km north. Find the total distance walked and the size of the displacement.

Solution: The distance is simply the total path length: 3 km + 4 km = 7 km. The displacement, however, is the straight-line arrow from the start to the end. Because the eastward and northward legs are at right angles, the straight-line distance is the hypotenuse of a right triangle with sides 3 km and 4 km. Using the Pythagorean theorem, the size of the displacement is the square root of (3 squared plus 4 squared), which is the square root of (9 + 16) = the square root of 25 = 5 km, directed toward the northeast. So the hiker walked a distance of 7 km but was displaced only 5 km from the start. The distance is larger, exactly as expected once the path bends.

Speed: how fast, without direction

Speed is how fast an object moves, calculated as distance divided by the time taken. It is a scalar, inheriting the directionless nature of distance. Speed answers the question "how many metres of path are covered each second?" and nothing more. The everyday number on a car's speedometer is a speed: it reads 60 km/h whether you are driving north, south, or around a bend.

The formula for average speed is the total distance divided by the total time:

average speed = total distance / total time

Speed is measured in metres per second (m/s) in SI units, though kilometres per hour and miles per hour are common in daily life. Because it comes from distance, average speed can never be negative and, over any real trip, is never smaller than the size of the average velocity.

Velocity: speed with a direction

Velocity is the rate of change of position, calculated as displacement divided by time. It is a vector, so it always carries a direction. Velocity answers a richer question than speed: not just "how fast?" but "how fast, and which way?" Driving at 60 km/h is a speed; driving at 60 km/h due north is a velocity. Two cars travelling the same road at the same speedometer reading but in opposite directions have the same speed and different velocities, because their directions differ.

The formula for average velocity uses displacement, not distance:

average velocity = displacement / total time

Because it is built from displacement, average velocity can be zero even when you have been moving the whole time, and it points in the direction of the overall change in position. This is the deepest and most tested idea in the lesson: over a round trip that returns to the start, the average velocity is exactly zero, while the average speed is not.

Worked example 2: a round trip

Question: You walk 300 m east in 200 s, then walk 300 m back west to your starting point in 100 s. Find the average speed and the average velocity for the whole trip.

Solution: First gather the totals. The total distance is the whole path length: 300 m + 300 m = 600 m. The total time is 200 s + 100 s = 300 s. The average speed is therefore 600 m / 300 s = 2 m/s. Now the displacement: because you finished exactly where you started, the straight-line change in position is zero. So the average velocity is 0 m / 300 s = 0 m/s. The average speed is 2 m/s while the average velocity is zero. That contrast is the entire point of separating these ideas: you genuinely moved (your feet covered 600 m of ground), yet your position did not change overall, so your velocity averaged to nothing.

Worked example 3: a straight-line trip

Question: A cyclist covers 150 m due north in 30 s without ever turning around. Find the average speed and the average velocity.

Solution: Because the motion is in a single straight line with no doubling back, the distance and the size of the displacement are equal: both are 150 m. The average speed is 150 m / 30 s = 5 m/s. The average velocity is 150 m north / 30 s = 5 m/s north. Here speed and velocity have the same size, 5 m/s, and differ only in that velocity also names the direction. This is the special case where the two quantities agree in magnitude: straight-line motion with no reversal.

Worked example 4: an unequal round trip

Question: A delivery drone flies 800 m east in 40 s, then 600 m west in 30 s. Find the total distance, the displacement, the average speed, and the average velocity.

Solution: The total distance is 800 m + 600 m = 1400 m. The displacement is the net change in position: taking east as positive, the drone ends up 800 - 600 = 200 m east of its start. The total time is 40 s + 30 s = 70 s. So the average speed is 1400 m / 70 s = 20 m/s, while the average velocity is 200 m / 70 s = 2.86 m/s east (to three significant figures). Notice how different the two numbers are: the average speed of 20 m/s reflects all the flying the drone did, while the far smaller average velocity reflects how little its overall position changed. Whenever a trip involves any reversal, expect the average speed to exceed the size of the average velocity.

Average versus instantaneous

Everything above concerns average quantities, which summarize a whole trip with a single number. But during a real journey your motion changes from moment to moment. Instantaneous velocity is how fast and in what direction you are moving at one exact instant, and instantaneous speed is just its magnitude. The number a car speedometer shows at any given second is an instantaneous speed. You might average 50 km/h over a whole commute (the average) while at various instants reading 0 km/h at a red light and 90 km/h on the motorway (instantaneous values).

The relationship is that instantaneous velocity is what you get when you measure the average velocity over a tiny interval of time, so short that the motion barely changes during it. In a full physics course this idea leads to calculus, but for now the key point is simply the distinction: an average smooths the whole trip into one figure, while an instantaneous value is a snapshot at a single moment.

Positive and negative in one dimension

When motion happens along a single straight line, we can capture direction with a plus or minus sign instead of writing "east" or "north." You first choose a positive direction, say east or "to the right," and then a velocity of +5 m/s means moving that way while a velocity of -5 m/s means moving the opposite way at the same speed. This sign convention is how vectors are handled in one dimension, and it is why a velocity can be negative while a speed (its size) never is. In worked example 4, choosing east as positive let us write the west leg as a subtraction, which is exactly the vector idea in action.

Real-world applications

The distance-displacement distinction is not a classroom trick; it shows up wherever motion is measured. A GPS unit in a car reports both the distance driven (useful for fuel and tolls) and the straight-line displacement to your destination (useful for "as the crow flies" estimates), and the two rarely match because roads are not straight. Air-traffic controllers care about an aircraft's velocity, direction and all, not merely its speed, because two planes with the same speed heading toward each other are in danger while two heading apart are safe. Athletes and coaches track average speed over a race but instantaneous speed at the finishing sprint. Even ocean navigation depends on velocity: a current that pushes a boat sideways changes its actual displacement even if its speed through the water is unchanged. In every one of these cases, confusing "how far traveled" with "how far displaced," or "how fast" with "how fast and which way," would give a wrong and sometimes dangerous answer.

Common misconceptions

• "Distance and displacement are the same thing." They agree only for straight-line motion with no reversal. The instant a path curves or doubles back, distance exceeds the size of the displacement, and over a round trip the displacement can be zero while the distance is large.
• "Speed and velocity are just two words for the same idea." Speed is a scalar (size only); velocity is a vector (size and direction). Two objects can share a speed and have different velocities because they move in different directions.
• "If I have been moving, my average velocity cannot be zero." It certainly can. Return to your starting point and your displacement, and therefore your average velocity, is exactly zero no matter how far you walked.
• "Displacement can be bigger than distance." Never. The straight line between two points is the shortest path, so the size of the displacement is always less than or equal to the distance.
• "A negative velocity means the object is slowing down." No. A negative sign only indicates direction (motion opposite to the chosen positive direction). An object moving steadily at -8 m/s is moving at a constant fast speed, not slowing at all.
• "The speedometer shows my velocity." It shows instantaneous speed only, with no direction, so strictly it displays a speed, not a velocity.

Recap

Kinematics describes motion, and it starts with four carefully distinguished quantities. Distance is the total path length, a scalar that only grows. Displacement is the straight-line change in position with a direction, a vector that can be zero over a round trip and is never larger than the distance. Speed is distance divided by time, a directionless scalar, found on a car speedometer. Velocity is displacement divided by time, a vector carrying a direction, which averages to zero over any trip that returns to its start. Average quantities summarize a whole journey, while instantaneous quantities are snapshots at a single moment. In one dimension, direction is recorded with a plus or minus sign, which is why a velocity can be negative but a speed cannot. Keep the scalar-versus-vector split in mind and every result in this lesson follows naturally.

Sources

1. OpenStax, "College Physics 2e," Chapter 2 (Kinematics), Sections 2.1 through 2.4 on displacement, distance, speed, and velocity. Free at openstax.org.
2. The Physics Classroom, "1-D Kinematics," Lesson 1 (Describing Motion with Words) and the tutorial on Distance and Displacement, at physicsclassroom.com.
3. HyperPhysics, "Motion, Distance, and Displacement," Georgia State University, at hyperphysics.phy-astr.gsu.edu.
4. OpenStax, "University Physics Volume 1," Chapter 3 (Motion Along a Straight Line), for the scalar and vector treatment of position and velocity. Free at openstax.org.

From steady motion to changing motion

In the last lesson you learned to describe motion at a steady velocity. But the world is rarely so tidy. Cars pull away from lights and brake for corners, balls speed up as they fall, planets curve around the Sun. Whenever velocity is not constant, a new quantity is needed to describe how it changes, and that quantity is acceleration. Acceleration is one of the most important ideas in all of physics because, as you will see in the module on forces, it is the direct link between motion and the forces that cause it. Master acceleration now and Newton's laws will feel almost obvious later.

The word "acceleration" in everyday speech usually means "speeding up," and the "accelerator" pedal in a car reinforces that. In physics the word is broader and more precise. Acceleration means any change in velocity, whether that change is getting faster, getting slower, or turning. Because velocity is a vector with both a size and a direction, changing either one counts as an acceleration. Holding that broader definition in mind is the key to this lesson.

Defining acceleration

Acceleration is the rate of change of velocity, meaning how much the velocity changes divided by the time it takes to change. In symbols:

a = (v - u) / t

Here u is the initial velocity (the velocity at the start of the time interval), v is the final velocity (the velocity at the end), and t is the time taken for the change. The top of the fraction, v - u, is the change in velocity, often written as the single symbol delta-v. The bottom is the time over which that change happened.

The units follow directly from the formula. Velocity is measured in metres per second (m/s) and time in seconds (s), so acceleration has units of (m/s) divided by s, which is metres per second squared, written m/s². This unit looks strange the first time you meet it, but it has a plain-English meaning: an acceleration of 3 m/s² means the velocity changes by 3 metres per second during each and every second. After one second the velocity has grown by 3 m/s; after two seconds by 6 m/s; and so on. Reading the unit as "metres per second, per second" makes its meaning clear.

Acceleration is a vector

Like velocity, acceleration has a direction as well as a size, so it is a vector. Its direction is the direction of the change in velocity, which is not always the direction of motion. When an object speeds up, the acceleration points the same way as the velocity. When an object slows down, the acceleration points opposite to the velocity. This is why the sign of an acceleration matters so much in one dimension, and why we must keep careful track of positive and negative.

Three ways to accelerate

Because velocity is a vector, there are three distinct ways an object can accelerate. Two of them surprise students who think acceleration only means speeding up.

• Speeding up. The speed increases while the direction stays the same. The acceleration points in the direction of motion. This is the everyday meaning of the word.
• Slowing down. The speed decreases. The acceleration points opposite to the motion. This is often called deceleration, and in one dimension it shows up as a negative acceleration when motion is taken as positive.
• Changing direction. Even if the speed never changes, turning is an acceleration, because the direction of the velocity vector is changing. A car rounding a bend at a steady 20 m/s is accelerating the entire time, and a planet in a circular orbit is constantly accelerating even though its speed may be nearly constant.

That third case is the one to remember. "Constant speed" is not the same as "no acceleration." Only constant velocity, meaning unchanging speed and unchanging direction, guarantees zero acceleration.

Worked example 1: speeding up

Question: A car speeds up from 10 m/s to 30 m/s in 4 s. Find its acceleration.

Solution: Identify the quantities: u = 10 m/s, v = 30 m/s, and t = 4 s. Apply the definition: a = (v - u) / t = (30 - 10) / 4 = 20 / 4 = 5 m/s². The positive answer tells you the car is speeding up, gaining 5 m/s of velocity each second. As a check, after the 4 seconds the velocity has grown by 5 m/s² times 4 s = 20 m/s, taking it from 10 to 30 m/s, which matches the given values.

Worked example 2: braking

Question: A cyclist slows from 12 m/s to a stop in 3 s. Find the acceleration.

Solution: Here u = 12 m/s, v = 0 m/s (a stop means the final velocity is zero), and t = 3 s. Then a = (0 - 12) / 3 = -12 / 3 = -4 m/s². The negative sign is meaningful: it shows the velocity is decreasing, so the acceleration points opposite to the motion. The cyclist is decelerating at 4 m/s². Notice that the size of the number, 4, tells you how fast the slowing happens, while the minus sign tells you the direction.

Worked example 3: solving for a different unknown

Question: A train accelerates from rest at a steady 0.5 m/s². How long does it take to reach 30 m/s?

Solution: This time the unknown is the time, so rearrange the definition. Starting from a = (v - u) / t and multiplying both sides by t then dividing by a gives t = (v - u) / a. Substitute u = 0, v = 30 m/s, and a = 0.5 m/s²: t = (30 - 0) / 0.5 = 60 s. The train needs a full minute to reach 30 m/s at that gentle acceleration. Being able to rearrange the formula to solve for whichever quantity is unknown is a skill you will use constantly.

Worked example 4: acceleration from a change of direction

Question: A ball moving east at 5 m/s bounces straight back off a wall and now moves west at 5 m/s. The contact lasts 0.02 s. Find the acceleration, taking east as positive.

Solution: The speed is unchanged at 5 m/s, but the velocity has reversed, so there is a large acceleration. With east positive, the initial velocity is u = +5 m/s and the final velocity is v = -5 m/s. The change in velocity is v - u = -5 - (+5) = -10 m/s. Then a = -10 / 0.02 = -500 m/s², directed west. This enormous acceleration, purely from a reversal of direction, is why a hard, fast bounce feels so violent even when the speed barely changes. It also shows vividly that changing direction is a genuine acceleration.

Free fall and the acceleration g

One acceleration appears so often in physics that it has its own symbol. Near the Earth's surface, any object falling with only gravity acting on it, a condition called free fall, accelerates downward at about 9.8 m/s². Physicists call this value g, the acceleration due to gravity. Drop a stone and its downward speed grows by 9.8 m/s every second: about 9.8 m/s after one second, about 19.6 m/s after two seconds, about 29.4 m/s after three. The same g governs a thrown ball at the top of its arc, a diver leaving a board, and the proverbial falling apple, which is why g reappears throughout this course.

A famous and surprising fact about free fall is that, ignoring air resistance, all objects accelerate at the same g regardless of their mass. A heavy hammer and a light feather dropped together in a vacuum hit the ground at the same instant. This was demonstrated dramatically on the Moon by an Apollo 15 astronaut, and it contradicts the natural but mistaken belief that heavier things always fall faster. Heavier objects are pulled harder by gravity, but they are also harder to accelerate, and these two effects cancel exactly.

Worked example 5: a falling stone

Question: A stone is dropped from rest. Ignoring air resistance and taking g as 9.8 m/s², what is its downward speed after 2.5 s?

Solution: Dropped from rest means u = 0. The acceleration is g = 9.8 m/s² downward, and the time is t = 2.5 s. Rearranging the definition to solve for the final velocity gives v = u + a times t = 0 + 9.8 times 2.5 = 24.5 m/s, directed downward. Each second added 9.8 m/s of downward speed, and after two and a half seconds that comes to 24.5 m/s.

Real-world applications

Acceleration is measured and managed everywhere. Car makers advertise "zero to sixty" times, which are really statements about acceleration; a sports car that reaches 27 m/s (about 60 mph) in 3 seconds has a much larger acceleration than a family car that takes 10 seconds. Engineers design roller coasters around the accelerations riders feel, because it is the changes in velocity, not the speed itself, that press you into your seat or lift you out of it. Aircraft carriers use catapults to give jets enormous accelerations for a short take-off. Smartphones contain tiny accelerometers that sense the acceleration of the phone to rotate the screen or count steps. Even safety devices depend on it: an airbag deploys when a sensor detects the sudden, large deceleration of a crash. In each case the useful quantity is not how fast something is going, but how quickly that velocity is changing.

Common misconceptions

• "Acceleration only means speeding up." It means any change in velocity: speeding up, slowing down, or changing direction. Slowing and turning are both accelerations.
• "Constant speed means no acceleration." Only constant velocity (unchanging speed and direction) means zero acceleration. Moving in a circle at constant speed is still accelerating, because the direction is changing.
• "A negative acceleration always means slowing down." A negative sign only indicates direction. If an object is already moving in the negative direction, a negative acceleration actually speeds it up. Slowing down happens whenever the acceleration points opposite to the velocity, whatever their signs.
• "Heavier objects fall with a larger acceleration." Ignoring air resistance, all objects fall with the same acceleration g, because the greater gravitational pull on a heavier object is exactly offset by its greater resistance to being accelerated.
• "If the velocity is momentarily zero, the acceleration must be zero too." Not so. A ball thrown straight up is momentarily at rest at the very top of its flight, yet its acceleration there is still g downward, which is precisely why it starts falling back down.

Recap

Acceleration is the rate of change of velocity, defined as the change in velocity divided by the time taken, a equals (v minus u) over t, and measured in metres per second squared. Because velocity is a vector, acceleration is a vector too, and there are three ways to accelerate: speeding up, slowing down (deceleration, a negative acceleration when motion is positive), and changing direction, which counts even at constant speed. The sign of an acceleration tells you its direction, not whether an object is fast or slow. Free fall is motion under gravity alone, with the special acceleration g of about 9.8 m/s² downward, the same for all objects regardless of mass when air resistance is ignored. Rearranging the defining equation lets you solve for velocity, time, or acceleration as needed.

Sources

1. OpenStax, "College Physics 2e," Chapter 2 (Kinematics), Sections 2.4 (Acceleration) and 2.7 (Falling Objects). Free at openstax.org.
2. The Physics Classroom, "1-D Kinematics," Lesson 1 (Acceleration) and the "Free Fall and the Acceleration of Gravity" tutorial, at physicsclassroom.com.
3. HyperPhysics, "Acceleration" and "Free Fall," Georgia State University, at hyperphysics.phy-astr.gsu.edu.
4. NIST, "The International System of Units (SI)," for the standard value of the acceleration due to gravity and SI units of acceleration, at nist.gov.

Why draw motion as a graph?

A paragraph describing a journey can be long and hard to follow: "the car sat still, then sped up, then cruised, then braked." A graph tells the same story in a single picture you can read at a glance, and it does something words cannot: it lets you extract exact numbers, like the speed at a certain moment or the total distance covered, straight off the page. Learning to read and draw motion graphs is one of the most powerful skills in introductory physics, because a graph packs the whole history of a motion into one shape.

Two kinds of graph do almost all the work in kinematics: the distance-time graph (sometimes drawn as position versus time) and the velocity-time graph. They look similar at first, but they mean very different things, and confusing them is the single most common mistake students make. The secret to both is knowing what two features tell you: the slope of the line, and, for velocity-time graphs, the area underneath it. Get those two ideas straight and every motion graph becomes readable.

What slope means

The slope of a line on any graph is its steepness, calculated as the "rise" (the change up the vertical axis) divided by the "run" (the change along the horizontal axis). A steep line has a large slope; a flat line has a slope of zero. The reason slope is so useful in motion graphs is that the rise-over-run happens to equal a meaningful physical quantity, and exactly which quantity depends on what the vertical axis measures. This is the through-line of the whole lesson: slope is always rise over run, and on a motion graph that ratio is a real physical rate.

Distance-time graphs

On a distance-time graph, time runs along the horizontal axis and distance up the vertical axis. Since slope is rise over run, here it is distance divided by time, which is exactly speed. So the slope of a distance-time graph tells you how fast the object is moving. The steeper the line, the faster the motion.

• A flat, horizontal line means distance is not changing, so the object is at rest (speed zero).
• A straight line sloping upward means the distance grows at a steady rate, so the object moves at constant speed.
• A steeper straight line means a faster constant speed.
• A curve that gets steeper over time means the slope is increasing, so the object is speeding up (accelerating). A curve that flattens out means it is slowing down.

Worked example 1: reading speed from a slope

Question: On a distance-time graph a straight line rises from 0 m to 40 m as time goes from 0 s to 8 s. What is the speed?

Solution: The slope is the rise divided by the run. The rise is 40 m - 0 m = 40 m and the run is 8 s - 0 s = 8 s, so the slope is 40 m / 8 s = 5 m/s. Because the line is straight, this speed is constant, so the object moves steadily at 5 m/s the whole time. Any straight segment can be read this way: pick two points on it, find the rise and run, and divide.

Worked example 2: a multi-stage journey

Question: A distance-time graph shows three straight segments. From 0 to 4 s the line rises from 0 m to 20 m. From 4 s to 7 s the line is flat at 20 m. From 7 s to 10 s it rises from 20 m to 50 m. Describe the motion in each stage.

Solution: In the first stage the slope is 20 m / 4 s = 5 m/s, a constant speed of 5 m/s. In the middle stage the line is flat, so the slope is zero and the object is at rest for 3 seconds. In the final stage the slope is (50 - 20) m / (10 - 7) s = 30 m / 3 s = 10 m/s, a faster constant speed of 10 m/s. Reading a graph in stages like this turns one picture into a full narrative: move, stop, then move faster.

Velocity-time graphs

On a velocity-time graph the vertical axis measures velocity instead of distance, and this one change alters the meaning of everything. Now two features carry information.

• The slope is rise over run, which here is velocity divided by time, and that is exactly acceleration. A steeper line means a larger acceleration.
• The area under the line, between the line and the time axis, equals the distance traveled. This is a new and powerful idea with no equivalent on a distance-time graph.

Reading the shapes: a horizontal line means the velocity is not changing, so the acceleration is zero and the object moves at constant velocity. A line sloping upward means the velocity is increasing, so the object is speeding up. A line sloping downward means the velocity is decreasing, so the object is slowing down. A line at zero velocity means the object is at rest.

Why area equals distance

The area rule is worth understanding, not just memorizing. Distance equals speed multiplied by time. On a velocity-time graph the height of the line is the speed and the width along the axis is the time, so height times width, which is the area of the region under the line, is exactly speed times time, which is distance. For a constant velocity the region is a rectangle; for a steadily changing velocity it is a triangle or a trapezoid. Whatever the shape, its area in the units of the axes gives the distance.

Worked example 3: distance from a rectangular area

Question: On a velocity-time graph an object moves at a constant 6 m/s for 10 s, shown as a horizontal line at a height of 6. How far does it travel?

Solution: The region under the line is a rectangle with height 6 m/s and width 10 s. Its area is height times width = 6 m/s times 10 s = 60 m. So the object travels 60 m. The area-equals-distance rule works for any shape; you simply break a complicated area into rectangles and triangles and add them up.

Worked example 4: distance from a triangular area

Question: An object starts at rest and its velocity rises steadily to 8 m/s over 4 s. Find its acceleration and the distance it covers.

Solution: The graph is a straight line from the origin up to 8 m/s at 4 s, forming a triangle with the time axis. The acceleration is the slope: (8 - 0) m/s / (4 - 0) s = 2 m/s². The distance is the area of the triangle: one half times base times height = one half times 4 s times 8 m/s = 16 m. So the object accelerates at 2 m/s² and covers 16 m while speeding up. Notice this single graph gave us both an acceleration (from the slope) and a distance (from the area), which is why velocity-time graphs are so valuable.

Worked example 5: a trapezoid

Question: A car accelerates from 4 m/s to 12 m/s over 6 s. Find the distance it travels during this time.

Solution: The velocity-time graph is a straight line rising from 4 to 12, and the region under it is a trapezoid. The area of a trapezoid is the average of the two parallel sides times the width: the average velocity is (4 + 12) / 2 = 8 m/s, and the time is 6 s, so the distance is 8 m/s times 6 s = 48 m. Equivalently you could split the region into a 4 by 6 rectangle (24 m) plus a triangle of area one half times 6 times 8 (24 m), which also totals 48 m. Both methods agree, a reassuring check.

Telling the two graphs apart

Because the two graphs look alike, keep a simple summary in mind. On a distance-time graph, a horizontal line means the object is stopped, and the slope is the speed. On a velocity-time graph, a horizontal line means constant velocity (still moving), the slope is the acceleration, and the area is the distance. The most dangerous trap is treating a flat line the same on both: flat on a distance-time graph means "not moving," but flat on a velocity-time graph means "moving steadily." Always check which quantity is on the vertical axis before you interpret anything.

Real-world applications

Motion graphs are the working language of anyone who studies movement. Sports scientists plot an athlete's velocity against time to see exactly when they reach top speed and how quickly they decelerate. Vehicle "black box" recorders and engineers analyzing crashes read velocity-time traces to reconstruct what happened in the seconds before impact, using the slope to find the braking deceleration and the area to find how far the vehicle traveled. Fitness apps draw pace and elevation graphs from GPS data. Train and transit systems design their acceleration and braking profiles, literally the slopes of velocity-time graphs, for both comfort and safety. In physics laboratories, plotting your data as a graph and reading its slope is often the cleanest way to measure a quantity like acceleration, because the slope of a straight-line fit averages out the small errors in individual points.

Common misconceptions

• "A flat line always means the object is stopped." Only on a distance-time graph. On a velocity-time graph a flat line means constant velocity, so the object is still moving, just not speeding up or slowing down.
• "A distance-time graph and a velocity-time graph of the same motion look the same." They usually look quite different. A constant velocity is a sloping line on a distance-time graph but a flat horizontal line on a velocity-time graph.
• "The area under a distance-time graph means something." It does not have a useful physical meaning. The area rule applies to velocity-time graphs, where area equals distance.
• "A steeper line on a velocity-time graph means faster." It means a larger acceleration, not a higher speed. Speed is the height of the line; steepness is the acceleration.
• "A line going downward on a velocity-time graph means the object is moving backward." A downward slope means the object is slowing down. It is moving backward only if the line actually drops below zero velocity.

Recap

Motion graphs turn a story of movement into a shape you can read for exact numbers. On a distance-time graph, time is horizontal and distance vertical, and the slope (rise over run) is the speed; a flat line means at rest, a straight slope means constant speed, and a curve means changing speed. On a velocity-time graph, the slope is the acceleration and the area under the line is the distance traveled; a flat line means constant velocity, an upward slope means speeding up, and a downward slope means slowing down. Area equals distance because height (speed) times width (time) is distance, and any region can be split into rectangles and triangles. The key discipline is to check which quantity is on the vertical axis before interpreting the graph, because a flat line means opposite things on the two kinds of plot.

Sources

1. OpenStax, "College Physics 2e," Chapter 2 (Kinematics), Sections 2.3 and 2.8 on graphical analysis of one-dimensional motion. Free at openstax.org.
2. The Physics Classroom, "1-D Kinematics," Lesson 3 (Describing Motion with Position-Time Graphs) and Lesson 4 (Velocity-Time Graphs), at physicsclassroom.com.
3. HyperPhysics, "Motion Graphs," Georgia State University, at hyperphysics.phy-astr.gsu.edu.
4. OpenStax, "University Physics Volume 1," Chapter 3 (Motion Along a Straight Line), for slope and area interpretations of motion graphs. Free at openstax.org.

Turning definitions into problem-solving tools

So far you have met velocity and acceleration as definitions. This lesson forges them into a small toolkit of equations, the kinematic equations, that can answer almost any question about motion in a straight line, provided the acceleration is constant. "Constant acceleration" means the velocity changes by the same amount every second, which is exactly the situation for a car accelerating steadily, a train braking evenly, or any object in free fall. Under that one condition, these equations let you predict where an object will be and how fast it will be going at any future time, which is precisely the kind of prediction that makes physics so powerful.

The equations connect five quantities, and it is worth naming them carefully because the symbols are used throughout physics. They are: displacement s (how far the object moves, in metres); initial velocity u (the velocity at the start, in m/s); final velocity v (the velocity at the end, in m/s); acceleration a (the constant acceleration, in m/s²); and time t (the duration, in seconds). The elegant thing is that you never need all five at once. Each equation involves four of the five quantities and leaves one out, so you choose the equation that contains the three things you know and the one thing you want.

The three main equations

Here are the three equations that do most of the work, along with which quantity each one leaves out. Knowing what each equation omits is the fastest way to pick the right one.

Equation	Leaves out	Use when you want
v = u + at	displacement	final velocity from time
s = ut + ½at²	final velocity	displacement from time
v² = u² + 2as	time	final velocity from distance

A fourth useful relation, s = ½(u + v)t, uses the average of the starting and ending velocities and leaves out the acceleration; it is handy when you know both velocities and the time but not the acceleration. These four together cover every combination you are likely to meet.

Where the equations come from

These formulas are not magic; they follow directly from the definitions in the previous lessons. The first, v = u + at, is simply the definition of acceleration rearranged: since acceleration is the change in velocity over time, the final velocity is the starting velocity plus the extra velocity gained, which is a times t. The second, s = ut + ½at², is the area under a velocity-time graph: a rectangle of height u and width t (giving ut) plus a triangle of the growing velocity (giving one half a t squared), which you saw in the graphing lesson. The third, v² = u² + 2as, comes from combining the first two to eliminate time. You do not need to derive them each time, but knowing they come from the definitions should reassure you that they are trustworthy, not arbitrary.

A reliable method for every problem

Kinematics problems become easy once you follow a fixed routine. First, list the quantities you are given, writing each with its symbol and unit, and note which quantity you are asked to find. Second, scan the table for the equation that contains exactly those letters, the three you know and the one you want, and leaves out the one you neither know nor need. Third, substitute the numbers with their units and solve algebraically. Keeping the units attached at every step catches most errors before they can spread. This "list, choose, substitute, solve" method is worth practicing until it is automatic.

Worked example 1: a launching car

Question: A car starts from rest and accelerates at a steady 3 m/s² for 5 s. Find its final velocity and how far it travels.

Solution: List the givens: starting from rest means u = 0, the acceleration is a = 3 m/s², and the time is t = 5 s. First find the final velocity. It depends on time, so use the equation that omits displacement: v = u + at = 0 + 3 times 5 = 15 m/s. Now find the distance. It also depends on time, so use the equation that omits final velocity: s = ut + ½at² = 0 + ½ times 3 times 5² = ½ times 3 times 25 = 37.5 m. So the car reaches 15 m/s having traveled 37.5 m. Because it started from rest, the u times t term vanished, which is a common simplification worth watching for.

Worked example 2: using the no-time equation

Question: A motorcycle accelerates from 8 m/s to 20 m/s over a distance of 56 m. Find its acceleration.

Solution: List the givens: u = 8 m/s, v = 20 m/s, and s = 56 m. The unknown is the acceleration, and notice that no time is given and none is wanted, so choose the equation that omits time: v² = u² + 2as. Substitute: 20² = 8² + 2 times a times 56, which is 400 = 64 + 112a. Subtract 64 from both sides: 336 = 112a. Divide by 112: a = 3 m/s². Recognizing when time is missing is the trigger to reach for this third equation.

Worked example 3: free fall

Question: A ball is dropped from rest and falls for 2 s. Taking g as 9.8 m/s² downward, find its speed and how far it falls.

Solution: Free fall is just constant-acceleration motion with a = g = 9.8 m/s², so the same equations apply. Dropped from rest means u = 0, and t = 2 s. Speed: v = u + at = 0 + 9.8 times 2 = 19.6 m/s. Distance fallen: s = ut + ½at² = 0 + ½ times 9.8 times 2² = ½ times 9.8 times 4 = 19.6 m. The ball is moving at 19.6 m/s after dropping 19.6 m. (It is a coincidence of the numbers that the speed and distance come out equal here; they are different quantities with different units.)

Worked example 4: a ball thrown upward

Question: A ball is thrown straight up at 19.6 m/s. Taking upward as positive and g as 9.8 m/s² downward (so a = -9.8 m/s²), how high does it rise before stopping?

Solution: At the highest point the ball is momentarily at rest, so the final velocity is v = 0. The givens are u = 19.6 m/s, v = 0, and a = -9.8 m/s², and the unknown is the displacement s, with time neither given nor wanted, so use v² = u² + 2as. Substitute: 0 = 19.6² + 2 times (-9.8) times s, which is 0 = 384.16 - 19.6s. Solving, 19.6s = 384.16, so s = 19.6 m. The careful use of signs, with the upward velocity positive and the downward acceleration negative, is what makes projectile problems work.

Worked example 5: braking distance

Question: A car travelling at 20 m/s brakes with a constant acceleration of -4 m/s². How far does it travel before stopping?

Solution: The givens are u = 20 m/s, v = 0 (it stops), and a = -4 m/s². Time is not given, so use v² = u² + 2as: 0 = 20² + 2 times (-4) times s, so 0 = 400 - 8s, giving s = 50 m. The car needs 50 m to stop. This kind of calculation is exactly how road-safety engineers work out stopping distances, and it shows why doubling your speed more than doubles your stopping distance, because the speed is squared in the equation.

Real-world applications

The kinematic equations are working tools far beyond the classroom. Traffic engineers use the braking-distance equation to set speed limits and design safe stopping sight-distances on roads. Aerospace engineers calculate how much runway a plane needs to reach take-off speed, an application of the very first equations you learned here. Sports analysts compute how high a jumper rises or how fast a ball leaves a bat. Amusement-park designers plan the speeds and distances of drops and launches. Anyone modelling the motion of a falling or thrown object, from a dropped tool on a construction site to a spacecraft firing its engines at a steady thrust, reaches for these same five symbols. The equations are a compact, reliable way to turn a few known facts about a motion into precise predictions of the rest.

Common misconceptions

• "These equations work for any motion." They require constant acceleration. If the acceleration changes during the motion, you must break the problem into stages, each with its own constant acceleration, or use more advanced methods.
• "Starting from rest, u can be left as anything." Starting from rest specifically means u = 0. Forgetting to set the initial velocity to zero is one of the most common errors.
• "At the top of its flight a thrown ball has zero acceleration because it is momentarily still." Its velocity is zero there, but its acceleration is still g downward. That is exactly why it does not hang in the air but begins to fall.
• "I can ignore signs and just use the numbers." Direction matters. For upward or downward motion you must choose a positive direction and give the acceleration and velocities the correct signs, or the answer will be wrong.
• "Doubling the speed doubles the stopping distance." Because v is squared in v² = u² + 2as, doubling the initial speed quadruples the stopping distance for the same braking acceleration.

Recap

When acceleration is constant, the kinematic equations connect displacement, initial velocity, final velocity, acceleration, and time. The three main ones are v = u + at (omits displacement), s = ut + one half a t squared (omits final velocity), and v squared = u squared + 2as (omits time), with s = one half (u + v) t as a useful fourth. They follow from the definitions of velocity and acceleration and from the area under a velocity-time graph. To solve a problem, list what you are given and what you want, pick the equation that contains exactly those letters, substitute with units, and solve. Free fall is simply constant-acceleration motion with a equal to g. Careful attention to signs is essential for upward and downward motion, and because velocity appears squared in the third equation, stopping distance grows with the square of the speed.

Sources

1. OpenStax, "College Physics 2e," Chapter 2 (Kinematics), Sections 2.5 (Motion Equations for Constant Acceleration) and 2.7 (Falling Objects). Free at openstax.org.
2. The Physics Classroom, "1-D Kinematics," Lesson 6 (Kinematic Equations) and the associated problem-solving guide, at physicsclassroom.com.
3. HyperPhysics, "Motion Equations for Constant Acceleration," Georgia State University, at hyperphysics.phy-astr.gsu.edu.
4. OpenStax, "University Physics Volume 1," Chapter 3 (Motion Along a Straight Line), for the derivation of the constant-acceleration equations. Free at openstax.org.

What makes things move and stop

Everything you have studied so far describes motion: how fast things go, how their velocity changes. Now we turn to the cause of it all. The branch of physics that asks why objects move the way they do is called dynamics, and at its heart is a single idea: force. A force is the influence that starts motion, stops it, or changes it. Understanding forces is the gateway to Newton's three laws, which together explain the motion of everything from a rolling ball to a planet. This lesson lays the groundwork by defining what a force is, naming the common forces you will meet again and again, and showing how to combine several forces into one.

The central skill of this entire module is learning to identify every force acting on an object and then add them up correctly. Almost every dynamics problem reduces to that: find the forces, combine them into a single net force, and see what motion results. So it is worth being careful and precise from the very start about what counts as a force and how forces add.

Force: a push or a pull

A force is a push or a pull on an object, an interaction that can change the object's motion. Forces can start a stationary object moving, bring a moving object to rest, speed it up, slow it down, or change its direction. They can also squeeze, stretch, or bend objects. A force always involves an interaction between two things: something does the pushing or pulling, and something is pushed or pulled.

Because a push or pull has both a strength and a direction, force is a vector. A push of 10 newtons to the right is a different force from a push of 10 newtons to the left, even though the strengths are equal, because their directions differ. This vector nature is why we must be so careful about direction when combining forces, and it is why arrows, not just numbers, are used to represent forces in diagrams.

The unit of force: the newton

The SI unit of force is the newton, with the symbol N, named in honour of Isaac Newton. As you will see in the next lesson, one newton is defined as the force needed to give a one-kilogram mass an acceleration of one metre per second squared, so 1 N equals 1 kg·m/s². To get an everyday feel for its size: one newton is roughly the weight of a small apple resting in your hand, and holding a 1 kg bag of sugar means supporting a force of about 9.8 N. These mental anchors help you judge whether a calculated force is reasonable.

The common forces

Only a handful of forces appear in most introductory problems. Learning to recognize them on sight is the first step in solving any dynamics question.

• Gravity, or weight: the downward pull of the Earth on every object with mass. It always acts vertically downward, toward the centre of the Earth. Its size is the object's weight, which you will calculate as mass times g.
• Normal force: the outward push a solid surface exerts on an object resting against it, always directed perpendicular ("normal") to the surface. It is what stops a book from falling through a table. On a flat horizontal surface it points straight up; on a ramp it points at right angles to the slope.
• Friction: a force that resists sliding between two surfaces in contact, always acting along the surface in the direction that opposes the motion or attempted motion. It is the subject of a later lesson.
• Tension: the pulling force transmitted along a rope, string, cable, or chain when it is pulled tight. Tension always pulls along the length of the rope, never pushes.
• Applied force: any push or pull you deliberately exert on an object, such as shoving a box or pressing a lever.
• Air resistance (drag): a friction-like force from the air, opposing an object moving through it. It is often ignored in simple problems but matters for fast-moving or falling objects.

Contact and non-contact forces

Forces fall into two broad families. A contact force requires the two objects to be touching: friction, tension, the normal force, and an applied push are all contact forces. A non-contact force acts across a distance with nothing in between; gravity is the great example, since the Earth pulls the Moon without touching it. Magnetic and electric forces, which you will meet in the electricity module, are also non-contact forces. Recognizing this split helps you avoid inventing forces that are not there, a common beginner mistake.

Net force: combining forces

Objects rarely feel just one force. A box being pushed across a floor feels an applied push, friction, gravity, and the normal force all at once. The single force that would have the same overall effect as all the individual forces combined is called the net force, also known as the resultant force. Finding the net force is the key step in predicting how an object will move.

Because forces are vectors, combining them means adding them as vectors, taking direction into account. In one dimension this is straightforward: choose a positive direction, then add the forces that point that way and subtract the ones that point the opposite way. Forces at right angles, like a horizontal push and vertical gravity, are combined separately along each direction, exactly as displacements were in the kinematics lesson.

Worked example 1: a tug of war

Question: In a tug of war, one team pulls the rope to the right with 400 N while the other pulls to the left with 350 N. Find the net force on the rope.

Solution: Choose right as the positive direction. The rightward pull is +400 N and the leftward pull is -350 N. The net force is the sum: 400 + (-350) = 50 N, positive, so 50 N to the right. The rope, and the whole contest, accelerates toward the stronger team. Notice the net force is much smaller than either individual pull, because the two forces largely cancel.

Worked example 2: forces in the same direction

Question: Two people push a stalled car in the same direction, one with 300 N and the other with 250 N, while friction resists with 200 N. Find the net force.

Solution: Take the pushing direction as positive. The two pushes add because they point the same way: 300 + 250 = 550 N forward. Friction opposes the motion, so it is negative: -200 N. The net force is 550 - 200 = 350 N in the direction of the push. The car accelerates forward. This shows the general recipe: add helpers, subtract opposers.

Balanced and unbalanced forces

When the forces on an object add up to zero, they are said to be balanced, and the net force is zero. A crucial consequence, which is really Newton's first law, is that a balanced object does not change its motion: it stays at rest if it was at rest, or keeps moving at constant velocity if it was already moving. A book lying on a table is a perfect example: gravity pulls it down with a certain force, the normal force pushes it up with an equal force, the two cancel, and the book stays still.

When the forces do not cancel, they are unbalanced, there is a non-zero net force, and the object accelerates in the direction of that net force. Every change in motion, every speeding up, slowing down, or turning, is the sign of an unbalanced force. This is the bridge to the next lesson, where Newton's second law tells you exactly how much acceleration a given net force produces.

Free-body diagrams

The single most useful tool for force problems is the free-body diagram. You draw the object as a simple dot or box, then draw an arrow for every force acting on it, with each arrow pointing in the direction the force acts and drawn longer for a larger force. You label each arrow with the type of force. Crucially, you include only forces acting on the object itself, not forces the object exerts on other things.

A good free-body diagram turns a confusing word problem into a clear picture from which the net force can be read almost by inspection: forces pointing opposite ways are candidates to subtract, forces pointing the same way add. Physicists and engineers draw one for nearly every problem, and you should too. The discipline of drawing every force, and only the real forces, prevents both the error of forgetting a force and the error of inventing one that is not there.

Real-world applications

Thinking in terms of forces and net force underlies all of engineering. A structural engineer designing a bridge adds up the forces on every beam, the weight it carries, the tension and compression in the supports, to make sure they balance so the bridge does not move or collapse. A crane operator relies on the tension in a cable exactly balancing the weight of a load to hold it steady in the air. Aircraft designers balance four forces, thrust and drag horizontally, lift and weight vertically, and the plane flies level only when each pair is balanced. Even standing still, your body is a balance of forces: gravity pulling you down and the normal force from the ground pushing up. Whenever something is held steady, it is because forces are balanced; whenever something starts, stops, or turns, it is because they are not.

Common misconceptions

• "A moving object must have a force pushing it along." No. An object already moving keeps moving at constant velocity with zero net force. A force is needed to change motion, not to maintain it. This is the heart of Newton's first law.
• "Force is a scalar, so I can just add the numbers." Force is a vector. You must account for direction, adding forces that point the same way and subtracting those that oppose.
• "The normal force always equals the weight." Only on a level surface with no other vertical forces. On a ramp, or when something also presses down or lifts up on the object, the normal force differs from the weight.
• "Balanced forces mean the object must be at rest." Balanced forces mean the motion does not change. An object already moving at constant velocity has balanced forces too.
• "Every force needs contact." Gravity, and the magnetic and electric forces, act at a distance with no contact at all. These are non-contact forces.

Recap

A force is a push or a pull, an interaction that can start, stop, or change motion, and because it has both size and direction it is a vector, measured in newtons (1 N is about the weight of a small apple). The common forces are gravity (weight), the normal force, friction, tension, applied forces, and air resistance, and they divide into contact forces and non-contact forces like gravity. Several forces usually act at once, and the net force is their vector sum: in one dimension, add forces in the positive direction and subtract those opposing. When forces are balanced the net force is zero and the motion does not change; when they are unbalanced there is a net force and the object accelerates. A free-body diagram, showing the object with a labelled arrow for every force acting on it, is the essential tool for keeping track of all this.

Sources

1. OpenStax, "College Physics 2e," Chapter 4 (Dynamics: Force and Newton's Laws of Motion), Sections 4.1 and 4.5 on force, net force, and free-body diagrams. Free at openstax.org.
2. The Physics Classroom, "Newton's Laws," Lesson 2 (Types of Forces) and the tutorial on Drawing Free-Body Diagrams, at physicsclassroom.com.
3. HyperPhysics, "Force" and "Standard Forces," Georgia State University, at hyperphysics.phy-astr.gsu.edu.
4. OpenStax, "University Physics Volume 1," Chapter 5 (Newton's Laws of Motion), for a fuller treatment of forces and free-body diagrams. Free at openstax.org.

The laws that founded modern physics

In 1687 Isaac Newton published three laws of motion that, together, explain how forces govern the movement of everything from a dropped pebble to the planets. They are among the most important ideas in all of science. This lesson covers the first two. The first law tells you what happens when there is no net force, and introduces the deep idea of inertia. The second law, the famous F = ma, tells you exactly what happens when there is a net force. In the previous lesson you learned to find the net force on an object; these laws now tell you what that net force does.

Newton's first law: the law of inertia

Newton's first law states that an object at rest stays at rest, and an object in motion continues moving at a constant velocity (constant speed in a straight line), unless acted upon by a net external force. In plain words: things keep doing what they are already doing unless something forces a change. A stationary object will not start moving on its own, and a moving object will not speed up, slow down, or turn on its own.

This built-in resistance to any change in motion is called inertia. Inertia is the reason a net force is needed to change motion at all. It explains why you lurch forward when a car brakes suddenly: your body, obeying the first law, tries to keep moving forward at the old speed even as the car slows beneath you, and it is the seatbelt that finally provides the force to stop you. The same inertia throws you back into your seat when a car accelerates hard, because your body resists the change from rest to motion.

The first law overturned two thousand years of belief. The ancient view, from Aristotle, was that a force is needed to keep an object moving, because everyday experience shows that a pushed object eventually stops. Newton, following Galileo, realized the object stops only because of the hidden force of friction. Remove friction, as nearly happens for a puck on smooth ice or a spacecraft in the vacuum of space, and the object glides on at constant velocity forever with no force at all. Recognizing that steady motion needs no force, and that friction is what really slows things down, is the great conceptual leap of the first law.

Mass measures inertia

How much inertia an object has depends on its mass. The more mass, the more inertia, and the harder it is to change the object's motion, whether starting it, stopping it, or turning it. A shopping cart full of groceries is far harder to get moving, and far harder to stop, than an empty one, precisely because it has more mass and therefore more inertia. This is why mass is sometimes called a measure of an object's inertia. It sets up the second law, which makes the relationship exact.

Newton's second law: F = ma

The first law says a net force changes motion; the second law says by exactly how much. It states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. In its famous equation form:

F = m × a

Here F is the net force in newtons, m is the mass in kilograms, and a is the resulting acceleration in metres per second squared. "Directly proportional to force" means that if you double the net force on a given object, you double its acceleration. "Inversely proportional to mass" means that if you double the mass while keeping the force the same, you halve the acceleration. The single equation captures both relationships at once.

The equation rearranges to solve for whichever quantity you need: a = F / m to find acceleration, and m = F / a to find mass. This flexibility is why F = ma is one of the most heavily used equations in all of physics. It is also where the newton comes from: one newton is defined as the force that gives a 1 kg mass an acceleration of 1 m/s², which is why 1 N equals 1 kg·m/s².

One vital detail: the F in the equation is always the net force, the single combined force after all the individual forces have been added as vectors. And because both force and acceleration are vectors, the acceleration always points in the same direction as the net force. A net force to the right produces an acceleration to the right.

Worked example 1: find the acceleration

Question: A net force of 20 N acts on a 4 kg cart. Find its acceleration.

Solution: The unknown is acceleration, so rearrange to a = F / m. Substitute the values: a = 20 N / 4 kg = 5 m/s², in the direction of the force. The cart's velocity increases by 5 m/s every second. Checking the units confirms the result: newtons divided by kilograms is (kg·m/s²) divided by kg, which is m/s², the correct unit for acceleration.

Worked example 2: find the force

Question: A 1000 kg car accelerates at 2 m/s². Find the net force driving it.

Solution: The unknown is force, so use the equation directly: F = m times a = 1000 kg times 2 m/s² = 2000 N. It takes a net force of 2000 N to give a car of that mass an acceleration of 2 m/s². This is the net force; the engine must actually produce even more than this to overcome friction and air resistance as well.

Worked example 3: find the mass

Question: A net force of 60 N produces an acceleration of 3 m/s². Find the mass of the object.

Solution: The unknown is mass, so rearrange to m = F / a = 60 N / 3 m/s² = 20 kg. The object has a mass of 20 kg. This shows how the second law can even be used to measure mass, by applying a known force and observing the resulting acceleration.

Worked example 4: combining net force with the second law

Question: A 5 kg box is pushed with a force of 40 N while friction resists it with 15 N. Find its acceleration.

Solution: First find the net force, because F in the second law is always the net force. Taking the push direction as positive, net force = 40 - 15 = 25 N. Now apply the second law: a = F / m = 25 N / 5 kg = 5 m/s², in the direction of the push. This two-step pattern, first combine forces into a net force, then divide by mass to get the acceleration, is the backbone of nearly every problem in this module.

Why the same force does more to a lighter object

Because a = F / m, the mass sits in the denominator, so the same push produces a large acceleration on a small mass and a small acceleration on a large mass. Push an empty shopping cart and it leaps forward; push a loaded one with the same effort and it barely moves. Flick a table-tennis ball and a bowling ball with the same force and the light ball rockets away while the heavy one hardly budges. The first law tells you that a net force is required to change motion; the second law tells you precisely how much change you get, and that the change shrinks as the mass grows.

Real-world applications

Newton's first two laws are everywhere in engineering and daily life. Seatbelts, airbags, and headrests are all designed around the first law: they exist to provide the force that stops your body's inertia from carrying it forward (or your head from snapping back) in a crash. Rocket and car engineers use the second law constantly, because F = ma tells them how much thrust or engine force is needed to accelerate a given mass. Heavier vehicles need more powerful engines to achieve the same acceleration, a direct consequence of the mass in the denominator. Sports equipment is tuned with these laws in mind: a lighter bat or racket can be swung to a higher acceleration and speed for the same muscular force. Even the layout of an emergency stop on a train, or the length of an aircraft-carrier catapult, is a second-law calculation of the force required to change a large mass's velocity in the available distance.

Common misconceptions

• "A moving object needs a continuous force to keep moving." This is the ancient Aristotelian error the first law corrects. With no net force, a moving object continues at constant velocity forever. Real objects slow down only because of friction and air resistance.
• "Heavier objects always fall faster." Under gravity alone, all masses accelerate at the same g, because although a heavier object feels more gravitational force, it also has more inertia, and the two effects cancel in a = F/m.
• "The F in F = ma is any single force." It is the net force, the vector sum of all forces. Using just one force when several act will give the wrong acceleration.
• "Mass and weight are the same." Mass is the amount of matter and the measure of inertia, in kilograms; weight is the gravitational force on that mass, in newtons. The next lesson makes the distinction precise.
• "Acceleration is in the direction of motion." Acceleration is in the direction of the net force, which may be opposite to the motion (when slowing down) or at an angle to it (when turning).

Recap

Newton's first law says an object at rest stays at rest and an object in motion stays in constant-velocity motion unless a net force acts, and this resistance to change is inertia, measured by mass. It corrected the old belief that motion needs a force; in fact steady motion needs none, and friction is what slows real objects. Newton's second law makes the effect of a net force exact: F = ma, so acceleration is proportional to the net force and inversely proportional to the mass. The equation rearranges to a = F/m and m = F/a, the F is always the net force, and the acceleration points along that net force. Because mass is in the denominator, the same force accelerates a light object much more than a heavy one. A typical problem combines the forces into a net force first, then applies the second law to find the acceleration.

Sources

1. OpenStax, "College Physics 2e," Chapter 4 (Dynamics: Force and Newton's Laws of Motion), Sections 4.2 (Newton's First Law) and 4.3 (Newton's Second Law). Free at openstax.org.
2. The Physics Classroom, "Newton's Laws," Lesson 1 (Newton's First Law) and Lesson 3 (Newton's Second Law), at physicsclassroom.com.
3. HyperPhysics, "Newton's Laws" and "Newton's Second Law," Georgia State University, at hyperphysics.phy-astr.gsu.edu.
4. OpenStax, "University Physics Volume 1," Chapter 5 (Newton's Laws of Motion). Free at openstax.org.

Completing Newton's laws, and clearing up a common confusion

This lesson finishes Newton's three laws with the third law, the famous rule about action and reaction, and then tackles one of the most confused pairs of ideas in all of physics: the difference between mass and weight. In everyday speech these two words are used interchangeably, and bathroom scales blur the line further by reporting "weight" in kilograms. In physics they are genuinely different quantities with different units, and keeping them straight is essential for everything that follows, from rockets to orbits. Two big ideas anchor the lesson: forces always come in equal and opposite pairs, and the pull of gravity on an object depends on where that object is.

Newton's third law: action and reaction

Newton's third law states that for every action there is an equal and opposite reaction. Stated more carefully: whenever one object exerts a force on a second object, the second object exerts a force of equal size but opposite direction back on the first. Forces never occur alone; they always come in pairs. If object A pushes on object B, then B pushes back on A just as hard, in the opposite direction.

Examples surround you. When you push against a wall, the wall pushes back on you with an equal force, which you feel in your hands. A swimmer pushes the water backward, and the water pushes the swimmer forward, propelling them through the pool. When you walk, your foot pushes backward on the ground, and the ground pushes forward on you, which is what actually moves you along. A rocket engine pushes exhaust gas downward and backward, and the gas pushes the rocket upward and forward, which is how rockets work even in the vacuum of space where there is nothing to "push off." Firing a gun kicks it backward (recoil) because the gun pushes the bullet forward and the bullet pushes the gun back.

The crucial point: the pair acts on different objects

The single most important thing to understand about the third law is that the two forces in an action-reaction pair act on two different objects. The action acts on one object; the reaction acts on the other. Because they act on different objects, they do not cancel each other out. This is the source of endless confusion, so it is worth dwelling on.

Consider the swimmer. The swimmer's push acts on the water, pushing it backward. The water's push acts on the swimmer, pushing them forward. These are equal and opposite, but they act on different bodies, so the swimmer is not left motionless; the force on the swimmer moves the swimmer. If the two forces acted on the same object they would cancel and nothing could ever accelerate. A frequent student error is to think action-reaction forces cancel; they cannot, because canceling requires two forces on the same object, while a third-law pair always straddles two objects. (Balanced forces, from the first lesson, are a different situation: those are two forces on the same object that happen to be equal and opposite.)

Worked example 1: identifying an action-reaction pair

Question: A book rests on a table. Name the third-law reaction to the force of the book pushing down on the table.

Solution: The action is the book pushing down on the table. The reaction is the table pushing up on the book with an equal and opposite force. Note carefully what the reaction is not: it is not the Earth's gravity pulling the book down. Gravity (Earth pulling book) and the table's push (table pushing book) both act on the book, so they are a balanced pair, not an action-reaction pair. The true reaction to "book pushes table down" is "table pushes book up." Pairing forces correctly means always swapping the two objects: if the action is A on B, the reaction is B on A.

Mass versus weight

Now to the second theme. These everyday words name two different physical quantities.

• Mass is the amount of matter in an object. It is measured in kilograms (kg), it is a scalar, and it is the same everywhere in the universe: on Earth, on the Moon, or floating in deep space, an object's mass does not change. Mass is also the measure of inertia, as you saw in the second law.
• Weight is the force of gravity acting on an object's mass. It is measured in newtons (N), it is a vector pointing downward toward the centre of the planet, and it changes with location, because the strength of gravity differs from place to place.

So mass is how much stuff there is, and weight is how hard gravity pulls on that stuff. A bag of sugar has the same mass on the Moon as on Earth, because it contains the same amount of matter, but it weighs about six times less on the Moon, because the Moon's gravity is weaker. Confusing the two leads to real errors, which is why engineers and scientists are so careful to keep newtons and kilograms distinct.

Calculating weight

Weight follows directly from Newton's second law, with gravity supplying the acceleration. Setting the acceleration equal to g in F = ma gives the weight formula:

W = m × g

where W is the weight in newtons, m is the mass in kilograms, and g is the gravitational field strength, about 9.8 m/s² near the Earth's surface. (Some textbooks round g to 10 m/s² for quick mental estimates; this course uses the more accurate 9.8.) The quantity g does double duty: it is both the acceleration of a freely falling object and the gravitational field strength that converts mass into weight, which is no coincidence, since a falling object accelerates precisely because of its weight.

Worked example 2: weight on Earth

Question: A student has a mass of 60 kg. Find the student's weight on Earth, taking g as 9.8 m/s².

Solution: Apply the formula: W = m times g = 60 kg times 9.8 m/s² = 588 N. The 60 kg of matter is pulled down by gravity with a force of 588 N. Notice the answer is in newtons, a force, not in kilograms; the 60 kg is the mass, the 588 N is the weight. Keeping the units distinct keeps the two ideas distinct.

Worked example 3: same mass, different weight

Question: On the Moon, the gravitational field strength is about 1.6 m/s². The same 60 kg student travels there. Find the student's mass and weight on the Moon.

Solution: The mass is unchanged: it is still 60 kg, because the student is made of the same amount of matter no matter where they are. The weight, however, uses the Moon's g: W = m times g = 60 kg times 1.6 m/s² = 96 N. So the weight drops from 588 N on Earth to just 96 N on the Moon, about one sixth as much, because the Moon's gravity is roughly one sixth of Earth's. This is exactly why the Apollo astronauts could bound across the lunar surface in their heavy suits: their mass (and inertia) was the same, but their weight was far smaller.

Worked example 4: finding mass from weight

Question: An object weighs 245 N on Earth. Find its mass (g = 9.8 m/s²).

Solution: Rearrange the weight formula to solve for mass: m = W / g = 245 N / 9.8 m/s² = 25 kg. The object has a mass of 25 kg. This is essentially what a scale does, though scales are usually calibrated to display the mass directly, hiding the division by g from the user.

Why you feel "weightless" in orbit

Astronauts on the International Space Station appear to float, and people often say they are "weightless" because there is "no gravity" in space. This is a misconception. Gravity at the height of the station is still about 90 percent as strong as at the ground, so the astronauts very much have weight. What they experience is free fall: the station and everyone in it are continually falling toward the Earth, but moving sideways so fast that they keep missing it, which is what an orbit is. Because they and their surroundings fall together at the same rate, there is nothing pushing up on them, so they feel weightless even though gravity is acting the whole time. This ties the third law and the weight idea together: weight is the gravitational force, and it does not vanish just because you cannot feel a surface pushing back.

Real-world applications

The third law is the working principle behind all propulsion. Rockets, jet engines, and even squid and octopuses move by pushing mass one way and being pushed the other way in return. Rifle recoil, the kick of a fire hose, and the backward roll of a skateboard when you step off it are all third-law effects. The mass-weight distinction matters wherever gravity varies: spacecraft engineers must track mass (which sets fuel needs and inertia) separately from weight (which changes en route to the Moon or Mars). Scales in commerce are calibrated for Earth's gravity, so a scale taken to a different altitude or planet would misreport unless recalibrated. Understanding that weight is a force, and mass is the unchanging amount of matter, is essential for anyone working in aerospace, and it clears up why the same object can "weigh" different amounts in different places while never changing how much stuff it contains.

Common misconceptions

• "Action-reaction forces cancel out." They never do, because they act on two different objects. Only forces on the same object can cancel. If the pair canceled, nothing could ever be propelled.
• "The reaction to an object's weight is the ground pushing up on it." No. The reaction to Earth pulling the object down is the object pulling the Earth up. The ground's push on the object is a separate force that happens to balance the weight.
• "Mass and weight are the same thing." Mass (kilograms) is the amount of matter and never changes with location; weight (newtons) is the gravitational force and changes with g.
• "There is no gravity in space, so astronauts are weightless." Gravity is still strong in low orbit. Astronauts feel weightless because they are in continual free fall, not because gravity is absent.
• "A bigger action force produces a smaller reaction." The action and reaction are always exactly equal in size, no matter how the masses differ. A truck hitting a fly pushes the fly exactly as hard as the fly pushes the truck; the fly simply accelerates far more because of its tiny mass.

Recap

Newton's third law says forces come in equal and opposite pairs: if A pushes on B, then B pushes on A with the same size force in the opposite direction. The two forces act on different objects, so they never cancel, which is why swimming, walking, and rockets work. Mass is the amount of matter, measured in kilograms, the same everywhere, and the measure of inertia; weight is the gravitational force on that mass, measured in newtons, and it changes with location. Weight is found from W = mg, with g about 9.8 m/s² near Earth and much smaller on the Moon, so an object keeps its mass but weighs less where gravity is weaker. Astronauts in orbit feel weightless not because gravity is gone but because they are in continual free fall.

Sources

1. OpenStax, "College Physics 2e," Chapter 4 (Dynamics: Force and Newton's Laws of Motion), Sections 4.4 (Newton's Third Law) and 4.5 (Normal, Tension, and Other Forces), plus the treatment of weight. Free at openstax.org.
2. The Physics Classroom, "Newton's Laws," Lesson 4 (Newton's Third Law of Motion) and the tutorial on Mass and Weight, at physicsclassroom.com.
3. HyperPhysics, "Newton's Third Law" and "Mass and Weight," Georgia State University, at hyperphysics.phy-astr.gsu.edu.
4. NIST, "The International System of Units (SI)," on the kilogram (mass) and the newton (force), at nist.gov.

The force that is always in the way

Every real object that slides, rolls, or even sits still against a surface experiences friction, the force that resists motion between surfaces in contact. Friction is the reason the ideal, frictionless world of the earlier lessons is only an approximation: in reality, a pushed box slows and stops, a rolling ball comes to rest, and machines waste energy as heat. Yet friction is also indispensable. Without it you could not walk, drive, hold a pencil, or even keep your clothes on. This lesson explains where friction comes from, distinguishes its two main types, gives the equation that predicts its size, and shows how it is both a help and a hindrance. Understanding friction turns the idealized problems of the last three lessons into descriptions of the real world.

Where friction comes from

Friction is a force that opposes the relative sliding of two surfaces in contact. It arises because no surface, however polished it looks, is perfectly smooth. Under a microscope even a mirror-like metal surface is a landscape of tiny bumps, ridges, and valleys. When two surfaces touch, these microscopic high points catch and interlock, and the materials also form weak molecular bonds where they meet. To slide one surface over the other, these interlockings must be broken and re-formed continuously, and the force required to do so is friction. Because friction resists sliding, it always acts along the surface, in the direction opposite to the motion (or the attempted motion).

Two kinds of friction

Friction behaves differently depending on whether the object is moving or still, giving two main types.

• Static friction acts on an object that is not yet moving, resisting the start of motion. Its special feature is that it adjusts itself to match whatever force is trying to move the object, up to a maximum value. Push gently on a heavy box and static friction pushes back exactly as hard, so the box stays put; push harder and static friction grows to match; push beyond its maximum and the box finally begins to slide. This is why a heavy box seems to resist until, suddenly, it "breaks free."
• Kinetic friction, also called sliding friction, acts on an object that is already moving. Unlike static friction, it has a roughly constant value regardless of speed. Importantly, kinetic friction is usually a little smaller than the maximum static friction between the same surfaces. This is why a box, once you get it moving, becomes noticeably easier to keep moving: you have dropped from fighting the larger static friction to fighting the smaller kinetic friction.

A closely related force is rolling friction, which resists a wheel or ball rolling over a surface. Rolling friction is much smaller than sliding friction, which is the whole reason wheels, rollers, and ball bearings were invented: rolling is far easier than dragging.

What friction depends on

The size of the friction force depends on just two things. The first is how hard the two surfaces are pressed together, measured by the normal force N (the perpendicular support force from the last few lessons). The second is how rough or sticky the particular pair of surfaces is, captured by a number called the coefficient of friction, written with the Greek letter mu. The coefficient has no units; it is just a ratio, typically between about 0.1 for slippery surfaces and around 1 for very grippy ones. The relationship is:

Fₛ = μ × N

where Fₛ is the friction force, μ is the coefficient of friction for that pair of surfaces, and N is the normal force. A rougher, stickier pairing has a larger mu; pressing the surfaces together harder increases N. Each pair of surfaces really has two coefficients, a static one (usually larger) governing the maximum static friction and a kinetic one (usually smaller) governing sliding friction, but they follow the same formula.

A surprising fact: area does not matter

Notice what does not appear in the equation: the area of contact. Within the simple model, the size of the touching area has no effect on the friction force. A brick dragged on its wide face experiences the same friction as the same brick dragged on its narrow end, as long as the normal force is unchanged. This seems wrong at first, but there is a reason: spreading the same weight over a larger area reduces the pressure at each point in proportion, so the total gripping force stays the same. This counterintuitive result is one of the most tested ideas about friction.

Worked example 1: friction on a sliding box

Question: A box presses on the floor with a normal force of 200 N, and the coefficient of kinetic friction between box and floor is 0.30. Find the friction force while the box slides.

Solution: Apply the formula directly: Fₛ = μ times N = 0.30 times 200 = 60 N. The kinetic friction force is 60 N, opposing the motion. To keep the box moving at constant velocity you would push with exactly 60 N (so the net force is zero); to make it accelerate you must push with more than 60 N.

Worked example 2: finding the coefficient

Question: It takes 45 N of friction to keep a crate sliding, and the crate's normal force on the floor is 150 N. Find the coefficient of kinetic friction.

Solution: Rearrange the formula to solve for mu: μ = Fₛ / N = 45 / 150 = 0.30. The coefficient of kinetic friction is 0.30, a typical value for many everyday materials. Because it is a ratio of two forces, the coefficient has no units, which is a useful check: if your "coefficient" came out with units attached, something went wrong.

Worked example 3: friction on a level floor with weight

Question: A 25 kg crate rests on a level floor with a coefficient of static friction of 0.40 (g = 9.8 m/s²). What is the maximum horizontal force you can apply before it starts to slide?

Solution: On a level floor with no other vertical forces, the normal force equals the weight: N = m times g = 25 times 9.8 = 245 N. The maximum static friction is then Fₛ = μ times N = 0.40 times 245 = 98 N. So you can push with up to 98 N and the crate will not move, because static friction rises to match your push; the instant you exceed 98 N, the crate breaks free and begins to slide. This shows how the normal force must often be found first, from the weight, before friction can be calculated.

Worked example 4: combining friction with Newton's second law

Question: The 25 kg crate above is now sliding, with a coefficient of kinetic friction of 0.30. If you push it horizontally with 120 N, what is its acceleration? (Use N = 245 N and g = 9.8 m/s².)

Solution: First find the kinetic friction: Fₛ = μ times N = 0.30 times 245 = 73.5 N, opposing the motion. Next find the net horizontal force: net force = 120 - 73.5 = 46.5 N in the direction of the push. Finally apply Newton's second law: a = F / m = 46.5 / 25 = 1.86 m/s² (to three significant figures). This worked example ties the whole module together: find the normal force from the weight, use it to find friction, subtract friction to get the net force, and divide by mass to find the acceleration.

Friction: friend and foe

Friction is neither simply good nor simply bad; it is both, and engineering is largely about controlling it. On the helpful side, friction is what lets you walk (your shoe grips the ground), lets car tyres grip the road to accelerate, turn, and brake, lets a pencil leave marks on paper, and lets a nail or screw hold in wood. Without friction the world would be an impossible ice rink where nothing could start, stop, or stay in place. On the harmful side, friction wastes energy as heat, wears down moving parts, and reduces the efficiency of every machine. To reduce unwanted friction, engineers use lubricants like oil, smooth or polished surfaces, and rolling elements such as wheels and ball bearings, which replace high sliding friction with low rolling friction. To increase helpful friction, they use rough treads on tyres, textured grips, and materials chosen for a high coefficient. Deciding when to fight friction and when to harness it is a genuine and constant engineering skill.

Real-world applications

Friction calculations shape countless designs. Car brakes convert a vehicle's motion into heat through friction, and anti-lock braking systems are tuned around the difference between static and kinetic friction, because a tyre grips best (with larger static friction) just before it starts to skid. Tyre and road engineers work hard to keep the coefficient of friction high, especially in the wet, to shorten stopping distances. Machine designers add lubrication and bearings to cut the friction that would otherwise waste fuel and wear engines out. Climbers, and the rubber on their shoes and ropes, depend on high friction for safety. Even the humble act of tying a knot relies on friction to hold. In every case the same simple model, friction equals mu times the normal force, gives a first, useful estimate of the forces involved.

Common misconceptions

• "Friction depends on the area of contact." In the simple model it does not. Only the normal force and the coefficient matter; a brick has the same friction on its large or small face for the same normal force.
• "Static and kinetic friction are equal." They usually are not. The maximum static friction is typically larger than the kinetic friction, which is why it is harder to start an object sliding than to keep it sliding.
• "The normal force always equals the weight." Only on a level surface with no other vertical forces. On a ramp, or when something presses down or lifts up on the object, N differs from the weight, so friction changes too.
• "Heavier objects always have more friction because they are heavy." It is the normal force, not the weight directly, that sets friction. On a level floor these are equal, but tilt the surface or add other vertical forces and the two part company.
• "Friction is always harmful and should always be reduced." Friction is essential for walking, gripping, and braking. The goal is to control it, increasing it where it helps and reducing it where it hinders.

Recap

Friction is a force that opposes sliding between surfaces in contact, arising from microscopic roughness and molecular bonding, and it always acts along the surface opposite to the motion. Static friction acts before motion starts and adjusts up to a maximum, while kinetic friction acts during sliding and is usually a bit smaller, which is why objects are harder to start than to keep moving. The friction force is given by F equals mu times N, where mu is the unitless coefficient of friction for the pair of surfaces and N is the normal force; the area of contact does not appear. On a level floor the normal force equals the weight, so friction is often found by first computing the weight, then the friction, then the net force, then the acceleration. Friction is both essential (walking, gripping, braking) and costly (wasted heat, wear), so engineering aims to control it rather than simply eliminate it.

Sources

1. OpenStax, "College Physics 2e," Chapter 5 (Further Applications of Newton's Laws: Friction, Drag, and Elasticity), Section 5.1 (Friction). Free at openstax.org.
2. The Physics Classroom, "Newton's Laws," Lesson 2 (Types of Forces: Friction) and the tutorial on the coefficient of friction, at physicsclassroom.com.
3. HyperPhysics, "Friction" and "Coefficients of Friction," Georgia State University, at hyperphysics.phy-astr.gsu.edu.
4. OpenStax, "University Physics Volume 1," Chapter 6 (Applications of Newton's Laws), for static and kinetic friction. Free at openstax.org.

A new way to think about motion

Newton's laws describe motion through forces and accelerations. But there is a second, equally powerful way to analyze how objects move and interact, especially in collisions and explosions: the idea of momentum. Momentum is a measure of "how much motion" an object has, and its great strength is that it is conserved, meaning it stays constant in any interaction where no outside force interferes. This conservation law lets you predict the outcome of a car crash, a game of pool, or the recoil of a gun without ever knowing the complicated, rapidly changing forces during the event. This lesson defines momentum, states the law of conservation of momentum, and introduces impulse, the link between force, time, and momentum change.

Defining momentum

Momentum is defined as the product of an object's mass and its velocity:

p = m × v

where p is the momentum, m is the mass in kilograms, and v is the velocity in metres per second. Momentum is measured in kilogram-metres per second (kg·m/s), a unit with no special name. Because it is built from velocity, which is a vector, momentum is itself a vector: it has a direction, always the same direction as the object's velocity. In one dimension we track this direction with a plus or minus sign, exactly as we did for velocity.

Momentum captures something that neither mass nor velocity alone can. A slow-moving truck and a fast-moving bullet can have similar momentum, because the truck's huge mass makes up for its low speed while the bullet's high speed makes up for its tiny mass. This is why momentum is often described as the "quantity of motion." A heavy, fast-moving object has a large momentum and is correspondingly hard to stop, which matches everyday experience: it is far harder to stop a loaded truck than a bicycle moving at the same speed, and harder to stop a fast bicycle than a slow one.

Worked example 1: momentum of a ball

Question: A 0.5 kg ball moves at 4 m/s. Find its momentum.

Solution: Apply the definition: p = m times v = 0.5 kg times 4 m/s = 2 kg·m/s, directed the same way the ball is moving. The momentum is a modest 2 kg·m/s. Notice the unit is kilogram-metres per second, which comes straight from multiplying kilograms by metres per second.

Worked example 2: comparing momenta

Question: A 1500 kg car moves at 2 m/s in a car park, and a 0.02 kg bullet flies at 400 m/s. Compare their momenta.

Solution: The car's momentum is p = 1500 times 2 = 3000 kg·m/s. The bullet's momentum is p = 0.02 times 400 = 8 kg·m/s. The slow car actually has far more momentum than the fast bullet, because its enormous mass outweighs the bullet's high speed. This shows that momentum depends on both quantities together, not on speed alone.

Conservation of momentum

The reason momentum is so useful is one of the deepest rules in physics: the law of conservation of momentum. It states that in any interaction (a collision, a push-apart, an explosion) where no net external force acts on the system, the total momentum before the interaction equals the total momentum after. The momentum may be redistributed among the objects, but the grand total is unchanged.

This law follows from Newton's third law. During a collision, the two objects push on each other with equal and opposite forces for exactly the same time, so one gains precisely the momentum the other loses, and the total stays fixed. Its power is that it lets you sidestep the messy details of the collision itself: you do not need to know how the forces varied during the crunch, only the masses and velocities before and after. Add up the momentum before, set it equal to the momentum after, and solve.

Worked example 3: a sticky (perfectly inelastic) collision

Question: A 2 kg cart moving at 3 m/s collides with a stationary 1 kg cart, and the two stick together. Find their common velocity afterward.

Solution: Compute the total momentum before the collision. The moving cart has 2 times 3 = 6 kg·m/s; the stationary cart has 1 times 0 = 0. So the total before is 6 kg·m/s. After the collision the carts move together as a single object of mass 2 + 1 = 3 kg at some common velocity v, giving a total momentum of 3 times v. Conservation of momentum sets these equal: 6 = 3v, so v = 2 m/s. The joined carts move off together at 2 m/s in the original direction. A collision in which objects stick together like this is called perfectly inelastic.

Worked example 4: recoil (an "explosion")

Question: A 60 kg person standing at rest on frictionless ice throws a 2 kg ball east at 9 m/s. Find the person's recoil velocity.

Solution: Before the throw, everything is at rest, so the total momentum is zero. After the throw, the total must still be zero, because no external horizontal force acted. Taking east as positive, the ball carries momentum 2 times 9 = 18 kg·m/s east. For the total to remain zero, the person must carry -18 kg·m/s (that is, 18 kg·m/s west). The person's velocity is therefore v = -18 / 60 = -0.3 m/s, or 0.3 m/s west. This is recoil: throwing something forward pushes you backward, exactly like the kick of a gun, and it is conservation of momentum in action.

Impulse: how momentum gets changed

To change an object's momentum you must apply a force, and the longer you apply it, the greater the change. The product of the force and the time for which it acts is called the impulse, and it equals the change in momentum:

F × t = Δp = m × v - m × u

where F is the force, t is the time it acts, and the change in momentum is the final momentum (m times v) minus the initial momentum (m times u). Impulse is measured in newton-seconds (N·s), which are exactly the same units as momentum, kg·m/s. This equation is really just Newton's second law rewritten: since F equals m times a, and a is the change in velocity over time, multiplying both sides by time gives force times time equals mass times change in velocity, which is the change in momentum.

The impulse idea explains a huge range of safety devices. The key insight is that a given change in momentum can be achieved with a large force over a short time, or a small force over a long time. Airbags, crumple zones, cushioned running shoes, and the practice of bending your knees when you land all work the same way: they lengthen the time t over which your momentum changes, which reduces the force F your body must endure for the same change in momentum. A longer stopping time means a gentler, safer stop.

Worked example 5: impulse from a kick

Question: A 0.4 kg ball at rest is kicked so that it leaves at 15 m/s. Find the impulse given to the ball.

Solution: The impulse equals the change in momentum: impulse = m times v - m times u = 0.4 times 15 - 0.4 times 0 = 6 kg·m/s, equivalently 6 N·s, in the direction of the kick. If the foot was in contact with the ball for, say, 0.05 s, the average force would be F = impulse / t = 6 / 0.05 = 120 N, showing how a short contact time demands a large force to deliver the impulse.

Worked example 6: cushioning a landing

Question: A 50 kg gymnast lands moving downward at 6 m/s and comes to rest. Compare the average force if she stops in 0.1 s on a hard floor versus 0.5 s on a soft mat.

Solution: The change in momentum is the same in both cases: Δp = m times v = 50 times 6 = 300 kg·m/s (the momentum goes from 300 to 0). On the hard floor, F = Δp / t = 300 / 0.1 = 3000 N. On the soft mat, F = 300 / 0.5 = 600 N. The soft mat cuts the force to a fifth by stretching the stopping time to five times as long. This is precisely why gym mats, airbags, and crumple zones save people from injury.

Real-world applications

Momentum and impulse are central to vehicle safety and to sport. Car crumple zones are engineered to collapse gradually, extending the collision time and slashing the peak force on the occupants; airbags and seatbelts do the same for the body. In sport, "following through" when hitting or throwing keeps the force on the ball for longer, delivering a larger impulse and hence more momentum and speed. Rocket propulsion is conservation of momentum on a grand scale: the rocket gains forward momentum equal and opposite to the momentum of the exhaust it throws backward. Police and engineers reconstruct traffic collisions using conservation of momentum, working backward from the wreckage to the speeds before impact. Even everyday acts like catching a ball more gently by drawing your hands back, or a boxer "riding" a punch, are applications of the impulse-momentum idea.

Common misconceptions

• "Momentum is the same as speed or velocity." Momentum is mass times velocity, so it also depends on mass. A slow truck can have far more momentum than a fast bullet.
• "Momentum is a scalar." It is a vector with direction. In one dimension opposite directions carry opposite signs, and they can cancel, as in the recoil example where the totals stayed at zero.
• "Momentum is only conserved in gentle collisions." It is conserved in every interaction with no net external force, including violent crashes and explosions, as long as you count the momentum of all the pieces.
• "A longer contact time means a bigger force." For a fixed change in momentum it is the opposite: a longer time means a smaller force. That is exactly why cushioning reduces injury.
• "Impulse and momentum have different units." They are the same: newton-seconds equal kilogram-metres per second, because impulse is literally a change in momentum.

Recap

Momentum is mass times velocity, p = mv, measured in kg·m/s, and it is a vector pointing along the velocity. It combines mass and speed, so a heavy slow object can carry more momentum than a light fast one. The law of conservation of momentum, which follows from Newton's third law, says the total momentum of a system is unchanged when no net external force acts, letting you solve collisions and explosions from the before-and-after values alone. In a sticky (perfectly inelastic) collision the objects move off together; in recoil, momenta are equal and opposite so the totals can stay zero. Impulse, force times time, equals the change in momentum (F t = mv minus mu) and shares momentum's units of N·s. Because a fixed momentum change can come from a small force over a long time, lengthening the stopping time reduces the force, which is the principle behind airbags, crumple zones, and cushioned landings.

Sources

1. OpenStax, "College Physics 2e," Chapter 8 (Linear Momentum and Collisions), Sections 8.1 (Linear Momentum and Force), 8.2 (Impulse), and 8.3 (Conservation of Momentum). Free at openstax.org.
2. The Physics Classroom, "Momentum and Collisions," Lessons 1 and 2 on momentum, impulse, and the conservation of momentum, at physicsclassroom.com.
3. HyperPhysics, "Momentum" and "Impulse of Force," Georgia State University, at hyperphysics.phy-astr.gsu.edu.
4. OpenStax, "University Physics Volume 1," Chapter 9 (Linear Momentum and Collisions). Free at openstax.org.

One of the biggest ideas in all of science

Energy is arguably the single most important concept in physics, uniting mechanics, heat, electricity, light, and even chemistry and biology under one accounting system. This lesson builds the idea from the ground up, starting with the precise physics meaning of work, which is the process that transfers energy, then defining energy itself and its two key mechanical forms, kinetic and potential. It ends with the great law of conservation of energy, which lets you solve problems that would be very hard using forces alone. The through-line is a chain of ideas: a force doing work transfers energy, energy is the stored capacity to do work, and the total amount of energy is never lost, only moved and transformed.

What work means in physics

In everyday language "work" means any effort or task. In physics it has a precise and narrower meaning. Work is done when a force moves an object through a distance in the direction of the force. It is calculated as:

W = F × d

where W is the work, F is the force, and d is the distance the object moves in the direction of that force. Work is measured in joules (J), where one joule is defined as one newton-metre: the work done when a force of one newton moves an object one metre. So doing work is precisely how energy is transferred from one object or store to another; when you do work on an object, you give it energy.

Two features of this definition surprise newcomers. First, if the object does not move, no work is done in the physics sense, no matter how much effort it takes. Holding a heavy suitcase still at arm's length is exhausting, your muscles burn energy, but because the suitcase does not move through any distance, the physical work done on the suitcase is zero. Second, work is zero if the force is perpendicular to the motion. The tension in a string whirling a ball in a circle does no work on the ball, because the tension points toward the centre while the ball moves along the circle, at right angles to the force. Only the part of a force along the direction of motion does work.

Worked example 1: work lifting a box

Question: You lift a box weighing 50 N straight up a height of 2 m at steady speed. How much work do you do?

Solution: To lift at steady speed you apply an upward force equal to the weight, 50 N, and the box moves 2 m in that same upward direction. So W = F times d = 50 N times 2 m = 100 J. You do 100 joules of work against gravity, and that energy is now stored in the box as gravitational potential energy, as we will see shortly.

Worked example 2: pushing against friction

Question: You push a crate 8 m across a floor with a steady horizontal force of 30 N. How much work do you do?

Solution: The force and the motion are both horizontal and in the same direction, so W = F times d = 30 N times 8 m = 240 J. You do 240 joules of work. If friction opposes the motion, that energy ends up as heat in the crate and floor, warming them very slightly, an example of energy being transformed rather than destroyed.

Energy: the capacity to do work

Energy is the capacity to do work. Anything that can exert a force over a distance, and so do work, possesses energy. Energy is measured in the same unit as work, the joule, precisely because work is the transfer of energy. Energy comes in many forms, heat, light, sound, chemical, electrical, nuclear, but in mechanics two forms dominate, and both can be calculated with simple formulas.

• Kinetic energy is the energy an object has because of its motion. It is given by KE = ½ × m × v², where m is the mass and v is the speed. Anything moving has kinetic energy, and a faster or heavier object has more.
• Gravitational potential energy is the energy an object has because of its height in a gravitational field. It is given by PE = m × g × h, where m is the mass, g is 9.8 m/s², and h is the height above some chosen reference level. Lifting an object stores potential energy in it; letting it fall releases that energy again.

Why kinetic energy depends on speed squared

The v-squared in the kinetic energy formula has a dramatic consequence worth pausing on. Because the speed is squared, doubling an object's speed does not double its kinetic energy; it quadruples it. Tripling the speed multiplies the energy by nine. This is why a car crash at 60 km/h is far more than twice as destructive as one at 30 km/h, and why stopping distances grow so quickly with speed (the brakes must remove four times the energy). Keeping this squared relationship in mind explains many real-world safety facts.

Worked example 3: kinetic energy

Question: A 2 kg ball moves at 3 m/s. Find its kinetic energy.

Solution: Apply the formula: KE = ½ times m times v² = ½ times 2 times 3² = ½ times 2 times 9 = 9 J. The ball has 9 joules of kinetic energy. As a check on the squared effect: if the ball sped up to 6 m/s, its kinetic energy would be one half times 2 times 36, which is 36 J, four times as much for double the speed.

Worked example 4: gravitational potential energy

Question: A 3 kg book sits on a shelf 2 m above the floor. Taking g as 9.8 m/s², find its gravitational potential energy relative to the floor.

Solution: Apply the formula: PE = m times g times h = 3 times 9.8 times 2 = 58.8 J. The book has 58.8 joules of gravitational potential energy relative to the floor. Notice "relative to the floor": potential energy is always measured from some chosen reference height, and only differences in height truly matter.

Conservation of energy

The law of conservation of energy is one of the most fundamental principles in all of science. It states that energy can never be created or destroyed, only transformed from one form into another or transferred from one object to another. The total amount of energy in a closed system stays constant. Every process you observe, a ball falling, a fire burning, a muscle contracting, is energy changing form while the total is preserved.

In mechanics this often takes the form of an interchange between kinetic and potential energy. When an object falls, its gravitational potential energy converts into kinetic energy: at the top it is all potential and no kinetic, at the bottom it is all kinetic and no potential, and, ignoring air resistance, the sum of the two stays constant throughout the fall. This is why a roller coaster moves fastest at the lowest point of the track, where the most potential energy has been converted to kinetic, and slowest at the highest point. Following the energy lets you solve problems that would be difficult with forces and kinematics alone, because you only need the start and end states, not the details in between.

Worked example 5: a falling object via energy

Question: A 2 kg ball is dropped from a height of 5 m. Ignoring air resistance and taking g as 9.8 m/s², find its speed just before it hits the ground, using energy conservation.

Solution: At the top the ball has potential energy and no kinetic energy: PE = m times g times h = 2 times 9.8 times 5 = 98 J. Just before landing, all of this has become kinetic energy, so KE = 98 J. Setting ½ times m times v² = 98 gives ½ times 2 times v² = 98, so v² = 98 and v = 9.9 m/s (to two significant figures). Energy conservation reached the answer in a couple of lines, without ever using a kinematic equation, and it would give the same result no matter what path the ball took down.

Real-world applications

The work-energy view organizes vast areas of technology. Power stations are, at heart, energy converters: a coal or gas plant turns chemical energy into heat, then into the kinetic energy of spinning turbines, then into electrical energy. A hydroelectric dam converts the gravitational potential energy of stored water into electricity. A hybrid or electric car recovers the kinetic energy of braking (which would otherwise become waste heat) and stores it in a battery, a process called regenerative braking, and it works precisely because energy can be transformed rather than lost. Roller coaster designers plan the whole ride as a budget of potential and kinetic energy. Nutrition labels report food energy in kilojoules or Calories, the chemical energy your body extracts. In every case the accounting is the same: energy changes form, but the total is always conserved, which is why the conservation law is such a powerful tool for engineers.

Common misconceptions

• "Holding something heavy does work." Not in physics. If the object does not move, the work done on it is zero, however tiring it feels. Your muscles use energy internally, but no mechanical work is done on the stationary object.
• "Doubling the speed doubles the kinetic energy." Because speed is squared, doubling the speed quadruples the kinetic energy. This is a frequently tested point.
• "A force always does work." A force does no work if it is perpendicular to the motion (like the tension on a ball swung in a circle) or if the object does not move.
• "Energy gets used up and disappears." Energy is never destroyed; it only changes form. When motion "disappears" to friction, the energy has become heat, not nothing.
• "Potential energy has one absolute value." Gravitational potential energy depends on the chosen reference height; only differences in PE are physically meaningful, which is why we say "relative to the floor" or "relative to the ground."

Recap

Work in physics is a force times the distance moved in the direction of the force, W = Fd, measured in joules, and it is the process that transfers energy; no motion or a perpendicular force means zero work. Energy is the capacity to do work, measured in joules, and its two main mechanical forms are kinetic energy, KE = one half m v squared, and gravitational potential energy, PE = mgh. Because speed is squared, kinetic energy grows dramatically with speed, quadrupling when speed doubles. The law of conservation of energy says energy is never created or destroyed, only transformed, so in a falling object potential energy converts to kinetic energy with the total held constant, which lets you solve motion problems from just the start and end states. This accounting underlies power stations, dams, regenerative braking, and roller coasters alike.

Sources

1. OpenStax, "College Physics 2e," Chapter 7 (Work, Energy, and Energy Resources), Sections 7.1 (Work), 7.2 (Kinetic Energy), 7.3 (Gravitational Potential Energy), and 7.6 (Conservation of Energy). Free at openstax.org.
2. The Physics Classroom, "Work, Energy, and Power," Lessons 1 and 2 on work, kinetic and potential energy, and energy conservation, at physicsclassroom.com.
3. HyperPhysics, "Work, Energy and Power" and "Conservation of Energy," Georgia State University, at hyperphysics.phy-astr.gsu.edu.
4. OpenStax, "University Physics Volume 1," Chapters 7 and 8 (Work and Kinetic Energy; Potential Energy and Conservation of Energy). Free at openstax.org.

Not just how much, but how fast

The last lesson was about work and energy: how much energy is transferred when a force acts. This lesson adds the dimension of time. Doing a certain amount of work quickly is very different from doing the same work slowly, and the quantity that captures this difference is power. Power is one of the most practical ideas in physics because it is the number stamped on almost every machine you own, from light bulbs to car engines. Understanding power lets you compare devices, size engines and motors, and make sense of your electricity bill. The central idea is simple: power is the rate at which work is done or energy is transferred, in other words, energy per unit time.

Defining power

Consider two cranes that each lift the same load to the same height, doing the same 1000 joules of work. One takes 1 second, the other takes 100 seconds. They do identical work, but the first is clearly the more powerful machine, because it does the work far faster. Power measures exactly this: how quickly work is done, or equivalently how quickly energy is transferred. It is defined as the work done divided by the time taken:

P = W / t

where P is the power, W is the work done (or energy transferred) in joules, and t is the time in seconds. Power is measured in watts (W), named after James Watt, the improver of the steam engine. One watt is defined as one joule per second: a machine has a power of one watt if it does one joule of work every second. A more powerful machine either does the same job in less time or a bigger job in the same time.

Take care not to confuse the two uses of the letter W in this topic: italic W often stands for work in joules, while the upright W is the symbol for the watt, the unit of power. Context makes the meaning clear, but it is a common source of confusion for beginners.

Worked example 1: power of a crane

Question: A crane does 6000 J of work lifting a load in 3 s. Find its power output.

Solution: Apply the definition: P = W / t = 6000 J / 3 s = 2000 W, which is 2 kilowatts. The crane transfers 2000 joules of energy every second. If a second crane did the same 6000 J of work in only 2 s, its power would be 3000 W, showing that less time for the same work means more power.

Worked example 2: climbing stairs

Question: A 60 kg student runs up a flight of stairs 5 m high in 10 s. Taking g as 9.8 m/s², find the power the student develops.

Solution: This problem combines energy and power. First find the work done, which equals the gain in gravitational potential energy in climbing: W = m times g times h = 60 times 9.8 times 5 = 2940 J. Then the power is that work divided by the time: P = W / t = 2940 J / 10 s = 294 W. The student develops about 294 watts, roughly the power of a bright household appliance, and a figure a fit person can sustain only briefly. If the student climbed the same stairs in 5 s instead, the power would double to 588 W, because the same work is done in half the time.

Everyday power ratings

Once you know that power is energy per second, the numbers on household devices come alive. A modern LED light bulb might be rated at 10 W, an older incandescent bulb at 60 W, a microwave oven at 1000 W, an electric kettle at around 2000 to 3000 W, and a small car engine at tens of thousands of watts. Because a watt is a joule per second, a 2000 W kettle transfers 2000 joules of energy into the water every second it runs, which is why kettles boil water so quickly and why they draw a lot of current. A higher power rating means energy is delivered faster.

For larger amounts of power the kilowatt (kW), equal to 1000 watts, is common, and for power stations the megawatt (a million watts) and gigawatt are used. A related unit that often confuses people is the kilowatt-hour (kWh), which appears on electricity bills. Despite its name it is a unit of energy, not power: it is the energy used by a 1 kW device running for one hour, which works out to 3.6 million joules. Your electricity bill charges you for the total energy (in kilowatt-hours) you consume, not for power directly.

Power as force times speed

There is a second, very useful formula for power that applies when a force moves something at a steady speed. Since work is force times distance, and power is work divided by time, power is force times distance divided by time. But distance divided by time is just speed, so:

P = F × v

where F is the force and v is the speed. This form explains a great deal about vehicles. To keep a car moving at a steady speed, the engine must provide a driving force that exactly balances air resistance and friction. At higher speed the same driving force delivers more power (because power is force times speed), and air resistance also grows, so a car needs dramatically more engine power to cruise at high speed than at low speed. It is also why a cyclist works far harder to gain the last few km/h of top speed.

Worked example 3: power to overcome drag

Question: A car moves at a steady 30 m/s against a total resistive force of 600 N. What power must the engine deliver to maintain this speed?

Solution: At steady speed the driving force equals the 600 N of resistance. Using the force-times-speed form: P = F times v = 600 N times 30 m/s = 18000 W, or 18 kilowatts. That is the power just to overcome resistance at that speed; the actual engine must produce more, because engines are not perfectly efficient. Note how the required power scales with speed: at 15 m/s against the same force it would be only 9000 W.

Efficiency: useful power out versus total power in

Real machines never turn all their input energy into useful output; some is always lost, usually as heat from friction. The efficiency of a machine is the fraction of the input power that comes out as useful power, often written as a percentage: efficiency equals useful power output divided by total power input. A motor that draws 1000 W but delivers only 800 W of useful mechanical power is 80 percent efficient, with the missing 200 W lost as heat. Efficiency is never more than 100 percent, because energy is conserved and some is always wasted, an idea that connects power directly back to the conservation of energy from the last lesson.

Worked example 4: efficiency

Question: An electric motor takes in 500 W of electrical power and delivers 400 W of useful mechanical power. Find its efficiency.

Solution: Efficiency is useful output divided by total input: 400 W / 500 W = 0.8, or 80 percent. The remaining 100 W (20 percent) is lost, mostly as heat in the windings and bearings. Real electric motors are often quite efficient, which is one reason electric vehicles waste less energy than petrol engines, whose efficiency is typically much lower.

Real-world applications

Power ratings drive countless practical decisions. Utility companies size power stations in megawatts and gigawatts to meet a city's demand, and the grid must supply power at exactly the rate it is used. Car and motorcycle buyers compare engine power (often quoted in kilowatts or horsepower, where one horsepower is about 746 W) because it largely determines acceleration and top speed. Appliance makers advertise power to signal how fast a kettle boils or how strongly a vacuum cleaner sucks. Electricity bills, charged in kilowatt-hours, reward using lower-power devices or running them for less time. Athletes and coaches measure the power a cyclist or rower can sustain, in watts, as a key performance metric. In every case power answers the practical question that energy alone cannot: not just how much work, but how fast.

Common misconceptions

• "Power and energy are the same thing." Energy (joules) is the total amount of work done; power (watts) is how fast it is done. A low-power device left on for a long time can use more energy than a high-power device used briefly.
• "A kilowatt-hour is a unit of power." It is a unit of energy, the energy used by a 1 kW device in one hour (3.6 million joules). Electricity bills charge for energy in kWh, not for power.
• "A more powerful machine does more work." Not necessarily. It does work faster. Two machines can do the same total work, but the more powerful one finishes sooner.
• "A car needs the same power at all speeds." Because power is force times speed and drag rises with speed, far more power is needed to cruise fast than slow.
• "Machines can be more than 100 percent efficient." Never. Energy is conserved and some is always lost as heat, so useful output can never exceed input.

Recap

Power is the rate of doing work or transferring energy, P = W / t, measured in watts, where one watt is one joule per second. A more powerful machine does the same work in less time or more work in the same time. Everyday devices carry power ratings, from a 10 W bulb to a multi-kilowatt kettle, and the kilowatt-hour on an electricity bill is a unit of energy (3.6 million joules), not power. When a force moves something at a steady speed, power also equals force times speed, P = Fv, which is why vehicles need much more power to go fast. Efficiency, the fraction of input power delivered as useful output, is always less than 100 percent because some energy is inevitably lost as heat, tying power back to the conservation of energy.

Sources

1. OpenStax, "College Physics 2e," Chapter 7 (Work, Energy, and Energy Resources), Section 7.7 (Power) and Section 7.8 (Work, Energy, and Power in Humans). Free at openstax.org.
2. The Physics Classroom, "Work, Energy, and Power," Lesson 1 (Power), at physicsclassroom.com.
3. HyperPhysics, "Power" and "Machine Efficiency," Georgia State University, at hyperphysics.phy-astr.gsu.edu.
4. NIST, "The International System of Units (SI)," for the definition of the watt, at nist.gov.

Making hard jobs possible

How can one person lift a car with a jack, pull a nail with a claw hammer, or split a log with an axe, tasks that seem to demand superhuman strength? The answer is the simple machine, a basic device that makes work easier by changing the size or the direction of a force. Simple machines are among the oldest technologies humanity possesses, and they are the building blocks of every complex machine, from a bicycle to a bulldozer. This lesson introduces the six classic simple machines, defines the crucial idea of mechanical advantage, and, most importantly, shows why a machine can multiply your force but can never create energy out of nothing. That last point ties the whole module together, because it is really the conservation of energy in disguise.

The six simple machines

A simple machine changes the size or direction of a force to make a task easier. There are six classic types, and almost every mechanical device is a combination of them.

• The lever: a rigid bar that pivots on a point, like a seesaw, crowbar, or bottle opener.
• The pulley: a wheel with a rope over it, used to lift loads and change the direction of a pull, as on a flagpole or crane.
• The wheel and axle: a large wheel fixed to a smaller axle, as in a doorknob, steering wheel, or windlass.
• The inclined plane, or ramp: a sloping surface that lets you raise a load gradually instead of lifting it straight up.
• The wedge: essentially a moving inclined plane, used to split or cut, as in an axe, knife, or chisel.
• The screw: an inclined plane wrapped around a cylinder, used to fasten or to raise, as in a screw, bolt, or corkscrew.

Crucially, none of these machines reduces the total work you must do. What they do is let you apply a smaller force over a longer distance, or a force in a more convenient direction. That trade, less force for more distance, is the heart of how every simple machine works.

Mechanical advantage

Mechanical advantage (MA) is the number that tells you how many times a machine multiplies your force. It is defined as the load force (the force the machine exerts on the object) divided by the effort force (the force you apply to the machine):

MA = load force / effort force

A mechanical advantage of 4 means the machine lets you move a load four times heavier than the force you put in: apply 50 N and the machine exerts 200 N on the load. This sounds like something for nothing, but there is always a catch, dictated by energy conservation: to gain a factor of 4 in force, you must move your end four times as far as the load moves. The machine multiplies force at the direct expense of distance. Some machines instead have a mechanical advantage less than 1, meaning they reduce force but increase speed or distance, or simply change a force's direction (a single fixed pulley has an MA of about 1 and just redirects your pull).

The lever and the law of the lever

The lever is the simplest and most illustrative of the machines. A lever is a rigid bar that turns on a fixed pivot called the fulcrum. You apply an effort at one point, and the lever moves a load at another. The turning effect of a force about the fulcrum is the force multiplied by its distance from the fulcrum (this product is called the moment or torque). For a lever balanced in equilibrium, the turning effect of the effort equals the turning effect of the load:

effort × effort arm = load × load arm

where the effort arm and load arm are the distances from the fulcrum to the effort and to the load. This is the law of the lever. It shows immediately why a long effort arm is powerful: if your arm is much longer than the load arm, a small effort can balance a large load. Placing the fulcrum close to the load makes the load arm short and the effort arm long, giving a large mechanical advantage, which is exactly how a crowbar or bottle opener works.

Worked example 1: a crowbar

Question: On a lever, the effort arm is 1.2 m, the load arm is 0.3 m, and the load weighs 200 N. Find the effort force needed to balance the load, and the mechanical advantage.

Solution: Apply the law of the lever: effort times 1.2 = 200 times 0.3, which is effort times 1.2 = 60. Solving, effort = 60 / 1.2 = 50 N. So a 50 N push balances a 200 N load. The mechanical advantage is load / effort = 200 / 50 = 4. Notice this equals the ratio of the arms, 1.2 / 0.3 = 4, which is no coincidence: for a lever the mechanical advantage equals the effort arm divided by the load arm.

Worked example 2: mechanical advantage of a ramp

Question: A ramp 6 m long is used to raise a load to a platform 1.5 m high. What is the ideal mechanical advantage of the ramp?

Solution: For an inclined plane, the ideal mechanical advantage is the length of the slope divided by the height raised: MA = slope length / height = 6 / 1.5 = 4. This means the load can be pushed up the ramp with one quarter of the force needed to lift it straight up, but you must push it along the full 6 m of slope rather than lifting it just 1.5 m. Once again, the force is reduced by exactly the factor that the distance is increased.

The inclined plane and the force-distance trade

A ramp, or inclined plane, lets you raise a load by pushing it up a gentle slope instead of lifting it straight up. The longer and gentler the ramp, the smaller the force needed, but the farther you must push. This is the clearest illustration of the universal rule of machines. Suppose you must raise a 300 N barrel to a height of 1 m. Lifting it straight up takes 300 N of force over 1 m, which is 300 J of work. Pushing it up a 3 m ramp to the same height takes only about 100 N of force (ignoring friction), but over 3 m of slope, which is again 300 J of work. The ramp cut the force to a third but tripled the distance, and the work came out the same.

Why machines cannot create energy

This equality of work is not an accident of the numbers; it is required by the conservation of energy from the previous lessons. A machine is not a source of energy. The work you get out can never exceed the work you put in, because that would mean creating energy from nothing, which is impossible. So a machine can trade a large force over a short distance for a small force over a long distance, or change a force's direction, but the product of force and distance, the work, is at best preserved. This is why there is always a "catch": every gain in force is paid for by an equal loss in distance.

In fact, real machines give back slightly less useful work than you put in, because friction between moving parts always wastes some energy as heat. This is why the mechanical advantage you actually get (the "actual mechanical advantage") is usually a little smaller than the ideal value calculated from the geometry, and why the efficiency of a real machine is always less than 100 percent, exactly as you saw for power in the last lesson. Oiling a machine reduces friction and brings its actual performance closer to the ideal.

Worked example 3: checking the work

Question: Using the crowbar from worked example 1 (MA of 4), the load rises 0.1 m. How far must the effort end move, and does the work balance?

Solution: Because the effort arm is four times the load arm, the effort end moves four times as far as the load: 4 times 0.1 = 0.4 m. Now check the work. Work done on the load is load times its distance = 200 N times 0.1 m = 20 J. Work done by the effort is effort times its distance = 50 N times 0.4 m = 20 J. They match exactly, confirming that the lever multiplies force fourfold but demands four times the distance, with the work conserved. This is the conservation of energy made concrete.

Real-world applications

Simple machines are everywhere, usually combined into more complex tools. A pair of scissors is two levers with two wedges (the blades). A bicycle combines wheels and axles, levers (the pedals and brakes), and screws holding it together, using gear ratios to trade force for speed. Construction relies on ramps, pulleys on cranes, and levers in countless tools. A car jack uses a screw or a system of levers to give a huge mechanical advantage, letting one person lift a tonne. Ramps make buildings accessible to wheelchairs by reducing the force needed to climb. Even the human body is full of levers: your forearm is a lever pivoting at the elbow, with the biceps providing the effort. Understanding mechanical advantage lets engineers design tools that let modest forces accomplish enormous tasks, always within the strict limit set by the conservation of energy.

Common misconceptions

• "A machine reduces the total work you have to do." No. It reduces the force but increases the distance, so the work is at best the same. Machines make work easier, not less.
• "A machine can give more energy out than you put in." Never. That would violate the conservation of energy. Real machines give out slightly less useful energy, because of friction.
• "Mechanical advantage means you get something for nothing." The gain in force is always paid for by an equal loss in distance moved. There is always a trade-off.
• "A longer ramp requires more total work." A longer, gentler ramp requires less force but over a greater distance, so the ideal work is unchanged (in reality a longer ramp adds a little more friction, hence slightly more work).
• "All simple machines multiply force." Some, like a single fixed pulley, mainly change a force's direction (MA about 1), and some trade force for speed or distance (MA less than 1), like the gears that let a bicycle go fast.

Recap

A simple machine changes the size or direction of a force to make a task easier, and the six classic types are the lever, pulley, wheel and axle, inclined plane, wedge, and screw. Mechanical advantage, the load force divided by the effort force, tells how many times the machine multiplies your force, but every gain in force is paid for by moving your end a proportionally greater distance. For a lever, effort times effort arm equals load times load arm, and the mechanical advantage equals the ratio of the arms; for a ramp, the ideal advantage is the slope length divided by the height. No machine can create energy: the work out never exceeds the work in, because of the conservation of energy, and real machines lose a little to friction, so their efficiency is always below 100 percent. Machines make hard jobs possible by trading force for distance, not by producing energy from nothing.

Sources

1. OpenStax, "College Physics 2e," Chapter 9 (Statics and Torque), Section 9.4 (Simple Machines), and Chapter 7 on the conservation of energy underlying machine efficiency. Free at openstax.org.
2. The Physics Classroom, "Work, Energy, and Power," and the tutorials on machines and mechanical advantage, at physicsclassroom.com.
3. HyperPhysics, "Simple Machines," "Lever," and "Inclined Plane," Georgia State University, at hyperphysics.phy-astr.gsu.edu.
4. OpenStax, "University Physics Volume 1," Chapter 12 (Static Equilibrium and Elasticity), for the physics of levers and torque. Free at openstax.org.

A different way for energy to travel

So far, energy has moved by objects moving: a thrown ball carries kinetic energy, a lifted weight stores potential energy. But there is another, subtler way for energy to travel across space, and it governs some of the most important phenomena in nature: sound, light, radio, and earthquakes. That way is the wave. Waves are how music reaches your ears, how sunlight crosses the emptiness of space to warm the Earth, and how a text message travels invisibly to your phone. This lesson introduces what a wave is, the two fundamental types, the handful of quantities that describe any wave, and the single most important equation of the topic, which links a wave's speed, frequency, and wavelength. Everything in the rest of this module, sound and light alike, is built on these ideas.

What a wave is: energy without matter transport

A wave is a disturbance that travels through space or through a material, carrying energy from one place to another without carrying the material along with it. This last point is the defining feature and the one students most often miss. When a wave passes through a medium, the particles of the medium vibrate about their positions and pass the disturbance along to their neighbours, but they do not travel with the wave.

The classic image is a cork floating on a pond. When a water wave passes, the cork bobs up and down in place; it does not get carried across the pond to the far bank. The water itself stays roughly where it is, merely rising and falling, while the wave, and the energy it carries, moves steadily outward. The same is true of a stadium "Mexican wave": the wave of standing spectators travels around the stadium, but each person stays in their own seat. The disturbance moves; the medium does not. This is why waves can transport energy over enormous distances without any matter making the journey.

Two types of waves

Waves come in two fundamental types, distinguished by the direction in which the medium vibrates relative to the direction the wave travels.

• In a transverse wave, the particles of the medium move up and down (or side to side) at right angles to the direction the wave travels. The wave moves horizontally while the medium moves vertically. Water waves (approximately), waves on a shaken rope, and light waves are all transverse. A transverse wave has visible high points called crests and low points called troughs.
• In a longitudinal wave, the particles of the medium move back and forth along the same direction the wave travels. This creates regions where the medium is squeezed together, called compressions, and regions where it is stretched apart, called rarefactions. Sound is the most important longitudinal wave; a shaken spring (a Slinky) pushed back and forth also carries one.

A simple way to remember the difference: in a transverse wave the vibration is aTRoss (across) the motion, while in a longitudinal wave the vibration is aLong the motion. Which type a wave is determines much of its behaviour, as you will see when comparing sound and light.

Describing a wave: the key quantities

Any wave, transverse or longitudinal, can be described by a small set of quantities. These are the vocabulary of the whole module.

• Wavelength, given the symbol of the Greek letter lambda, is the length of one complete wave, for example the distance from one crest to the next crest (or one compression to the next). It is measured in metres.
• Frequency, symbol f, is the number of complete waves that pass a fixed point each second. It is measured in hertz (Hz), where one hertz is one wave per second.
• Amplitude is the maximum displacement of the medium from its rest position, for a transverse wave the height of a crest above the middle line. Amplitude indicates how much energy the wave carries: a bigger amplitude means more energy (a louder sound or a brighter light).
• Period, symbol T, is the time taken for one complete wave to pass a point, measured in seconds. Period and frequency are reciprocals of each other: T = 1 / f. A wave with a high frequency has a short period, and vice versa.
• Wave speed, symbol v, is how fast the disturbance (and its energy) travels through the medium, in metres per second.

Amplitude and frequency are independent

A subtle but important point: amplitude and frequency are entirely independent properties of a wave, and changing one does not change the other. A sound can be loud (large amplitude) yet low-pitched (low frequency), like a bass drum, or quiet yet high-pitched, like a distant whistle. Amplitude controls how much energy the wave carries; frequency controls how rapidly it vibrates. Keeping these two separate in your mind prevents a common confusion, especially in the next lesson where amplitude governs loudness and frequency governs pitch.

The wave equation

The three quantities speed, frequency, and wavelength are tied together by the single most important relationship in the study of waves, the wave equation:

v = f × λ

where v is the wave speed in m/s, f is the frequency in Hz, and lambda is the wavelength in m. The logic behind it is straightforward: in each second, f complete waves pass a point, and each wave is lambda metres long, so the total length passing per second, which is the speed, is f times lambda. The equation rearranges to f = v / λ and λ = v / f, letting you find any one of the three from the other two.

The equation carries a powerful consequence. For waves travelling at a fixed speed in a given medium, frequency and wavelength are inversely related: if one goes up, the other must go down to keep the product equal to the constant speed. High-frequency waves have short wavelengths, and low-frequency waves have long wavelengths. This trade-off appears again and again, from the colours of light to the notes of music.

Worked example 1: a water wave

Question: A wave has a frequency of 5 Hz and a wavelength of 0.4 m. Find its speed.

Solution: Apply the wave equation directly: v = f times λ = 5 Hz times 0.4 m = 2 m/s. The wave travels at 2 metres per second. Checking the reasoning: 5 waves pass each second, each 0.4 m long, so 5 times 0.4 = 2 m of wave passes each second, which is a speed of 2 m/s.

Worked example 2: finding a wavelength

Question: A wave travels at 12 m/s with a frequency of 3 Hz. Find its wavelength.

Solution: Rearrange the wave equation to solve for wavelength: λ = v / f = 12 m/s / 3 Hz = 4 m. Each complete wave is 4 metres long. Because the speed is fixed, a lower frequency would give an even longer wavelength, and a higher frequency a shorter one.

Worked example 3: frequency and period

Question: A wave on a string has a period of 0.02 s and a wavelength of 0.5 m. Find its frequency and its speed.

Solution: First find the frequency from the period: f = 1 / T = 1 / 0.02 = 50 Hz. Then use the wave equation for the speed: v = f times λ = 50 times 0.5 = 25 m/s. So the wave vibrates 50 times per second and travels at 25 m/s. This shows how period, frequency, wavelength, and speed all interlock through two simple relationships.

Real-world applications

Waves and the wave equation underpin much of modern technology. Radio and television broadcasts, mobile phone signals, and wifi are all electromagnetic waves, and stations are assigned specific frequencies (and hence wavelengths) so their signals do not overlap; the dial on an old radio literally selects a frequency. Musical instruments produce sound waves whose frequencies we hear as musical notes, and tuning an instrument means adjusting those frequencies precisely. Medical ultrasound uses high-frequency sound waves to image a baby before birth. Seismologists study the waves from earthquakes, both transverse and longitudinal, to locate the quake and to probe the structure deep inside the Earth. Even microwave ovens heat food using electromagnetic waves of a particular frequency chosen to agitate water molecules. In every case, engineers use v = f times lambda to relate the speed, frequency, and wavelength of the waves they work with.

Common misconceptions

• "A wave carries the medium along with it." It does not. The medium vibrates in place and passes the disturbance on, like the cork bobbing on the pond or people in a stadium wave staying in their seats. Only energy travels.
• "Sound and light are the same kind of wave." Sound is a longitudinal wave that needs a medium; light is a transverse wave that can travel through a vacuum. They are fundamentally different.
• "Amplitude and frequency are linked." They are independent. A wave can have large amplitude and low frequency, or small amplitude and high frequency, in any combination.
• "Higher frequency always means faster." Not so. In a given medium the speed is fixed, and a higher frequency simply means a shorter wavelength. Frequency and speed are different things.
• "Wavelength is the height of the wave." Wavelength is the horizontal distance of one full cycle (crest to crest); the height from the middle to a crest is the amplitude.

Recap

A wave is a disturbance that carries energy through space or a medium without carrying the medium along, as the bobbing cork shows. Transverse waves vibrate the medium at right angles to their travel (light, water, rope waves), with crests and troughs, while longitudinal waves vibrate it along their travel (sound), with compressions and rarefactions. Any wave is described by its wavelength (length of one cycle, symbol lambda), frequency (cycles per second, in hertz), amplitude (maximum displacement, related to energy), period (time for one cycle, T = 1/f), and speed. The wave equation v = f times lambda links speed, frequency, and wavelength, and it means that at a fixed speed, higher frequency gives shorter wavelength. Amplitude and frequency are independent. These ideas are the foundation for understanding sound and light in the lessons that follow.

Sources

1. OpenStax, "College Physics 2e," Chapter 16 (Oscillatory Motion and Waves), Sections 16.9 (Waves) and 16.10 (Superposition and Interference). Free at openstax.org.
2. The Physics Classroom, "Waves," Lessons 1 and 2 on the nature of a wave, wave properties, and the speed of a wave, at physicsclassroom.com.
3. HyperPhysics, "Waves" and "Wave Motion," Georgia State University, at hyperphysics.phy-astr.gsu.edu.
4. OpenStax, "University Physics Volume 1," Chapter 16 (Waves), for a fuller treatment of the wave equation and wave properties. Free at openstax.org.

The physics of what you hear

Every sound you have ever heard, a voice, a song, a slammed door, a whisper, is a physical wave travelling through the air to your ears. Sound is the most familiar wave in daily life, and it is a perfect example of the longitudinal waves introduced in the last lesson. This lesson explains what sound really is, how fast it travels and why that speed depends on the material, and how the two things you notice about a sound, its pitch and its loudness, connect directly to the wave properties of frequency and amplitude. Along the way you will learn to use the speed of sound to measure distances, the same principle behind sonar, echolocation, and hearing thunder after lightning.

What sound is

Sound is a longitudinal wave made of vibrations travelling through a material medium. It begins whenever something vibrates: a plucked guitar string, a struck drum skin, a loudspeaker cone, or your own vocal cords. As the vibrating object moves back and forth, it alternately pushes the neighbouring air molecules together and pulls them apart, creating the compressions (squeezed-together regions) and rarefactions (spread-apart regions) of a longitudinal wave. These pressure variations spread outward through the air in all directions, and when they reach your ear they make your eardrum vibrate, which your brain interprets as sound.

Because sound is passed along by the collisions of particles, it absolutely requires a medium: some material, whether gas, liquid, or solid, whose particles can carry the vibration. This leads to one of the most important facts about sound: sound cannot travel through a vacuum. Empty space has no particles to pass the disturbance along, so in the vacuum of space there is complete silence. The dramatic explosions heard in science-fiction films are physically impossible; in reality, no one in space could hear a distant blast. This is the sharpest contrast between sound and light, which crosses empty space perfectly well.

Speed of sound and why it varies

Sound travels at about 340 metres per second in air at everyday temperatures (it rises slightly in warmer air). That is fast by human standards but far slower than many people expect, and crucially it is far slower than light. The speed of sound depends strongly on the medium: sound travels faster in liquids than in gases, and faster still in solids. In water it moves at roughly 1500 m/s, and in steel at about 5000 m/s, roughly fifteen times its speed in air.

The reason is that sound travels faster when the particles of the medium are closer together and more tightly bound, so they pass the vibration to their neighbours more quickly. In a solid the particles are packed tightly and strongly connected, so the disturbance races through; in a gas they are far apart and weakly interacting, so it dawdles. This is why you can hear an approaching train sooner by listening to the rail than through the air, and why sea creatures can communicate over huge distances underwater. The contrast with light, which travels at about 300 million m/s, explains a everyday observation: in a thunderstorm you see the lightning flash almost instantly but hear the thunder seconds later, because the light arrives essentially at once while the sound crawls to you at 340 m/s.

Pitch and loudness

When you listen to a sound you naturally notice two things about it: how high or low it is, and how loud it is. Each corresponds directly to a wave property from the previous lesson.

• Pitch is how high or low a sound seems, and it is determined by the wave's frequency. A higher frequency produces a higher pitch. A tiny whistle or a soprano's voice has a high frequency and high pitch; a bass drum or a tuba has a low frequency and low pitch. Doubling the frequency raises the pitch by exactly one musical octave.
• Loudness is how strong or intense a sound seems, and it is determined by the wave's amplitude. A larger amplitude means the wave carries more energy and the sound is louder; a smaller amplitude means a quieter sound. Turning up the volume increases the amplitude of the sound waves.

Because frequency and amplitude are independent (as you learned last lesson), pitch and loudness are independent too. A sound can be high and quiet, high and loud, low and quiet, or low and loud, in any combination. Loudness is often measured on the decibel (dB) scale, where a whisper is around 30 dB, normal conversation around 60 dB, and a rock concert or jet engine well over 100 dB, loud enough to damage hearing.

The range of human hearing

Human ears can detect only a limited range of frequencies, typically from about 20 Hz to 20,000 Hz (20 kHz), though the upper limit falls with age. Sound with a frequency above 20,000 Hz is called ultrasound; humans cannot hear it, but many animals can, and it has valuable uses in medical imaging (such as scans of an unborn baby), in cleaning delicate objects, and in range-finding. Sound below about 20 Hz is called infrasound; it too is inaudible to us, but elephants and whales use it to communicate over long distances. Bats and dolphins navigate by echolocation, emitting high-frequency sound and listening for the echoes, a natural sonar.

Worked example 1: how far is the lightning?

Question: You hear thunder 3 seconds after seeing the lightning flash. Taking the speed of sound as 340 m/s, how far away did the lightning strike?

Solution: Because light travels so fast, the flash reaches you almost instantly, so the 3-second delay is essentially the time the sound took to travel from the strike to you. Using distance equals speed times time: distance = 340 m/s times 3 s = 1020 m, roughly 1 kilometre. A handy rule of thumb follows: since sound covers about 340 m per second, it takes roughly 3 seconds to travel a kilometre, so counting the seconds between flash and thunder and dividing by three gives the distance in kilometres.

Worked example 2: an echo off a cliff

Question: You shout toward a cliff and hear the echo 2 seconds later. With the speed of sound at 340 m/s, how far away is the cliff?

Solution: The key insight is that in those 2 seconds the sound travelled all the way to the cliff and back, so the total path is 340 m/s times 2 s = 680 m. But that is the round trip; the distance to the cliff is only half of it: 680 / 2 = 340 m. So the cliff is 340 metres away. This "halve the round trip" method is exactly how sonar measures the depth of the sea and how ultrasound machines and reversing-car sensors measure distance: send a pulse, time the echo, multiply by the speed, and halve.

Worked example 3: sonar depth sounding

Question: A ship sends a sonar pulse straight down and receives the echo from the seabed 0.4 s later. Taking the speed of sound in seawater as 1500 m/s, how deep is the water?

Solution: The pulse travels down and back in 0.4 s, so the round-trip distance is 1500 m/s times 0.4 s = 600 m. The depth is half of this: 600 / 2 = 300 m. The water is 300 metres deep. Note we used the speed of sound in water, not in air, because that is the medium the wave travelled through; using 340 m/s would give a badly wrong depth.

Real-world applications

The physics of sound is put to work everywhere. Sonar lets ships map the seafloor and detect submarines, and fishing boats find shoals of fish, all by timing echoes. Medical ultrasound builds images of organs and unborn babies using reflected high-frequency sound, harmlessly and without radiation. Bats and dolphins hunt by biological echolocation. Musicians and instrument makers manipulate frequency to produce pitch and amplitude to produce volume, and concert halls are designed around how sound reflects and travels. Noise engineers use the decibel scale to set safe limits and design ear protection, because sustained sound above about 85 dB can permanently damage hearing. Even reversing sensors on cars and ultrasonic distance meters used by builders rely on the same send-a-pulse-and-time-the-echo principle you used in the worked examples.

Common misconceptions

• "Sound can travel through empty space." It cannot. Sound needs a material medium (gas, liquid, or solid) to carry the vibration, so space is silent. This is the biggest single difference between sound and light.
• "Sound travels faster in air than in solids because air is thin and easy to move through." The opposite is true. Sound travels faster where particles are closer and more tightly bound, so it is fastest in solids and slowest in gases.
• "Loudness and pitch are the same thing." They are independent. Pitch depends on frequency; loudness depends on amplitude. A sound can be high-pitched and quiet, or low-pitched and loud.
• "For an echo, the distance equals speed times time." That gives the round-trip distance. The distance to the reflecting surface is half of it, because the sound travels there and back.
• "Everyone can hear all sounds." Human hearing is limited to roughly 20 Hz to 20,000 Hz. Ultrasound and infrasound exist all around us but are inaudible to people, though many animals detect them.

Recap

Sound is a longitudinal wave of compressions and rarefactions travelling through a material medium, produced by something vibrating, and because it needs particles to carry it, sound cannot travel through a vacuum. Its speed is about 340 m/s in air but much greater in liquids and solids, where particles are closer and more tightly bound, and far slower than light, which is why thunder lags behind lightning. Pitch is set by frequency (higher frequency, higher pitch) and loudness by amplitude (larger amplitude, louder), and the two are independent. Human hearing spans about 20 Hz to 20,000 Hz, with ultrasound above and infrasound below. The speed of sound lets you measure distances: for an echo, multiply the speed by the time and then halve it, because the sound travels to the surface and back, which is the basis of sonar and ultrasound.

Sources

1. OpenStax, "College Physics 2e," Chapter 17 (Physics of Hearing), Sections 17.1 (Sound), 17.2 (Speed of Sound), and 17.3 (Sound Intensity and Sound Level). Free at openstax.org.
2. The Physics Classroom, "Sound Waves and Music," Lessons 1 through 3 on the nature, speed, and properties of sound, at physicsclassroom.com.
3. HyperPhysics, "Sound" and "Speed of Sound," Georgia State University, at hyperphysics.phy-astr.gsu.edu.
4. OpenStax, "University Physics Volume 1," Chapter 17 (Sound). Free at openstax.org.

The fastest thing in the universe

Light is how we see the world, and it is also one of the most remarkable phenomena in physics. Unlike sound, light can cross the empty vacuum of space, which is how the Sun's light and warmth reach the Earth across 150 million kilometres of nothing. This lesson, the study of light and its behaviour, is called optics. It covers what light is, the two ways light changes direction, reflection and refraction, and how these explain everyday sights from mirrors and lenses to rainbows. The governing ideas are simple and elegant: light travels in straight lines until it meets a surface, where it either bounces off in a predictable way or bends as its speed changes.

Light as an electromagnetic wave

Light is a transverse wave, and crucially it is an electromagnetic wave, meaning it consists of vibrating electric and magnetic fields rather than a vibrating material. Because it is made of fields and not of moving particles, light needs no medium and can travel through a perfect vacuum. This is the fundamental difference from sound: sound must have a material to carry it, while light travels best through empty space.

Visible light is only a small slice of a much larger family called the electromagnetic spectrum, which also includes, in order of increasing frequency, radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays. These are all the same kind of wave, differing only in frequency and wavelength. In a vacuum, every one of them travels at the same extraordinary speed: about 300,000,000 metres per second, written 3 × 10⁸ m/s. This speed, usually denoted c, is the fastest that anything in the universe can travel; nothing carrying information or energy can exceed it. It is so fast that light crosses a room in a few billionths of a second, which is why in a thunderstorm the lightning flash appears instantly while the sound crawls behind.

Reflection

When light strikes a surface it can bounce off, a process called reflection. Reflection follows a precise and simple rule, the law of reflection. To state it we first draw the normal, an imaginary line perpendicular to the surface at the point where the light hits. The angle between the incoming ray and the normal is the angle of incidence, and the angle between the outgoing ray and the normal is the angle of reflection. The law says these two angles are always equal:

angle of incidence = angle of reflection

The law of reflection explains the difference between a mirror and a wall. A smooth, polished surface like a mirror reflects all the parallel rays of light in the same organized way, so they stay ordered and form a clear image; this is called specular reflection. A rough surface, like a painted wall or a sheet of paper, has countless tiny facets pointing every which way, so it scatters the light in all directions (diffuse reflection). That is why you can see your face in a mirror but not in a wall, even though both are reflecting light: the mirror preserves the image and the wall scrambles it.

Worked example 1: reflection angle

Question: A ray of light strikes a flat mirror at 30 degrees to the normal. Find the angle of reflection.

Solution: The law of reflection states that the angle of reflection equals the angle of incidence. Since the ray arrives at 30 degrees from the normal, it leaves at 30 degrees from the normal, on the other side. Note that the angle is always measured from the normal, not from the surface itself; a ray at 30 degrees to the normal is at 60 degrees to the mirror's surface, and beginners sometimes measure from the wrong reference.

Refraction

Refraction is the bending of light as it passes from one transparent material into another, for example from air into water or from air into glass. The cause of refraction is a change in the light's speed. Light travels fastest in a vacuum (and nearly as fast in air), but it slows down when it enters a denser transparent material like water or glass, because the material's atoms interact with the electromagnetic wave. When light meets the boundary at an angle, one side of the wave slows before the other, and this uneven slowing swings the ray to a new direction, just as a car veers if its wheels on one side hit mud and slow down first. Entering a denser medium, light bends toward the normal; leaving it for a less dense medium, light bends away from the normal.

Refraction explains many everyday sights. A straw standing in a glass of water looks bent or broken at the water's surface, because the light from the underwater part is refracted as it leaves the water. A swimming pool looks shallower than it really is for the same reason, and a coin at the bottom of a mug can seem to shift when water is poured in. Most importantly, refraction is the principle behind every lens: the curved glass of eyeglasses, cameras, microscopes, and telescopes is shaped to refract light in a controlled way, bending rays to form magnified or focused images. Your own eye contains a lens that refracts incoming light onto the retina.

Colour and the spectrum

White light looks colourless, but it is actually a mixture of all the colours combined. The colours differ in wavelength: red light has the longest wavelength (and lowest frequency) of the visible range, and violet light has the shortest wavelength (and highest frequency), with orange, yellow, green, and blue in between. Because refraction depends slightly on wavelength, each colour bends by a slightly different amount when it passes through glass.

This is what happens in a prism. When white light enters a triangular glass prism, all the colours slow and bend, but by slightly different amounts, so they fan out and emerge separated into a band of colours called the spectrum: red, orange, yellow, green, blue, indigo, violet. Red bends the least because it slows the least; violet bends the most. This spreading of white light into its colours is called dispersion. A rainbow is nature's prism: sunlight enters millions of tiny raindrops, which refract, reflect inside, and refract again, dispersing the light into the familiar arc of colours. Isaac Newton was the first to show, with prisms, that white light is a mixture and that the prism separates rather than creates the colours.

Worked example 2: reasoning about refraction

Question: A ray of light passes from air into a block of glass, striking the surface at an angle to the normal. Does the ray bend toward or away from the normal, and why?

Solution: Glass is denser (optically) than air, so light slows down on entering it. When light slows entering a denser medium, it bends toward the normal. So the refracted ray inside the glass makes a smaller angle with the normal than the incoming ray did. When the ray later exits the glass back into air it speeds up again and bends away from the normal, so if the glass has parallel faces the emerging ray is parallel to the original but shifted sideways. Reasoning from the change in speed lets you predict the bending direction every time.

Real-world applications

Optics is one of the most useful branches of physics. Mirrors rely on the law of reflection: flat mirrors for everyday use, curved mirrors for car wing mirrors, telescopes, and headlight reflectors. Lenses, which work by refraction, correct vision in eyeglasses and contact lenses, magnify in microscopes to reveal cells and microbes, and gather distant light in telescopes and cameras. Optical fibres carry internet and telephone signals as pulses of light bouncing along thin glass strands by total internal reflection, a special case of the reflection and refraction ideas here, moving vast amounts of data across oceans. Prisms and diffraction gratings split light into spectra, which astronomers use to determine what distant stars are made of. Even the colours of a soap bubble, an oil slick, or a rainbow are optics at work. Understanding reflection, refraction, and dispersion lets engineers design the instruments that extend human sight from the microscopic to the astronomical.

Common misconceptions

• "Light needs a medium to travel, like sound." It does not. Light is an electromagnetic wave that travels through the vacuum of space, which is how sunlight reaches Earth. This is the key difference from sound.
• "Angles in optics are measured from the surface." They are measured from the normal, the line perpendicular to the surface. Measuring from the surface itself gives the wrong angle.
• "A prism adds or creates the colours in white light." The colours are already present in white light. The prism only separates them by bending each wavelength differently.
• "Light bends during refraction because it hits the surface." Light bends specifically because it changes speed on entering a new medium. If it enters straight along the normal, its speed still changes but it does not bend, because both sides of the wave slow at once.
• "You can see your face in any surface that reflects light." Only smooth surfaces give a clear image (specular reflection). Rough surfaces scatter light in all directions (diffuse reflection), so no image forms, even though light is reflected.

Recap

Light is a transverse electromagnetic wave that, unlike sound, travels through a vacuum, which is why the Sun's light reaches us; it is the visible part of the electromagnetic spectrum, and in a vacuum all such waves travel at about 3 times 10 to the 8 metres per second, the fastest speed possible. Reflection follows the law of reflection: the angle of incidence equals the angle of reflection, both measured from the normal, and smooth surfaces form images while rough ones scatter light. Refraction is the bending of light as it changes speed passing between media, bending toward the normal when it slows entering a denser material, and it explains bent straws, shallow-looking pools, and how lenses focus light. Because refraction depends on wavelength, a prism disperses white light into the spectrum from red (bends least) to violet (bends most), and a rainbow is nature's version of the same effect.

Sources

1. OpenStax, "College Physics 2e," Chapter 25 (Geometric Optics), Sections 25.1 (The Ray Aspect of Light), 25.2 (Law of Reflection), and 25.3 (The Law of Refraction). Free at openstax.org.
2. The Physics Classroom, "Reflection and the Ray Model of Light" and "Refraction and the Ray Model of Light," at physicsclassroom.com.
3. HyperPhysics, "Reflection," "Refraction," and "Dispersion," Georgia State University, at hyperphysics.phy-astr.gsu.edu.
4. OpenStax, "University Physics Volume 3," Chapter 1 (The Nature of Light) and Chapter 2 (Geometric Optics and Image Formation). Free at openstax.org.

The force behind modern life

Electricity powers almost everything in the modern world: lights, phones, computers, motors, and the vast grid that connects them. Yet electricity rests on a single, invisible property of matter called electric charge. This lesson builds the foundations of electricity from that starting point. It explains what charge is and how charges interact, what it means for charge to flow as an electric current, what voltage is and why it is needed to make current flow, and why a circuit must be a complete loop. A helpful water analogy runs throughout, making these invisible ideas concrete. Everything in the electricity module, including Ohm's law and circuits, is built on the concepts introduced here.

Electric charge

All matter is made of atoms, and atoms contain tiny particles that carry electric charge, a fundamental property of matter. There are two kinds of charge. Protons, found in the nucleus of every atom, carry positive charge. Electrons, which orbit the nucleus, carry negative charge. In a normal atom the number of protons and electrons is equal, so the positive and negative charges balance and the atom is electrically neutral.

The way charges interact is captured in one simple rule: like charges repel, and opposite charges attract. Two positive charges push each other apart, two negative charges push each other apart, but a positive and a negative charge pull toward each other. This attraction and repulsion is a genuine force, called the electric force, and it acts at a distance without the charges touching, much like gravity. It is the electric force that holds electrons to atoms and binds atoms into materials.

Static electricity

Electrons can be transferred from one object to another, usually by rubbing or contact. When you rub a balloon on your hair, electrons are scraped off the hair onto the balloon. The balloon, now with extra electrons, becomes negatively charged, while the hair, having lost electrons, becomes positively charged. The oppositely charged balloon and hair then attract, which is why your hair stands up toward the balloon, and the charged balloon sticks to a neutral wall. This build-up of charge that is not moving is called static electricity. Lightning is static electricity on a giant scale: charge builds up in storm clouds until it discharges in an enormous spark. The key point is that static electricity is charge that has accumulated but is not yet flowing.

Electric current

When charge is made to flow, rather than sitting still, we have an electric current. In a metal wire, the flowing charges are the loose outer electrons of the metal atoms, which are free to drift through the material. Current is defined as the rate at which charge flows past a point, in other words how much charge passes each second. It is measured in amperes (A), usually shortened to amps. One ampere is defined as one coulomb of charge passing a point each second, where a coulomb is a fixed, very large number of electrons (about 6.24 times 10 to the 18). A larger current simply means more charge flowing past every second, like more water per second through a pipe.

A historical curiosity worth knowing: by long-standing convention, "conventional current" is taken to flow from the positive terminal of a battery to the negative terminal, the direction a positive charge would move. In metal wires the actual moving charges are electrons, which flow the opposite way. This does not affect the calculations in this course, but it explains why current arrows in diagrams point from plus to minus even though the electrons are going the other way.

Voltage: the push that drives current

Charges do not flow on their own; they need something to push them, just as water will not flow through a pipe without a pump or a height difference. That push is provided by voltage, also called potential difference. Voltage is the energy given to each unit of charge by a source such as a battery or power supply, and it is measured in volts (V). You can think of voltage as the electrical "pressure" that drives current around a circuit: the higher the voltage, the harder the charges are pushed, and the more current flows through a given wire.

A battery acts like a pump for charge. Chemical reactions inside it raise charges to a higher energy at one terminal, and as those charges travel around the circuit they release that energy in the components, lighting a bulb or spinning a motor, before returning to the battery to be pumped again. A 9-volt battery gives each unit of charge more energy than a 1.5-volt battery, which is why it can drive a stronger current.

A circuit needs a complete loop

Current can only flow around a complete circuit, an unbroken conducting loop leading from one terminal of the battery, through the connecting wires and components, and back to the other terminal. If the loop is broken anywhere, the current stops everywhere in that loop. This is exactly what a switch does: closing the switch completes the loop and current flows, lighting the bulb; opening the switch breaks the loop and the current, and the light, stops. A break can also be accidental, such as a snapped wire or a blown fuse, with the same effect. The requirement of a complete loop is one of the most basic and important facts about electric circuits.

The water analogy

Because charge and current are invisible, a comparison with water flowing through pipes makes them easier to picture, and the analogy is worth learning because it recurs throughout the module. Voltage is like the pressure supplied by a pump (or by water held at a height): it drives the flow. Current is like the rate at which water flows through the pipes: how much passes each second. A narrow section of pipe that restricts the flow is like electrical resistance, the opposition to current that is the subject of the next lesson. The analogy makes the relationships intuitive: raise the pressure (voltage) and more water (current) flows; narrow the pipe (increase resistance) and less flows. The battery is the pump, the wires are the pipes, and a break in the circuit is like a cut pipe that stops all flow. Like all analogies it is imperfect, but it is a reliable guide to how circuits behave.

Conductors and insulators

Materials differ enormously in how easily they let charge flow. A conductor is a material through which charge flows easily, because it has many free electrons. Metals are the best everyday conductors, especially copper and aluminium, which is why electrical wires are made of them. An insulator is a material that strongly resists the flow of charge, because its electrons are tightly bound and not free to move. Plastic, glass, rubber, and dry wood are good insulators. This is why wires are made of copper (to carry the current) wrapped in plastic (to keep the current confined and to protect you from electric shock). The distinction between conductors and insulators is what makes it possible to guide electricity safely to where it is needed and keep it away from where it is not.

Real-world applications

These basic ideas underlie all electrical technology. Every device you plug in or switch on depends on driving a current around a complete circuit with a voltage source. Batteries in phones, laptops, and cars store chemical energy and supply the voltage that drives their circuits. Household wiring delivers current at a standard mains voltage, guided by copper conductors and protected by plastic insulation, with switches to complete or break loops and fuses or breakers to stop dangerous currents. Static electricity, the same phenomenon as the balloon in your hair, must be controlled in electronics factories, where a static discharge can destroy delicate microchips, and it is the reason for lightning rods on tall buildings. Understanding charge, current, voltage, conductors, and insulators is the essential first step toward understanding, and safely using, all of electricity.

Common misconceptions

• "Current is used up as it goes around a circuit." It is not. The same current that leaves the battery returns to it; charge is not consumed. What the components use is energy, delivered by the voltage, not the charge itself.
• "Voltage flows through a circuit." Voltage does not flow; it is the push, the potential difference between two points. Current is what flows. Saying "voltage flows" confuses the pressure with the flow.
• "Like charges attract." The opposite: like charges repel and unlike charges attract. This is the reverse of what some students first guess.
• "A circuit works even if the loop is broken somewhere else." No. A single break anywhere in a simple loop stops the current everywhere in that loop, which is exactly how a switch turns a device off.
• "Metals conduct because charge passes through the empty space." Metals conduct because they contain many free electrons that can drift; insulators lack these free charges, which is why they block the flow.

Recap

Electric charge is a fundamental property of matter, carried as positive charge by protons and negative charge by electrons, and the rule is that like charges repel while opposite charges attract. Charge that builds up without flowing is static electricity, as when a rubbed balloon sticks to a wall. When charge flows it is an electric current, the rate of charge past a point, measured in amperes (one amp is one coulomb per second). Charges need a push to flow, provided by voltage (potential difference) in volts, which acts like the pressure from a pump. Current flows only around a complete, unbroken loop, which a switch opens or closes. In the water analogy, voltage is pressure, current is flow rate, and resistance is a narrow pipe. Conductors like copper carry charge easily, while insulators like plastic block it, which is why wires are copper wrapped in plastic.

Sources

1. OpenStax, "College Physics 2e," Chapter 18 (Electric Charge and Electric Field) and Chapter 20 (Electric Current, Resistance, and Ohm's Law), Sections 20.1 (Current) and 20.2 (Ohm's Law). Free at openstax.org.
2. The Physics Classroom, "Static Electricity" and "Electric Circuits," Lessons 1 and 2 on charge, current, and electric potential difference, at physicsclassroom.com.
3. HyperPhysics, "Electric Charge," "Electric Current," and "Voltage," Georgia State University, at hyperphysics.phy-astr.gsu.edu.
4. OpenStax, "University Physics Volume 2," Chapters 5 and 9 (Electric Charges and Fields; Current and Resistance). Free at openstax.org.

The most useful equation in electronics

The last lesson introduced voltage (the push), current (the flow), and hinted at resistance (the opposition to flow). This lesson makes the relationship between the three exact, through Ohm's law, one of the most important and widely used equations in all of electronics. Understanding Ohm's law lets you predict how much current will flow in a circuit, choose the right components, and see why some devices draw far more power than others. The lesson defines resistance precisely, presents Ohm's law and its rearrangements, works through examples of finding voltage, current, and resistance, and finishes with the formula for electrical power, which explains your electricity bill.

Resistance: opposition to the flow of current

Resistance is the opposition that a component or material offers to the flow of electric current. Every real conductor resists the flow to some degree, converting some electrical energy into heat as the charges push through. Resistance is measured in ohms, whose symbol is the Greek capital letter omega. The more resistance in a path, the harder it is for current to flow through it for a given voltage.

What determines a wire's resistance? Several factors: a longer wire has more resistance (more material to push through), a thicker wire has less resistance (a wider path for the charge, like a wider pipe), and the material matters (copper resists far less than, say, the nichrome wire used in heaters). Temperature also plays a role, with most metals resisting more when hot. A component manufactured to have a specific, controlled resistance is called a resistor, and resistors are used everywhere in electronics to limit and control current. Resistance is also what usefully converts electrical energy into heat and light in appliances such as toasters, kettles, electric heaters, and the filament of an old-style light bulb.

Ohm's law

For many components, especially metal wires and resistors at constant temperature, experiment shows that the current through the component is directly proportional to the voltage across it. Double the voltage and the current doubles. This proportional relationship is Ohm's law, and it is written:

V = I × R

where V is the voltage across the component in volts, I is the current through it in amperes, and R is its resistance in ohms. This single equation ties together the three central quantities of electricity. Because it is an equation with three variables, it can be rearranged to solve for whichever one you do not know, given the other two. The three forms are summarized here:

To find	Use
Voltage	V = I × R
Current	I = V / R
Resistance	R = V / I

These three are the same equation written three ways, not three separate laws. The form I = V / R makes the physics especially clear: current increases with voltage (more push, more flow) and decreases with resistance (more opposition, less flow), exactly matching the water analogy of pressure and a narrow pipe.

Worked example 1: find the current

Question: A 12 V battery is connected across a 4 ohm resistor. Find the current through the resistor.

Solution: The unknown is current, so use the form I = V / R = 12 V / 4 ohms = 3 A. A current of 3 amperes flows through the resistor. As a check with the water analogy: a bigger battery voltage would push more current, and a larger resistance would allow less, both of which the equation reflects.

Worked example 2: find the resistance

Question: A current of 2 A flows through a component when 10 V is applied across it. Find its resistance.

Solution: The unknown is resistance, so use R = V / I = 10 V / 2 A = 5 ohms. The component has a resistance of 5 ohms. This is, in fact, how resistance is measured in practice: apply a known voltage, measure the resulting current, and divide.

Worked example 3: find the voltage

Question: A current of 0.5 A flows through a 20 ohm resistor. Find the voltage across it.

Solution: The unknown is voltage, so use the basic form V = I times R = 0.5 A times 20 ohms = 10 V. There is a 10-volt potential difference across the resistor. Notice how a modest current through a fairly large resistance still needs a sizeable voltage to drive it.

Ohmic and non-ohmic components

Ohm's law is a very good description for metal wires and ordinary resistors at a steady temperature; such components are called ohmic, and a graph of current against voltage for them is a straight line through the origin. But not everything obeys Ohm's law. A filament light bulb heats up as more current flows, and since its resistance rises with temperature, doubling the voltage does not quite double the current; it is a non-ohmic component. Devices like diodes deliberately let current flow easily one way and hardly at all the other way, so they are strongly non-ohmic. Knowing that Ohm's law holds only for ohmic components at constant temperature keeps you from misapplying it, though for this course the resistors you meet can be treated as ohmic.

Electrical power

Recall from the energy module that power is the rate of transferring energy. In an electrical component, the power transferred (usually appearing as heat and light) is the voltage across it multiplied by the current through it:

P = V × I

where P is the power in watts, V is the voltage in volts, and I is the current in amperes. This makes sense from the definitions: voltage is energy per unit charge, current is charge per second, so their product is energy per second, which is power. Combining this with Ohm's law also gives the useful alternatives P = I² times R and P = V² / R, handy when you know the resistance and only one of voltage or current.

Worked example 4: electrical power

Question: A device draws a current of 2 A when connected to a 12 V supply. Find the power it uses.

Solution: Apply the power formula: P = V times I = 12 V times 2 A = 24 W. The device transfers 24 joules of energy every second. This is why high-power appliances such as electric heaters and kettles draw a large current at mains voltage: because power is voltage times current, delivering a lot of power at a fixed voltage requires a large current, which is also why such appliances need thick wires and their own robust circuits.

Worked example 5: power from resistance

Question: A 6 ohm heating element carries a current of 5 A. Find the power it dissipates as heat.

Solution: Here it is convenient to use the form that involves current and resistance: P = I² times R = 5² times 6 = 25 times 6 = 150 W. The element gives out 150 watts of heat. You could also find the voltage first (V = IR = 5 times 6 = 30 V) and then use P = VI = 30 times 5 = 150 W, which gives the same answer, a good consistency check.

Real-world applications

Ohm's law and the power formula are the daily tools of every electrician and electronics engineer. Choosing a resistor to protect a component, sizing a wire so it can carry the expected current without overheating, and designing anything from a phone charger to a power grid all rest on V = IR and P = VI. Fuses and circuit breakers are rated in amperes and are chosen using these relationships to cut the current before a wire overheats and starts a fire. The reason mains electricity is transmitted across the country at very high voltage is a direct consequence of P = I squared R: raising the voltage lets the same power be delivered with a smaller current, which drastically reduces the heat wasted in the transmission wires. Even reading the label on an appliance, where the power in watts is stated, lets you use these formulas to work out the current it will draw and whether a circuit can safely supply it.

Common misconceptions

• "Ohm's law applies to every component." It holds well for metal wires and resistors at constant temperature (ohmic components), but not for filament bulbs, diodes, and other non-ohmic devices whose resistance changes.
• "Resistance depends only on the material." It also depends on length (longer means more resistance), thickness (thicker means less), and temperature (hotter usually means more for metals).
• "More current always means more voltage." Only if the resistance is fixed. For a given voltage, a smaller resistance gives a larger current; current depends on both voltage and resistance.
• "Power and current are the same." Power is voltage times current (P = VI). Two devices can draw the same current but use different power if they run at different voltages.
• "High voltage always means high power or danger by itself." Power depends on both voltage and current. A high voltage with a tiny current (like static shocks) carries little energy, while a moderate voltage driving a large current can be very powerful and dangerous.

Recap

Resistance is the opposition to current flow, measured in ohms, and it depends on a conductor's length, thickness, material, and temperature; a component with a set resistance is a resistor, and resistance turns electrical energy into heat and light. Ohm's law, V = IR, links voltage, current, and resistance, and rearranges to I = V / R and R = V / I, so any one can be found from the other two. It applies to ohmic components (metal wires and resistors at constant temperature) but not to non-ohmic ones like filament bulbs and diodes. Electrical power is P = VI (equivalently I squared R or V squared over R), which is why high-power appliances draw large currents, and why the grid uses high voltage to cut current and reduce wasted heat in transmission.

Sources

1. OpenStax, "College Physics 2e," Chapter 20 (Electric Current, Resistance, and Ohm's Law), Sections 20.2 (Ohm's Law: Resistance and Simple Circuits) and 20.3 (Resistance and Resistivity), and 20.4 (Electric Power and Energy). Free at openstax.org.
2. The Physics Classroom, "Electric Circuits," Lesson 3 (Ohm's Law) and Lesson 4 (Electrical Power), at physicsclassroom.com.
3. HyperPhysics, "Ohm's Law," "Resistance," and "Electric Power," Georgia State University, at hyperphysics.phy-astr.gsu.edu.
4. OpenStax, "University Physics Volume 2," Chapter 9 (Current and Resistance) and Chapter 10 (Direct-Current Circuits). Free at openstax.org.

Wiring more than one component

A circuit with a single bulb is simple, but real devices contain many components, and how they are connected matters enormously. There are two fundamental ways to wire components together: in series or in parallel. The choice determines how the current is shared, how the voltage is divided, what the total resistance becomes, and, most visibly, what happens when one component fails. This lesson compares the two arrangements, shows how to calculate the total resistance in series, and explains why your home is wired in parallel while old holiday lights were wired in series. These ideas build directly on Ohm's law from the last lesson.

Series circuits

In a series circuit, the components are connected one after another in a single loop, so there is only one path for the current to follow. Because there is just one path, the same current flows through every component in the series; the charge that goes through the first resistor must go through the second and the third, with nowhere else to go. The voltage of the supply, however, is shared out among the components, with more voltage dropped across a larger resistance.

Since the charges must push their way through each resistor in turn, the resistances add up: the total resistance of a series circuit is simply the sum of the individual resistances:

Rₜₒₜₔₗ = R₁ + R₂ + R₃ + ...

Adding more components in series therefore increases the total resistance, which reduces the current from the supply. The great weakness of a series circuit is its single loop: if any one component fails or a wire breaks anywhere in the loop, the whole circuit is broken and every component stops working. This is exactly why old strings of holiday lights, wired in series, would all go dark when a single bulb burned out, and finding the one dead bulb was a frustrating chore.

Worked example 1: series resistance and current

Question: Resistors of 2 ohms, 3 ohms, and 5 ohms are connected in series across a 20 V battery. Find the total resistance and the current.

Solution: First add the resistances, because in series they sum: R total = 2 + 3 + 5 = 10 ohms. Then apply Ohm's law to the whole circuit to find the current: I = V / R = 20 V / 10 ohms = 2 A. This same 2 A flows through all three resistors, because there is only one path. You could go further and find the voltage across each: for the 5 ohm resistor, V = IR = 2 times 5 = 10 V, and the three voltages (4 V, 6 V, 10 V) add up to the 20 V of the battery, as they must in a series circuit.

Parallel circuits

In a parallel circuit, the components are connected side by side across the same two points, so the current has more than one path, or branch, to follow. Two consequences follow, and they are the mirror image of the series case. First, each branch receives the full voltage of the supply, because each branch connects directly across the same two terminals. Second, the current from the battery splits among the branches, with more current flowing through the branch of lower resistance, and the branch currents add up to the total current drawn from the supply.

Two practical features make parallel wiring extremely useful:

• Independence: if one branch breaks, the others keep working, because each branch still forms its own complete loop back to the battery. Only the broken branch loses its current.
• Lower total resistance: adding more parallel branches actually lowers the total resistance, because you are giving the current more separate paths to flow through (like opening more checkout lanes in a shop). Lower total resistance means the battery supplies a larger total current.

This is precisely why homes are wired in parallel. Every light and socket is connected across the same mains voltage, so each receives the full voltage it needs, and switching one off, or one bulb blowing, does not interrupt the rest of the house. Imagine if your home were wired in series: turning off one light would plunge every room into darkness.

Worked example 2: a broken branch

Question: Three identical bulbs are wired in parallel across a battery, and the middle bulb burns out. What happens to the other two bulbs?

Solution: Because each bulb sits on its own branch, with its own complete loop back to the battery, the other two bulbs stay lit at full brightness. Only the broken branch loses its current; the intact branches are unaffected because their loops are still complete and they still receive the full supply voltage. Had the three bulbs been wired in series instead, the broken filament would have cut the single loop and all three would have gone dark. This contrast is the clearest demonstration of the practical difference between the two arrangements.

Worked example 3: comparing brightness

Question: Two identical bulbs are connected first in series and then in parallel across the same battery. In which arrangement is each bulb brighter, and why?

Solution: In series, the two bulbs' resistances add, giving a larger total resistance and so a smaller current, and each bulb receives only half the battery voltage. In parallel, each bulb receives the full battery voltage and a larger current flows through each. Since a bulb's brightness depends on the power it dissipates (P = VI), each bulb is brighter in the parallel arrangement. This is another reason parallel wiring is preferred for lighting: each device gets full voltage and full performance regardless of the others.

Choosing an arrangement

Each arrangement has its place. Series wiring is simple, uses a single shared current, and is the right choice when you genuinely want everything to switch on or off together, or when you deliberately want to limit the current (a resistor placed in series limits current to protect a component). Parallel wiring keeps every device independent and supplied at the full voltage, which is why it dominates real household and automotive electrical systems. Many real circuits combine both, with some components in series and others in parallel, but the two pure cases here are the essential building blocks for understanding any of them.

Real-world applications

The series-versus-parallel choice shapes electrical design everywhere. Household and building wiring is parallel so that every appliance gets the full mains voltage and works independently, with each circuit protected by its own fuse or breaker. Car electrical systems are parallel too, so headlights, radio, and wipers all run at the full battery voltage and one failing does not disable the others. Modern holiday lights often use clever wiring so a single failure no longer kills the whole string, a direct response to the weakness of the old series design. Series connections appear where a shared current is wanted or current must be limited, such as a resistor in series with an LED to keep it from burning out, or batteries stacked in series to add their voltages (which is why several 1.5 V cells in a torch give a higher total voltage). Understanding both arrangements lets an engineer or electrician predict how a circuit behaves and design it to be safe and reliable.

Common misconceptions

• "Current gets used up as it passes through series components." No. In a series circuit the same current flows through every component; it is the voltage that is shared, not the current.
• "Adding resistors in parallel increases the total resistance." It decreases it. More parallel paths let more total current flow, so the combined resistance is less than any single branch.
• "In a parallel circuit, each branch gets a share of the voltage." Each branch gets the full supply voltage. It is the current that divides among the branches.
• "If one bulb goes out, the rest must go out too." That is true only for a series circuit. In a parallel circuit the other branches keep working, which is why homes are wired in parallel.
• "Series and parallel bulbs are equally bright on the same battery." Parallel bulbs are brighter, because each gets the full voltage, while series bulbs share the voltage and carry a smaller current.

Recap

Components can be wired in series (a single loop, one path) or in parallel (side by side, multiple branches). In a series circuit the same current flows through every component, the supply voltage is shared, and resistances add (R total = R1 + R2 + ...), but a single break stops everything, as in old holiday lights. In a parallel circuit each branch gets the full supply voltage, the current splits among the branches, adding branches lowers the total resistance and raises the total current, and a break in one branch leaves the others working. Homes are wired in parallel so every device gets full voltage and runs independently, and parallel bulbs shine brighter than the same bulbs in series. Both arrangements build directly on Ohm's law.

Sources

1. OpenStax, "College Physics 2e," Chapter 21 (Circuits and DC Instruments), Sections 21.1 (Resistors in Series and Parallel). Free at openstax.org.
2. The Physics Classroom, "Electric Circuits," Lesson 4 (Series Circuits) and Lesson 5 (Parallel Circuits), at physicsclassroom.com.
3. HyperPhysics, "Series and Parallel Resistances" and "DC Circuits," Georgia State University, at hyperphysics.phy-astr.gsu.edu.
4. OpenStax, "University Physics Volume 2," Chapter 10 (Direct-Current Circuits). Free at openstax.org.

The deep link between electricity and magnetism

This final lesson explores magnetism and its remarkable connection to electricity, one of the most important discoveries in the history of science. On its own, magnetism explains compasses, fridge magnets, and the behaviour of iron. But the truly powerful idea is that electricity and magnetism are two aspects of a single phenomenon: an electric current creates magnetism, and, in reverse, a moving magnet creates electricity. This link, called electromagnetism, is the basis of the electric motors, generators, and transformers that run the modern world. The lesson covers magnetic poles and fields, why the Earth acts as a magnet, how a current produces magnetism, and how motors and generators exploit the connection, tying together the whole electricity module.

Magnets and poles

A magnet is an object that produces a magnetic force and has two ends called poles: a north pole and a south pole. The behaviour of poles mirrors the rule for electric charge almost exactly: like poles repel and opposite poles attract. Push two north poles together and they resist; bring a north pole near a south pole and they snap together. Magnets attract certain materials, especially iron, nickel, and cobalt, which is why a magnet picks up iron nails but not a copper coin or a plastic ruler.

One striking property sets magnets apart from electric charges. Whereas a positive and a negative charge can exist separately, a magnet's two poles cannot be separated. If you cut a bar magnet in half, you do not get one north piece and one south piece; instead each half becomes a complete new magnet with its own north and south pole. Cut those halves again and the same thing happens. No matter how finely you divide it, you can never isolate a single magnetic pole. Magnetic poles always come in pairs.

Magnetic fields

Surrounding every magnet is a magnetic field, the region of space where the magnet's force can be felt by other magnets or magnetic materials. Although the field is invisible, physicists represent it with field lines that, by convention, run from the north pole to the south pole outside the magnet. The lines show both the direction of the force and its strength: where the field lines are close together the field is strong (near the poles), and where they spread apart the field is weaker. A classic demonstration sprinkles iron filings on paper over a magnet; the filings line up along the field lines, making the invisible field visible as a pattern of curves looping from pole to pole.

The Earth is a giant magnet

The Earth itself behaves like an enormous bar magnet, with a magnetic field produced by the churning molten iron in its core. This is why a compass works. A compass needle is simply a small, lightweight magnet free to rotate, and it turns until it lines up with the Earth's magnetic field, one end pointing roughly toward the geographic north. Sailors, explorers, and hikers have navigated by this effect for many centuries, long before anyone understood what caused it. (A subtle point: the compass needle's north-seeking end is attracted toward a magnetic pole that lies near the geographic north, which by the pole rule is actually a magnetic south pole, but for navigation what matters is simply that the needle reliably points north.)

Electromagnetism: current makes magnetism

For most of history, electricity and magnetism were thought to be unrelated. Then, in 1820, Hans Christian Oersted noticed that a compass needle moved whenever an electric current flowed in a nearby wire. This was the pivotal discovery that electricity and magnetism are linked. The fact is this: an electric current flowing through a wire creates a magnetic field around that wire. A single straight wire produces a fairly weak field that circles around it, but the effect can be greatly strengthened.

If you wind the wire into a coil of many loops, the magnetic fields of the individual loops add together to make a much stronger and more useful magnet called an electromagnet. Adding more turns of wire, or increasing the current, makes the electromagnet stronger, and placing a piece of iron (an iron core) inside the coil strengthens the field dramatically, because the iron itself becomes magnetized.

The great advantage of an electromagnet over an ordinary permanent magnet is control: it is magnetic only while the current is flowing. Switch the current off and the magnetism vanishes almost instantly; switch it on and the magnetism returns. This on-off control, and the ability to vary the strength by changing the current, is why electromagnets rather than permanent magnets are used wherever magnetism must be turned on and off: in the giant scrapyard cranes that pick up a car and then drop it by cutting the current, in the electromagnet that rings a doorbell, in the read-write heads of hard drives, and in the loudspeakers and motors all around you.

Worked example: strengthening an electromagnet

Question: You have built a simple electromagnet from a coil of wire connected to a battery. Name three ways to make it stronger, and explain why each works.

Solution: First, increase the current, by using a higher-voltage battery, because a larger current produces a stronger magnetic field. Second, add more turns of wire to the coil, because the fields of the individual loops add together, so more loops give a stronger combined field. Third, insert an iron core into the coil, because the iron becomes magnetized and greatly amplifies the field. Any of these increases the electromagnet's strength, and combining all three makes a powerful magnet. This reasoning is exactly how engineers design electromagnets for a required strength.

Electric motors: magnetism makes motion

The link between electricity and magnetism does mechanical work in the electric motor. The principle is that a wire carrying a current, when placed in a magnetic field, experiences a force. If a coil of wire carrying current is placed between the poles of a magnet, one side of the coil is pushed up and the other pushed down, and this pair of opposite forces makes the coil rotate. By arranging for the current to reverse at the right moment each half-turn, the coil is kept spinning continuously. This spinning coil is the heart of an electric motor, which converts electrical energy into rotational motion. Motors are everywhere: in fans, washing machines, power tools, hard drives, and the electric cars that are reshaping transport.

Generators: motion makes electricity

Perhaps the most important consequence of electromagnetism is the reverse of the motor effect, discovered by Michael Faraday. If you move a magnet near a coil of wire (or move a coil near a magnet), a voltage is induced in the coil, and if the coil is part of a circuit, a current flows. This is called electromagnetic induction, and the device that uses it is a generator. A generator is essentially a motor run backwards: instead of putting electricity in to get motion out, you put motion in (spinning a coil in a magnetic field) to get electricity out.

This is how almost all of the world's electricity is produced. In a power station, some energy source, burning coal or gas, falling water at a dam, nuclear heat, or a wind turbine, is used to spin coils inside magnets, and the induced current is sent out along the power lines. The electricity that lit the bulbs and drove the circuits earlier in this module was almost certainly generated this way, which ties the entire topic together: currents create magnetism, and moving magnets create currents, in an endless useful loop.

Real-world applications

Electromagnetism is arguably the most economically important idea in physics, because it underpins the generation and use of electrical energy. Every power station on Earth uses generators to turn mechanical motion into electricity through electromagnetic induction. Every electric motor, from the tiny one that vibrates a phone to the huge ones in trains and electric cars, uses the motor effect to turn electricity into motion. Transformers, which change voltage for efficient transmission across the grid, rely on the changing magnetic field of an alternating current. Electromagnets lift scrap, focus beams in particle accelerators and MRI scanners, and levitate maglev trains. Loudspeakers and headphones use electromagnets to push a cone or membrane and make sound. Data is stored magnetically on hard drives. The discovery that a moving charge makes magnetism, and a moving magnet makes current, quite literally powers modern civilization.

Common misconceptions

• "You can isolate a single magnetic pole." You cannot. Cutting a magnet always produces new magnets, each with both a north and a south pole. Poles always come in pairs.
• "Magnets attract all metals." Only certain metals, chiefly iron, nickel, and cobalt, are strongly attracted. Copper, aluminium, and gold are not, and non-metals are not at all.
• "An electromagnet stays magnetic after the current is switched off." A pure electromagnet loses its magnetism when the current stops; that on-off control is its main advantage. (An iron core may keep a little residual magnetism, but the strong field disappears.)
• "Electricity and magnetism are separate, unrelated things." They are deeply linked. A current creates a magnetic field, and a moving magnet induces a current. This unity is the basis of motors and generators.
• "A generator creates energy from nothing." It does not. A generator converts mechanical energy (the motion that spins it) into electrical energy, obeying the conservation of energy; something must do work to turn it.

Recap

A magnet has a north and a south pole, and like poles repel while opposite poles attract; poles always come in pairs, since cutting a magnet makes two new complete magnets. Around every magnet is a magnetic field, drawn as field lines from north to south, closer together where the field is stronger, and the Earth's own field makes a compass point north. Electricity and magnetism are linked: a current in a wire creates a magnetic field, and winding the wire into a coil with an iron core makes a strong, switchable electromagnet. A current-carrying coil in a magnetic field feels a force and spins, which is the electric motor (electricity to motion); moving a magnet near a coil induces a current, which is the generator (motion to electricity) that produces almost all our electrical power. This mutual link between currents and magnets underlies motors, generators, transformers, and much of modern technology.

Sources

1. OpenStax, "College Physics 2e," Chapter 22 (Magnetism), Sections 22.1 through 22.4, and Chapter 23 (Electromagnetic Induction, AC Circuits, and Electrical Technologies). Free at openstax.org.
2. The Physics Classroom, resources on "Magnetism" and "Electromagnetism," at physicsclassroom.com.
3. HyperPhysics, "Magnetic Field," "Electromagnet," and "Electromagnetic Induction," Georgia State University, at hyperphysics.phy-astr.gsu.edu.
4. OpenStax, "University Physics Volume 2," Chapters 11 through 13 (Magnetic Forces and Fields; Sources of Magnetic Fields; Electromagnetic Induction). Free at openstax.org.