It sounds like this“A car going twice as fast has twice the kinetic energy.”
Thinking twice the speed means twice the energy
Why it feels right
Speed is a plain, straight-line thing. Twice as fast covers twice the distance in the same time, and the speedometer moves evenly. It seems natural for a quantity that comes from speed, such as energy, to double when speed doubles.
Why it’s wrong
Kinetic energy is half the mass times the speed squared. Doubling the speed quadruples the energy, and tripling it multiplies the energy by nine. A 1,000 kg car has 34.7 kJ at 30 km/h and 139 kJ at 60 km/h.
A worked example
Think about what it takes to get up to speed. Push a car with a constant force F. It accelerates steadily, so after time t its speed is v = a t and the distance it has covered is half a t squared. Since t = v / a, the distance is v squared divided by 2a.
That is the key: the distance needed to reach a speed grows with the square of the speed. To reach twice the speed takes four times the distance, because you need twice the time and, on average, twice the speed during it.
Energy is force times distance, so four times the distance at the same force means four times the energy. With F = m a, the work is m a x v squared over 2a, which simplifies to half m v squared. For a 1,000 kg car, 30 km/h is 8.33 m/s and its energy is half x 1,000 x 69.4 = 34.7 kJ. At 60 km/h, 16.67 m/s, it is 139 kJ: four times as much.
How to catch yourself
- You say a car at 60 has 'twice' the energy of one at 30.
- You treat energy as a quantity that scales directly with speed.
- You are surprised that small speed increases cost so much more energy.
What to hold instead
Reaching a speed takes a distance that grows with the speed squared, and energy is force times distance, so kinetic energy is half the mass times the speed squared. Doubling speed quadruples the energy.
Where you’ll meet it
- Comparing the fuel needed to drive at 60 and at 120 km/h.
- Understanding why high-speed crashes are so much worse.
- Reading about the energy in a fast bullet.
Questions people ask
- Why is kinetic energy proportional to speed squared?
- Because a constant force needs a distance proportional to speed squared to reach a speed, and energy is force times distance. Putting that together gives half m v squared.
- How much more energy does a car have at 100 km/h than at 50 km/h?
- Four times as much, since doubling speed quadruples kinetic energy. Going from 50 to 100 km/h costs three times the energy that getting to 50 did.