Trading Height for Speed
- Physics
- Ages 14–18 · Grades 9–12
- Hands-on experiment
What you’ll find out
- Gravitational potential energy mgh converts to kinetic energy ½mv², so v = √(2gh) when nothing else takes a share.
- Speed at the bottom does not depend on mass or on the shape of the path, and grows with the square root of height.
- Energy "lost" to friction and air is transferred to thermal energy; efficiency is the share that does what we want.
No sign-up and no real names. In the lesson you check your understanding as you read, run the experiment yourself, and practise until it sticks.
Height is stored energy
Lift a ball and you do work against gravity. That work is stored as gravitational potential energy, equal to mgh: mass, times gravitational field strength (about 9.8 newtons per kilogram), times height. Let the ball slide down a smooth ramp and the stored energy becomes kinetic energy, the energy of motion, equal to ½mv². If nothing else takes a share, every joule of potential energy becomes a joule of kinetic energy, so mgh = ½mv². Two surprising things follow. First, the mass cancels, so the speed at the bottom is v = √(2gh), whatever the object weighs. From 1 metre, anything arrives at about 4.4 metres per second. Second, speed grows with the square root of height: four times the height gives only twice the speed, about 8.9 metres per second from 4 metres. The shape of the path does not matter either, only the height. That is why a roller coaster’s first hill is its tallest: every later hill must be lower, or the car could not climb it.
Where the missing energy goes
Real pendulums and real rides never quite keep the books balanced. A pendulum swings up almost to its starting height, then a little less, then less again, until it hangs still. The energy has not vanished — energy is never created or destroyed — but at each swing some of it is transferred, by friction at the pivot and air pushing on the bob, into thermal energy: a slight warming of the pivot, the bob and the air. That energy still exists, but it is spread thinly among countless particles and cannot lift the bob again. This is called dissipation, and the share that does what we want is the efficiency. Brakes work the same way. A 1,000 kilogram car at 20 metres per second carries 200,000 joules of kinetic energy, and to stop it the brakes must turn all of that into heat. That is why brakes can overheat on a long mountain descent. Electric and hybrid cars recover some of it with regenerative braking, charging the battery instead of heating the brakes.
Curriculum links: NGSS HS-PS3-1 · NGSS HS-PS3-2