High School · Physics · Unit 3

Energy and Work

Work, energy, power and momentum: the bookkeeping behind every push, fall and crash.

A roller coaster has no engine on its way down, yet it races through the dip and climbs the next hill. Where does the speed come from, and where does it go? The answer is energy: it moves from place to place and changes form, but it never just appears or vanishes.

On this page you'll learn what physicists mean by work, kinetic energy and potential energy, why energy is conserved, how power measures how fast work gets done, and how momentum explains airbags and collisions. Then try worked problems, a roller-coaster calculator, the story of James Joule, and a quiz.

Start with work Jump to the quiz

Key words first

Energy

The ability to make things happen: to move something, lift it, warm it or light it up. Energy is measured in joules (J).

Work

Energy passed to an object by a force that moves it. Push a box across the floor and your push does work on the box. Work is also measured in joules.

Joule (J)

One joule is the work done by a 1 N force pushing something 1 m: 1 J = 1 N·m. Lifting an apple (about 1 N) one meter takes about 1 J.

Power

How fast work is done or energy is used, in watts (W). One watt is one joule per second.

Momentum

Mass times velocity, in kg·m/s. A loaded truck and a fast baseball both have a lot of momentum, and both are hard to stop.

Idea 1: Work, or "force times distance"

In everyday speech, "work" means effort. In physics it has an exact meaning: work is done when a force moves an object, and it equals the force times the distance moved in the direction of the force.

W = F × d

Push a box with 40 N across 5 m of floor and you do 40 × 5 = 200 J of work on it. That 200 J doesn't disappear. It becomes the box's motion and, through friction, a little warmth in the floor and the box.

Only the part of the force along the motion counts. If you carry a bag across a level room, your hand pushes up but the bag moves sideways, so your hand does no work on the bag. (In more advanced courses you'll write this as W = Fd cos θ, where θ is the angle between the force and the motion.)

Top: the push and the motion point the same way, so the push does work. Bottom: an upward force on a bag carried sideways does no work on it. Not to scale.

Where you see work

Lifting a backpack onto a shelf: your upward force moves it upward, so you do work against gravity.
Pushing a stalled car: the farther you push it, the more work you do.
Holding a barbell overhead: it feels exhausting, but if the bar doesn't move, you do no work on the bar. Your muscles are using energy inside your body to stay tense.

Idea 2: Kinetic and potential energy

Energy comes in many forms: motion, height, heat, light, sound, chemical energy in food and fuel, and electrical energy. This unit focuses on two forms of mechanical energy.

Kinetic energy (KE): energy of motion

KE = ½mv²

Anything moving has kinetic energy. It grows with mass, and it grows with the square of speed. Double the speed and the kinetic energy becomes four times as large.

Gravitational potential energy (PE): energy of height

PE = mgh

Lift an object to height h and you store energy in it. Here g ≈ 9.8 m/s² on Earth, so mg is the object's weight and mgh is just weight × height, the work you did lifting it.

Height is always measured from a level you choose, such as the floor or the ground. Only changes in potential energy matter, so pick whatever zero level makes the problem easiest.

Why speed matters so much

Because kinetic energy depends on v², a small increase in speed means a big increase in energy. Here is a 1,200 kg car:

KE = ½ × 1,200 kg × v². Three times the speed means nine times the energy.
Speed	About	Kinetic energy
10 m/s	36 km/h (22 mph)	60,000 J
20 m/s	72 km/h (45 mph)	240,000 J (4 times as much)
30 m/s	108 km/h (67 mph)	540,000 J (9 times as much)

To stop, the brakes must do enough work to remove all that energy. With the same braking force, stopping distance grows with the square of the speed too, which is a big reason speed limits near schools are low.

Myth: "energy gets used up." Energy is never destroyed. When we say a battery is "used up," its chemical energy has been changed into other forms, like light and heat, that are harder to use again.

Idea 3: Conservation of energy

This is the law of conservation of energy: energy cannot be created or destroyed. It can only move from one object to another or change from one form to another. The total amount stays the same.

Follow a roller coaster car with no engine. At the top of the first hill (A) it is barely moving, so its energy is almost all potential. As it drops, height turns into speed: PE goes down and KE goes up by the same amount. At the bottom (C) it is fastest. Climbing the next hill (B), speed turns back into height.

With no friction, PE + KE would be exactly the same at every point:

mgh = ½mv² so v = √(2gh)

Notice the mass cancels. Without friction or air resistance, a full car and an empty car reach the same speed at the bottom.

In a real coaster, friction and air resistance turn some of the energy into heat and sound, so the total mechanical energy slowly drops. The energy isn't lost; it has warmed the wheels, the track and the air. That's why, on a classic coaster pulled up by a chain, no later hill can be as tall as the first.

Energy at points A, C and B with no friction. The dashed line is the total, which stays the same. Not to scale.

Where you see energy conserved

A swinging pendulum or a playground swing: fastest at the bottom, slowest at the ends, trading KE and PE twice every swing.
A skateboarder in a half-pipe rises almost to the starting height on the other side. The small shortfall went into heat and sound.
Hydroelectric dams let high water fall through turbines, turning gravitational potential energy into electrical energy.

Idea 4: Power, or "how fast"

Walking up a flight of stairs and running up the same stairs take the same work: you lift the same body through the same height. Running feels harder because you do that work in less time. That rate is power:

P = W ÷ t

Power is measured in watts (W), where 1 W = 1 J/s. A 10 W LED bulb turns 10 joules of electrical energy into light and heat every second.

Car engines are often rated in horsepower, an older unit James Watt defined as 550 foot-pounds per second. One horsepower is about 746 W.

Same stairs, same work. Less time means more power. Not to scale.

Idea 5: Momentum and impulse

Momentum is "mass in motion":

p = m × v

Unlike energy, momentum has a direction. A 1,500 kg car at 20 m/s east has 30,000 kg·m/s of momentum east.

To change momentum you need a force acting for some time. Force × time is called impulse, and it equals the change in momentum:

F × Δt = Δp

This is Newton's second law in another form. It explains a lot of safety design: if something must lose a fixed amount of momentum, stretching out the time makes the force smaller.

Same area under each curve means the same impulse, so the same change in momentum. Not to scale.

Where you see impulse

Airbags and crumple zones make a crash stop take longer, so the force on the people inside is smaller.
Catching a raw egg: pull your hands back as it lands and it's less likely to break.
Landing from a jump: bending your knees stretches out the stopping time and protects your joints. Gym mats do the same.
Follow-through in baseball, golf and tennis keeps the force on the ball for longer, giving it more momentum.

Idea 6: Momentum in collisions

When two objects collide, Newton's third law says they push on each other with equal and opposite forces for the same amount of time. So whatever momentum one gains, the other loses. That gives us the law of conservation of momentum:

If no outside net force acts on a group of objects, their total momentum stays the same.

m₁v₁ + m₂v₂ = total before = total after

In the diagram, a 2 kg cart at 3 m/s has 6 kg·m/s of momentum. It hits a 1 kg cart at rest and they lock together. Now 3 kg must share the same 6 kg·m/s, so they move at 2 m/s.

A perfectly inelastic collision: the carts stick, and total momentum stays 6 kg·m/s. Not to scale.

Elastic and inelastic collisions

Momentum is conserved in every collision (with no outside net force). Kinetic energy is a different story:

Type	What happens	Momentum	Kinetic energy
Elastic	Objects bounce apart with no lasting dents or heating, like billiard balls (almost) or the steel balls in a Newton's cradle (almost)	Conserved	Conserved
Inelastic	Objects bounce but get dented, warmed or noisy, like a car crash or a dropped basketball	Conserved	Some changes to heat, sound and bending
Perfectly inelastic	Objects stick together, like the carts above or a football tackle	Conserved	The most is changed to other forms

In the cart example, the kinetic energy before is ½ × 2 × 3² = 9 J, and after it's ½ × 3 × 2² = 6 J. The missing 3 J went into sound, heat and squashing the coupler. Total energy is still conserved; it just isn't all kinetic anymore.

Worked examples

Try each one on paper first, then open the solution. Use g = 9.8 m/s².

1. Pushing a box

You push a box with a steady 40 N force for 5 m across the floor. How much work do you do?

Show solution

W = F × d = 40 N × 5 m = 200 J

2. Lifting a backpack

You lift an 8 kg backpack straight up 1.5 m onto a shelf at a steady speed. How much work do you do, and how much potential energy does it gain?

Show solution

Force needed = weight = 8 × 9.8 = 78.4 N
W = 78.4 N × 1.5 m ≈ 118 J
PE = mgh = 8 × 9.8 × 1.5 ≈ 118 J

Same number: the work you did is stored as potential energy.

3. Kinetic energy of a car

A 1,200 kg car moves at 15 m/s. What is its kinetic energy?

Show solution

KE = ½ × 1,200 × 15² = ½ × 1,200 × 225 = 135,000 J

4. Double the speed

The same car speeds up to 30 m/s. How many times more kinetic energy does it have?

Show solution

KE = ½ × 1,200 × 30² = 540,000 J
540,000 ÷ 135,000 = 4 times

Twice the speed, four times the energy, because of the v².

5. Roller coaster drop

A coaster car starts almost at rest at the top of a 45 m drop. Ignoring friction, how fast is it going at the bottom?

Show solution

v = √(2gh) = √(2 × 9.8 × 45) = √882 ≈ 29.7 m/s

That's about 107 km/h (66 mph). The car's mass doesn't matter.

6. Power on the stairs

A 60 kg student runs up stairs 3 m high in 4 s. What is their power output?

Show solution

W = mgh = 60 × 9.8 × 3 = 1,764 J
P = W ÷ t = 1,764 J ÷ 4 s = 441 W

That's about 441 ÷ 746 ≈ 0.6 horsepower.

7. Catching a baseball

A 0.145 kg baseball at 40 m/s is caught and stopped. Find the change in momentum, then the average force if it stops in 0.01 s, and if a soft glove stretches the stop to 0.05 s.

Show solution

Δp = 0.145 × 40 = 5.8 kg·m/s
0.01 s: F = 5.8 ÷ 0.01 = 580 N
0.05 s: F = 5.8 ÷ 0.05 = 116 N

Five times as long, one-fifth the force. That's why catchers' mitts are padded.

8. Carts that stick

A 2 kg cart at 3 m/s hits a 1 kg cart at rest and they stick. How fast do they move, and how much kinetic energy changes into other forms?

Show solution

Momentum before = 2 × 3 = 6 kg·m/s
v = 6 ÷ 3 kg = 2 m/s
KE before = 9 J, after = 6 J, so 3 J becomes heat, sound and bending

Try it: the roller-coaster calculator

Height ↔ speed at the bottom

Things to try

Find the speed at the bottom of a 45 m drop. Then change the mass from 500 kg to 1,000 kg. What changes, and what doesn't?
How tall must a drop be to reach 30 m/s at the bottom? What if you want twice that speed?
Switch to the Moon. Is a 45 m drop faster or slower there?

How do we know?

Energy conservation isn't just a nice rule; it has been tested for nearly two centuries. In the 1840s James Joule measured the heat produced when a known amount of work stirred water, and found the same exchange rate between work and heat again and again (see the history below). If energy could appear from nothing, we could build a "perpetual motion" machine that runs forever with no fuel. Nobody ever has.

You can test these ideas in a classroom. Roll a cart down a ramp and time it with a motion sensor: its speed at the bottom comes close to √(2gh), a little less because of friction. Crash two carts together on a low-friction track and the total momentum before and after matches, even when the carts stick or bounce.

Energy or momentum? Use both. Momentum is always conserved in a collision (with no outside net force), so it tells you the speeds after a crash. Total energy is also always conserved, but kinetic energy alone is only kept in elastic collisions.

A short history: Joule and the joule

For a long time, people thought heat was an invisible fluid that flowed from hot things to cold ones. In the 1840s several scientists, working separately, argued that heat, motion and electricity are different forms of one thing that can change from one form into another but can't be destroyed. (They called it "force"; we now call it energy.)

The German doctor Julius Robert Mayer estimated how much work equals a given amount of heat in 1842, and in 1845 stated clearly that work and heat convert into each other but are never destroyed. In 1847 Hermann von Helmholtz published a book, On the Conservation of Force, that applied the idea to heat, light, electricity and chemistry.

The careful measurements came from James Prescott Joule (1818–1889) of Manchester, England. In his best-known experiment, falling weights turned a paddle wheel inside a container of water. The paddles churned the water and warmed it very slightly, and Joule measured that tiny temperature rise. In a paper sent to the Royal Society in 1849 and printed in 1850, he concluded that warming a pound of water by 1 °F

"requires for its evolution the expenditure of a mechanical force represented by the fall of 772 lbs. through the space of one foot."J. P. Joule, "On the Mechanical Equivalent of Heat," Philosophical Transactions of the Royal Society (1850).

That's within about 1 percent of the modern value (about 778 foot-pounds).

A simplified, original drawing of Joule's paddle-wheel experiment, not his exact apparatus. Not to scale.

Why the units are called the joule and the watt

The watt honors James Watt (1736–1819), the Scottish engineer whose improved steam engines helped power the Industrial Revolution, and who defined the horsepower. In 1882 the engineer William Siemens proposed naming units after Watt and Joule. On 31 August 1889, an International Electrical Congress in Paris adopted the joule as the unit of work and the watt as one joule per second. Joule was still alive to see it: he died a few weeks later, on 11 October 1889. Today they are official SI units: 1 J = 1 N·m and 1 W = 1 J/s.

1842 Mayer publishes an estimate of how much work is equivalent to a given amount of heat.
1843 Joule announces his first value for the "mechanical equivalent of heat," from electrical experiments.
1845 Mayer states that work and heat convert into each other but are never destroyed.
1847 Helmholtz publishes On the Conservation of Force.
1849–1850 Joule's paddle-wheel paper, giving 772 foot-pounds, is sent to the Royal Society and printed.
1882 William Siemens proposes the names watt and joule.
1889 The International Electrical Congress in Paris adopts the joule and the watt.

Test yourself

The Energy Check

Talk about it

A ball is dropped and bounces back to only half its starting height. Did it lose energy? Where did the "missing" energy go?
You push hard against a brick wall for a minute and you're tired. Did you do any work on the wall? Why do you feel tired anyway?
A bowling ball and a ping-pong ball roll toward you with the same momentum. Which one is moving faster? Which would you rather stop with your foot?
Why do you think it took until the 1840s, long after Newton, for scientists to see heat as a form of energy?

Sources

OpenStax, Physics (high school), Chapter 9 "Work, Energy, and Simple Machines": 9.1 Work, Power, and the Work–Energy Theorem, 9.2 Mechanical Energy and Conservation of Energy.
OpenStax, Physics (high school), Chapter 8 "Momentum": 8.1 Linear Momentum, Force, and Impulse, 8.2 Conservation of Momentum, 8.3 Elastic and Inelastic Collisions.
J. P. Joule, "On the Mechanical Equivalent of Heat," Philosophical Transactions of the Royal Society of London 140 (1850), received June 1849, doi:10.1098/rstl.1850.0004 (paddle-wheel method, the 772 foot-pound result and the quoted conclusion; also his 1843 value). https://royalsocietypublishing.org/doi/10.1098/rstl.1850.0004
Gregory S. Girolami, "A Brief History of Thermodynamics, As Illustrated by Books and People," Journal of Chemical & Engineering Data 65 (2020): 298–311, doi:10.1021/acs.jced.9b00515 (Mayer 1842 and 1845, Helmholtz 1847). Author's PDF
Carmen J. Giunta, "Watt's in a name? Units of power and energy," Substantia 3, no. 2, suppl. 1 (2019): 13–26, doi:10.13128/Substantia-403 (James Watt and the horsepower, Siemens's 1882 proposal, the 1889 Paris congress and Joule's death that October). https://riviste.fupress.net/index.php/subs/article/download/403/293/2272
Frank A. Wolff, "The so-called international electrical units," Bulletin of the Bureau of Standards 1, no. 1 (1904): 39, on the 1889 Paris definitions of the joule and the watt. https://nvlpubs.nist.gov/nistpubs/bulletin/01/nbsbulletinv1n1p39_A2b.pdf
NIST, Guide for the Use of the International System of Units (SP 811), Chapter 4 (1 J = 1 N·m, 1 W = 1 J/s). https://www.nist.gov/pml/special-publication-811/nist-guide-si-chapter-4-two-classes-si-units-and-si-prefixes
NIST, SP 811 Appendix B.9, conversion factors (1 horsepower of 550 ft·lbf/s = 745.7 W; the British thermal unit and foot-pound-force, used for the modern 778 foot-pound comparison). https://www.nist.gov/pml/special-publication-811/nist-guide-si-appendix-b-conversion-factors/nist-guide-si-appendix-b9
NASA Goddard Space Flight Center, "Moon Fact Sheet" (surface gravity 1.62 m/s², used in the calculator). https://nssdc.gsfc.nasa.gov/planetary/factsheet/moonfact.html

Credits: all diagrams on this page, including the drawing of Joule's apparatus and the share image, are original SVG made for Jugami. The Joule quotation is from his public-domain 1850 paper.