Energy is one of the most useful ideas in physics because it is conserved: it changes form but the total stays the same. Many problems that would be hard with forces and kinematics become a few lines with energy. This guide covers work, kinetic and potential energy, conservation and power.
Work
Work is done when a force moves an object: W = Fd cos θ, where F is the force, d is the displacement and θ is the angle between them. Work is measured in joules (J); 1 J = 1 N·m.
- Pushing a box 5 m with 20 N in the direction it moves: W = 20 × 5 = 100 J.
- If the force is perpendicular to the motion (θ = 90°), no work is done. Carrying a backpack horizontally at constant speed does no work on it, even though your arms get tired.
- Friction usually does negative work, because it points opposite the motion.
Kinetic energy
Kinetic energy is energy of motion: KE = ½mv². Because speed is squared, doubling the speed quadruples the kinetic energy. That is why braking distance grows so quickly with speed.
The work-energy theorem says the net work on an object equals its change in kinetic energy: W_net = ΔKE.
Gravitational potential energy
Near Earth's surface, PE = mgh, where h is the height above a reference level you choose. Only changes in PE matter, so you can set h = 0 wherever is convenient.
Conservation of mechanical energy
If only gravity (and other conservative forces) do work, total mechanical energy stays constant:
KE₁ + PE₁ = KE₂ + PE₂
If friction or air resistance acts, some energy becomes heat, so the mechanical energy decreases, but the total energy, including heat, is still conserved.
Worked example 1: roller coaster
A 500 kg roller coaster car starts from rest at the top of a 30 m hill. Ignoring friction, how fast is it moving at the bottom?
- At the top: KE = 0 and PE = mgh = 500 × 9.8 × 30 = 147,000 J.
- At the bottom: PE = 0, so KE = 147,000 J.
- ½(500)v² = 147,000, so v² = 588 and v ≈ 24.2 m/s.
Notice the mass cancels: v = √(2gh) = √(2 × 9.8 × 30). Any car released from that height would reach the same speed without friction.
Worked example 2: with friction
A 2.0 kg block slides down a ramp from a height of 1.5 m and reaches the bottom at 4.0 m/s. How much energy was lost to friction?
- Starting energy: PE = 2.0 × 9.8 × 1.5 = 29.4 J.
- Final energy: KE = ½ × 2.0 × 4.0² = 16 J.
- Energy lost to friction: 29.4 − 16 = 13.4 J.
Power
Power is the rate of doing work: P = W / t, measured in watts (1 W = 1 J/s). For something moving at constant speed under a force F, P = Fv.
Worked example 3: lifting
A motor lifts a 60 kg crate 12 m in 8.0 s at constant speed. What is its power output?
- Work = mgh = 60 × 9.8 × 12 = 7,056 J.
- P = 7,056 / 8.0 ≈ 882 W.
Common mistakes
- Forgetting to square the velocity in KE.
- Counting work done by a force perpendicular to the motion.
- Using different reference heights for PE in the same problem.
- Assuming energy is conserved when friction is present and the problem gives information about it.
Practice questions
- How much kinetic energy does a 1,200 kg car have at 15 m/s?
- A ball is dropped from 20 m. How fast is it going just before it hits the ground, ignoring air resistance?
- How much work is done lifting a 3.0 kg book 2.0 m?
- A 75 W light bulb runs for 2 minutes. How much energy does it use?
Answers: (1) ½ × 1200 × 225 = 135,000 J. (2) √(2 × 9.8 × 20) ≈ 19.8 m/s. (3) 3.0 × 9.8 × 2.0 = 58.8 J. (4) 75 × 120 = 9,000 J.
Last reviewed 2026-10-23 by the BrainBoost team.