Work, kinetic energy and potential energy explained

Key ideas

W = F d \cos\theta

$W$ is work, in joules (J). $1\ \text{J} = 1\ \text{N·m}$ .
$F$ is the size of the force, in newtons (N), and $d$ is the distance moved, in meters (m).
$\theta$ is the angle between the force and the direction of motion. Force along the motion gives positive work, against it gives negative work, and at 90° the work is zero.

KE = \tfrac{1}{2} m v^2 \qquad PE_g = m g h

$KE$ is kinetic energy and $PE_g$ is gravitational potential energy, both in joules (J).
$m$ is mass (kg), $v$ is speed (m/s), $g = 9.8\ \text{m/s}^2$ and $h$ is height above a level you choose as zero (m).

W_{\text{net}} = \Delta KE \qquad KE_1 + PE_1 = KE_2 + PE_2 \ \text{(no friction)}

Power is how fast work is done: $P = W/t$ , in watts (W), where $1\ \text{W} = 1\ \text{J/s}$ .

Worked examples

Example 1: work with a force at an angle

Problem You pull a sled 4.00 m across flat snow with a rope. The rope pulls with 50.0 N at 30.0° above the ground. How much work does the rope do?

Only the part of the force along the motion does work.
$W = Fd\cos\theta = (50.0\ \text{N})(4.00\ \text{m})\cos 30.0^\circ$
Calculate.
$W = 200.\ \text{J} \times 0.866 = 173.2\ \text{J}$

Answer 173 J

Example 2: kinetic energy

Problem What is the kinetic energy of a 0.50 kg ball moving at 12 m/s?

Square the speed first.
$KE = \tfrac{1}{2}(0.50\ \text{kg})(12\ \text{m/s})^2 = \tfrac{1}{2}(0.50)(144)\ \text{J}$
Multiply.
$KE = 36\ \text{J}$

Answer 36 J

Example 3: conservation of energy on a roller coaster

Problem A coaster car starts from rest at the top of a 45.0 m drop. Ignoring friction, how fast is it going at the bottom?

All the potential energy at the top becomes kinetic energy at the bottom.
$mgh = \tfrac{1}{2}mv^2$
Mass cancels, so the answer does not depend on how heavy the car is.
$v = \sqrt{2gh} = \sqrt{2(9.8\ \text{m/s}^2)(45.0\ \text{m})} = 29.70\ \text{m/s}$

Answer 29.7 m/s

Example 4: power climbing stairs

Problem A 60.0 kg student runs up stairs 4.50 m high in 6.00 s. What power does the student put out?

Work against gravity equals the gain in potential energy.
$W = mgh = (60.0)(9.8)(4.50) = 2646\ \text{J}$
Divide by time.
$P = \frac{2646\ \text{J}}{6.00\ \text{s}} = 441\ \text{W}$

Answer 441 W

Common mistakes and how to fix them

Counting work when nothing moves. Holding a heavy box still does no work on it, because $d = 0$ .
Forgetting to square the speed. Doubling the speed makes KE four times bigger, not twice. Fix: square $v$ first.
Using conservation of energy with friction. Friction turns some energy into heat. Fix: include the friction work, or only use $KE + PE$ = constant when friction is ignored.
Forgetting the angle. Fix: $\cos\theta$ uses the angle between force and motion. A force at right angles to the motion does zero work.

Practice problems

How much work does it take to lift a 2.0 kg book 1.5 m at a steady speed?
1. 3.0 J
2. 29 J
3. 15 J
4. 2.9 J
Show answer
Answer: 29 J
The lifting force equals the weight, $2.0 \times 9.8 = 19.6$ N. $W = 19.6 \times 1.5 = 29.4$ J, which is 29 J.
If a car's speed doubles, what happens to its kinetic energy?
1. It doubles
2. It stays the same
3. It becomes 4 times as large
4. It halves
Show answer
Answer: It becomes 4 times as large
KE depends on $v^2$ , and $2^2 = 4$ .
A ball is dropped from 5.0 m. Ignoring air resistance, how fast is it moving just before it hits the ground?
Show answer
Answer: 9.9 m/s
$v = \sqrt{2gh} = \sqrt{2 \times 9.8 \times 5.0} = \sqrt{98} = 9.90$ , which is 9.9 m/s.
A box slides across a flat floor. How much work does the normal force do on it?
1. 0 J
2. $mgd$
3. $-mgd$
4. It depends on the friction
Show answer
Answer: 0 J
The normal force points up and the motion is sideways, so $\theta = 90^\circ$ and $\cos 90^\circ = 0$ .
A 1200 kg car speeds up from 10. m/s to 20. m/s. How much net work was done on it?
Show answer
Answer: $1.8 \times 10^5$ J
$W_{\text{net}} = \Delta KE = \tfrac{1}{2}(1200)(20.^2 - 10.^2) = 600 \times 300 = 180{,}000$ J.

Frequently asked questions

Is energy ever destroyed?

No. Energy changes form but the total stays the same. When friction slows a sliding box, its kinetic energy becomes thermal energy (the surfaces warm up). When a book falls, potential energy becomes kinetic energy, then sound and heat when it lands.

Why does the zero level for height not matter?

Only changes in potential energy affect the motion. If you measure height from the floor or from the table, PE values change, but the difference between two points stays the same. Pick whatever level makes the numbers simple, usually the lowest point.

What is the difference between work and power?

Work is how much energy is transferred. Power is how fast it is transferred. Walking or running up the same stairs does the same work, but running does it in less time, so it takes more power.

Sources

OpenStax College Physics 2e, 7.1 Work: The Scientific Definition, accessed October 1, 2026
OpenStax College Physics 2e, 7.2 Kinetic Energy and the Work-Energy Theorem, accessed October 1, 2026
OpenStax College Physics 2e, 7.3 Gravitational Potential Energy, accessed October 1, 2026
OpenStax College Physics 2e, 7.7 Power, accessed October 1, 2026

Work and energy

Key ideas

Worked examples

Common mistakes and how to fix them

Frequently asked questions

Sources

Try asking Ducky

Key ideas

Worked examples

Common mistakes and how to fix them

Frequently asked questions

Sources

Related

Try asking Ducky