Kinetic energy

Anything that moves has energy in its kinetic store. The kinetic energy depends on the mass and, even more, on the speed: KE = ½mv². Doubling the mass doubles the kinetic energy, but doubling the speed makes it four times bigger. That is why a small rise in a car’s speed needs a much longer distance to stop.

Key facts

  • Kinetic energy KE = ½mv².
  • Doubling speed makes KE four times bigger; doubling mass doubles it.
  • To find a speed from kinetic energy, rearrange: v = √(2KE ÷ m).

Worked examples

Example 1

A 0.5 kg ball moves at 4 m/s. Find its kinetic energy.

  1. Square the speed first. 4² = 16
  2. Then multiply by half the mass. E = ½ × 0.5 × 16 = 4 J

Answer: 4 J

Example 2

A 60 kg runner moves at 5.0 m/s. Find her kinetic energy.

  1. E = ½ × m × v² = ½ × 60 × 5.0²
  2. E = 30 × 25 = 750 J

Answer: 750 J

Example 3

A 1,200 kg car travels at 15 m/s. Find its kinetic energy.

  1. Use KE = ½mv². m = 1,200 kg, v = 15 m/s
  2. Square the speed first. 15² = 225
  3. Then multiply. KE = ½ × 1,200 × 225 = 135,000 J

Answer: 135,000 J (135 kJ)

Example 4

A 1,000 kg car travels at 20 m/s. What is its kinetic energy in kJ?

  1. ½ × 1,000 × 20² = 200,000 J
  2. = 200 kJ

Answer: 200 kJ

Example 5

Two trolleys roll along. One has 3 times the mass of the other (everything else the same). How does its kinetic energy compare?

  1. Kinetic energy KE = ½mv².
  2. KE is proportional to the mass.
  3. Mass × 3 → KE × 3, so 3 times as much.

Answer: 3 times as much

Example 6

Two trolleys roll along. One has 4 times the speed of the other (everything else the same). How does its kinetic energy compare?

  1. Kinetic energy KE = ½mv².
  2. KE depends on the speed squared.
  3. Speed × 4 → KE × 4² = 16, so 16 times as much.

Answer: 16 times as much

Example 7

A car speeds up from 60 km/h to 80 km/h. By what factor does its kinetic energy increase? Give your answer to 2 decimal places.

  1. KE ∝ v², so the factor is (v₂ ÷ v₁)².
  2. The speeds are both in km/h, so the units cancel.
  3. (80 ÷ 60)² = 1.778

Answer: 1.78

Example 8

An object moving at 18 m/s has 1,053 J of kinetic energy. What is its mass?

  1. Speed 18 m/s, KE = 1,053 J.
  2. Rearrange KE = ½mv² to m = 2KE ÷ v².
  3. m = 2 × 1,053 ÷ 18² = 6.5 kg

Answer: 6.5 kg

Example 9

A 0.058 kg tennis ball has 8.38 J of kinetic energy. How fast is it moving? Give your answer to 1 decimal place.

  1. Mass 0.058 kg, KE = 8.38 J.
  2. Rearrange KE = ½mv² to v = √(2KE ÷ m).
  3. v = √(2 × 8.38 ÷ 0.058) = 17.00 m/s

Answer: 17.0 m/s