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Kinetic energy

Energy and power · level 5, lesson 3 of 6

A question to start

Perseverance weighs a tonne but creeps along at 4 cm/s. A pebble from space hits Mars at 10 km/s. Which carries more energy?

The idea

The kinetic energy of a mass \(m\) moving at speed \(v\) is

\[E_k = \tfrac12 m v^2.\]

Where does it come from? Push a mass \(m\) from rest with a steady force \(F\) over a distance \(d\). The work done is \(Fd\). The acceleration is \(a = F/m\), and from the motion equations \(v^2 = 2ad\). So

\[W = Fd = (ma)\left(\frac{v^2}{2a}\right) = \tfrac12 mv^2.\]

The work done on it becomes its kinetic energy. Because of the \(v^2\), speed matters far more than mass: double the speed, four times the energy.

Moving thing Mass Speed Kinetic energy
Perseverance at top speed 1,025 kg 0.042 m/s 0.9 J
A person walking 70 kg 1.4 m/s 69 J
A car on a motorway 1,500 kg 30 m/s 675 kJ
A 1 kg rock hitting Mars from space 1 kg 10 km/s 50 MJ
v2vE4Espeed

Kinetic energy does not depend on gravity: a 10 kg cart at 2 m/s has 20 J on Mars or on Earth. A collision on Mars at the same speed is just as hard as on Earth.

Worked examples

1. A walker. \(\tfrac12 \times 70 \times 1.4^2 = 68.6\) J.

2. A meteorite. A 1 kg rock at 10,000 m/s: \(\tfrac12 \times 1 \times 10^8 = 5\times10^7\) J \(= 50\) MJ, about the energy in a litre and a half of petrol, released in an instant.

3. Finding a speed. A 0.5 kg ball has 9 J of kinetic energy. \(v = \sqrt{2E/m} = \sqrt{36} = 6\) m/s.

Try it

  1. Find the kinetic energy of a 2 kg box sliding at 3 m/s.
  2. A buggy of 800 kg drives at 5 m/s on Mars. Find its kinetic energy.
  3. The buggy speeds up to 10 m/s. By what factor does its kinetic energy grow?
  4. What speed gives a 60 kg runner 3,000 J of kinetic energy?
Answers
  1. \(\tfrac12 \times 2 \times 9 = 9\) J.
  2. \(\tfrac12 \times 800 \times 25 = 10{,}000\) J.
  3. Four times: 40,000 J.
  4. \(v = \sqrt{2 \times 3000/60} = \sqrt{100} = 10\) m/s.

Common mistake

Forgetting to square the speed. \(\tfrac12 mv^2\), not \(\tfrac12 mv\). Check units: kg × (m/s)² = kg m²/s² = J.

Where it leads