g = 9.8 m/s². Heights are measured from the lowest point of the track (h = 0). With friction off, KE + PE stays exactly the same.
Energy is never made or destroyed. It moves from one form (or store) to another. Kinetic and potential energy swap back and forth, and friction or air resistance slowly turns some of the energy into thermal energy. Tap Try it to watch the first four in the lab.
The motor does work to pull the cart up the lift hill. After that, no engine is needed: PE becomes KE on each drop and KE becomes PE on each climb. The brakes turn the last of the KE into thermal energy.
At each end of the swing the bob stops for an instant (all PE). At the bottom it moves fastest (all KE). Air resistance makes each swing a tiny bit smaller.
Dropping in from the top, the skater speeds up as PE turns into KE, then slows down going up the other side. To go higher than the start, the skater must add energy by pumping with the legs.
The sled speeds up going down, then friction with the snow slows it on the flat until it stops. Its energy has not vanished: it warmed the snow and the runners very slightly.
When the ball hits the floor it squashes and stores elastic energy, then springs back. Each bounce is lower because some energy becomes thermal energy and sound.
Your muscles do work to bend the bow, which stores elastic potential energy. When you let go, the bow does work on the arrow and the energy becomes the arrow’s kinetic energy.
Water held high behind the dam has gravitational PE. As it falls through pipes it speeds up (KE) and spins turbines that drive generators, making electrical energy.
Brakes use friction to turn the car’s kinetic energy into thermal energy, so the brake discs get hot. Electric and hybrid cars can turn some of it back into electrical energy (regenerative braking).
The ball leaves your hand fast, slows as KE turns into PE, stops for an instant at the top (all PE), then falls and speeds up again. Without air resistance it comes back at the speed you threw it.
How does an object get its energy in the first place? Something does work on it: a force pushes or pulls it through a distance. Work = force × distance, measured in joules (J). Power is how fast work is done: power = work ÷ time, measured in watts (W). 1 W = 1 J every second.
Measure your own power: time yourself running up a flight of stairs and measure its height.
The diagram teachers use for kinetic and potential energy: where is PE largest, where is KE largest, where does one turn into the other? Switch between the labeled diagram, a blank one to test your memory, and a quiz where you place each label yourself. You can also download both versions or print a label worksheet.
Tap a word, then tap the numbered box it belongs to (on the diagram or in the list below). Tap a filled box to take the word back. On a phone, swipe the diagram sideways to see every number.
Ten questions on KE, PE, conservation of energy, work and power (g = 9.8 m/s²).
Tap an example, then tap the box it belongs in. Check when all twelve are placed.
The cart starts at rest at A (20 m) and there is no friction. Tap the points from fastest to slowest.
Anything that is moving has kinetic energy: a rolling ball, the wind, a running dog. Faster things have more, and heavier things moving at the same speed have more. Energy can also be stored, ready to be used: a book on a high shelf, a stretched rubber band, a drawn bow. This stored energy is potential energy. Lift something higher and it stores more; let it fall and the stored energy turns into motion.
Kinetic energy KE = ½ × m × v² (mass in kg, speed in m/s, energy in joules). Because the speed is squared, going twice as fast means four times the KE. Gravitational potential energy PE = m × g × h, where g = 9.8 m/s² on Earth and h is the height above a level you choose as zero. A 50 kg skater 4 m up has PE = 50 × 9.8 × 4 = 1,960 J. At the bottom of a frictionless ramp all of it is KE: ½ × 50 × v² = 1,960 J, so v ≈ 8.85 m/s.
With no friction, KE + PE = constant (conservation of mechanical energy), so m g h = ½ m v² and the mass cancels: v = √(2 g h). Every object dropped from the same height reaches the same speed. Energy gets in or out by work: W = F × d (more exactly F d cos θ). The work–energy theorem says the net work done on an object equals its change in KE. Friction does negative work, turning mechanical energy into thermal energy, so with friction KE + PE + thermal = constant. Power P = W ÷ t is the rate of doing work (watts).
Skate half-pipe: switch to the bar chart and watch the blue KE bar and the green PE bar trade places while the yellow total stays still. Change the mass: the speed at the bottom does not change. Roller coaster: start below 14 m and see the cart fail to get over hill C. Turn friction on and watch the orange thermal bar grow until everything stops. Then in the Work & Power Lab, make climber B heavier but slower: who does more work, and who is more powerful?
| Quantity | Formula | Unit | In words |
|---|---|---|---|
| Kinetic energy | KE = ½ m v² | joule (J) | energy of motion; double the speed → 4× the KE |
| Gravitational potential energy | PE = m g h | J | energy stored by height; g = 9.8 m/s² (9.8 N/kg) on Earth |
| Elastic potential energy | PE = ½ k x² | J | energy stored in a stretched or squashed spring (k = spring constant) |
| Conservation (no friction) | KE + PE = constant | J | energy changes form; the total stays the same |
| Speed after dropping h | v = √(2 g h) | m/s | from m g h = ½ m v²; the mass cancels |
| With friction | KE + PE + thermal = constant | J | thermal energy = work done against friction |
| Work | W = F × d (= F d cos θ) | J = N·m | force × distance moved in the direction of the force |
| Work–energy theorem | Wnet = ΔKE | J | net work done on an object = its change in KE |
| Power | P = W ÷ t | watt (W) = J/s | how fast work is done (or energy is transferred) |
For a constant force W = F d cos θ, where θ is the angle between the force and the displacement. A force at 90° to the motion (the normal force on a flat floor, the string of a pendulum, the pull toward the center in circular motion) does no work. For a force that changes, W = ∫F dx, the area under the force–distance graph. For a spring, F = kx rises from 0 to kx, so the area is a triangle: W = ½ k x², the elastic PE.
Start from Newton’s second law, F = ma. Using a dx = v dv, ∫F dx = m∫v dv = ½ m vf² − ½ m vi². So Wnet = ΔKE. Gravity and springs are conservative forces: their work equals −ΔPE, so if only they do work, ΔKE + ΔPE = 0 and mechanical energy is conserved.
Kinetic friction is f = μ N, and it does negative work −f d, so Δ(KE + PE) = −f d and the thermal energy rises by f d. On a curved track the normal force is not just m g cos θ: N = m(g cos θ + v²/r) in a dip (bigger) and m(g cos θ − v²/r) over a hump (smaller). The lab uses this, which is why friction slows the cart most in the valleys.
Instantaneous power is P = dW/dt = F v cos θ; at constant power a car’s driving force falls as its speed rises (F = P/v). For the pendulum the period for small swings is T = 2π√(L/g) = 2.84 s for L = 2 m. Bigger swings take longer: the lab gives 2.89 s for a 30° release and 3.05 s for 60°, matching the exact formula.
Kinetic energy is the energy an object has because it is moving. Potential energy is stored energy an object has because of its position or shape, such as being high up (gravitational) or being stretched or squashed (elastic). A roller coaster at the top of a hill has lots of potential energy; at the bottom it has lots of kinetic energy.
Key takeaway: kinetic = moving; potential = stored.KE = ½ m v², with the mass m in kilograms and the speed v in meters per second, giving joules. A 2 kg ball at 3 m/s has KE = ½ × 2 × 9 = 9 J. Because the speed is squared, doubling the speed makes the kinetic energy four times bigger.
Key takeaway: KE = ½ m v²; twice as fast = four times the KE.PE = m g h: mass (kg) × gravitational field strength (g = 9.8 m/s², also written 9.8 N/kg) × height (m) above a chosen zero level. A 5 kg box on a 2 m shelf has PE = 5 × 9.8 × 2 = 98 J.
Key takeaway: PE = m g h with g = 9.8 m/s².A chain does work to pull the cart up the first hill, giving it lots of PE. On the way down PE turns into KE and the cart speeds up; going up the next hill KE turns back into PE and it slows down. The lowest point has the most KE (fastest), the highest point the most PE. Each hill must be lower than the first because friction slowly turns some energy into thermal energy.
Key takeaway: down: PE → KE; up: KE → PE.Energy cannot be created or destroyed, only changed from one form to another. In the lab with friction off, KE + PE stays exactly the same. With friction on, KE + PE goes down but KE + PE + thermal energy stays the same.
Key takeaway: the total energy never changes.It becomes thermal energy (the surfaces warm up) and a little sound. Rub your hands together and you feel it. The energy is not destroyed, but it is spread out and hard to use again, which is why people often call it “lost”.
Key takeaway: friction turns mechanical energy into thermal energy.Kinetic: a moving car, a flying arrow, wind, a flowing river, a rolling ball, a running dog. Potential: a book on a high shelf, water behind a dam, a stretched rubber band, a drawn bow, a compressed spring, a skier standing at the top of a slope.
Key takeaway: moving things have KE; high, stretched or squashed things store PE.Work is the energy transferred when a force moves something: W = F × d, in joules. Power is how fast the work is done: P = W ÷ t, in watts. Two people who climb the same stairs with the same weight do the same work, but the one who runs up faster has more power.
Key takeaway: work = how much; power = how fast.No. A heavier object has more PE (m g h) but needs more energy to reach each speed (½ m v²). The mass cancels: v = √(2 g h). From 4 m every object reaches 8.85 m/s at the bottom. Try changing the mass in the lab: the speed does not change, only the energies do.
Key takeaway: v = √(2gh): speed does not depend on mass.Only when the force points along the motion. In general W = F d cos θ, where θ is the angle between the force and the direction of motion. A force at right angles to the motion, like the string of a pendulum or the normal force on a flat floor, does no work, and pushing a wall that does not move does no work on the wall.
Key takeaway: only the part of the force along the motion does work.