Why does kinetic energy grow with speed squared?
Twice the speed, twice the energy? Guess first, then work out why it is four times, and what that does to braking distance and crashes.
- 1A push adds energy over the distance it acts5 min
- 2Reaching a speed takes distance that grows as speed squared7 min
- 3Braking distance grows with speed squared6 min
- 4Momentum and energy answer different questions5 min
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The ideas people most often get wrong on the way, in their own words. Each step starts from one of them.
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The energy a force transfers is the force times the distance it acts through, called work. A 100 N push for 2 m transfers 200 J. The same push for 8 m transfers 800 J, four times as much. That energy ends up as the object's kinetic energy, so a longer push gives a faster object.
Pushed steadily, an object reaches speed v after a distance of v squared over twice the acceleration. Twice the speed needs four times the distance, and energy is force times distance, so it needs four times the energy. That is why kinetic energy is half m v squared, and doubling speed quadruples it.
A steady braking force removes energy in proportion to distance, and kinetic energy goes as speed squared. So doubling speed quadruples the braking distance: about 5 m at 30 km/h and 20 m at 60 km/h, with 7 m per second squared of deceleration. Reaction distance is extra, and grows only in step with speed.
Momentum, mass times speed, is what a force changes over time. Kinetic energy, half mass times speed squared, is what it changes over distance. A 2 kg ball at 3 m/s and a 1 kg ball at 6 m/s share a momentum of 6 but have 9 J and 18 J of energy. Speed matters more than mass for energy.
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