Kinetic and Potential Energy

Kinetic energy — the energy an object has because of its motion:

KE = ½ m v²

with energy in joules, mass in kilograms and speed in m/s.

The squared term matters more than anything else in this topic. Doubling the speed quadruples the kinetic energy; tripling it multiplies the energy by nine. This is why a small increase in speed makes such a large difference to braking distance and to the severity of a crash, and it is the physics behind speed limits.

Squaring only the speed, not the mass, is the usual slip: in ½ × 1200 × 20² the 20 is squared first to give 400, and only then is everything multiplied.

Gravitational potential energy — the energy an object has because of its height:

GPE = m g h

with mass in kilograms, g the gravitational field strength (10 N/kg on Earth) and height in metres.

Height is measured vertically, from whatever level you choose as zero. For an object on a slope, the height is the vertical rise, not the distance along the slope.

Transfers between the two. For a falling object with no air resistance, the gravitational potential store empties into the kinetic store:

m g h = ½ m v²

The mass cancels from both sides, which gives v = √(2gh) — and tells you something worth noticing: the speed reached in a fall does not depend on the mass. A heavy ball and a light ball dropped together land together, provided air resistance is negligible. This is what Galileo is supposed to have demonstrated from the leaning tower.

Example: a 0.50 kg ball is dropped from 1.8 m. Its GPE is mgh = 0.50 × 10 × 1.8 = 9.0 J. All of it becomes kinetic energy, so 9.0 = ½ × 0.50 × v², giving v² = 36 and v = 6.0 m/s.

Where the energy goes in practice. A real ball does not bounce back to its starting height: some of the store has gone to the thermal store of the ball and the ground, and some into the sound you hear. The total is still conserved.