Roll a bowling ball down the lane and it flattens the pins. Let the wind push a turbine and it lights a town. Miss the brakes and a car crumples a lamp-post like paper. In every case something moving does the damage — the pins, the turbine and the lamp-post all felt the same thing: the energy carried by a moving object.

That energy of motion is called kinetic energy (from the Greek kinesis, "movement"). Anything that is moving has it; the instant something stops, its kinetic energy is gone (it has been passed on to whatever it hit, or drained away as heat and sound). The faster a thing goes, and the more mass it has, the more kinetic energy it carries — and, as we're about to see, speed matters far more than you'd guess.

The formula

Kinetic energy depends on just two things: how much mass the object has (m, in kilograms) and how fast it is going (v, its speed in metres per second). They combine like this:

The answer comes out in joules (J) — the same unit we use for every kind of energy, so kinetic energy can be swapped for other stores and for the work done by a force. Reading the formula piece by piece:

• m — the mass in kg. Double the mass and you double the energy.
• v — the speed in m/s. But v is squared, so its effect is far stronger (next card).
• \tfrac12 — a fixed one-half that always sits out front.

The squared speed changes everything

Here is the single most important fact on this page. Because it is v^2 and not just v, doubling the speed does not double the kinetic energy — it quadruples it. Squaring means the speed gets multiplied by itself, so twice the speed gives 2^2 = 4 times the energy.

Take one object and leave its mass alone, so only the speed changes. Watch what its kinetic energy does:

• Speed \times 1  →  energy \times 1
• Speed \times 2  →  energy \times 4  (2^2)
• Speed \times 3  →  energy \times 9  (3^2)
• Speed \times 4  →  energy \times 16  (4^2)

This one line of maths is why speed limits exist. A car at 40 mph carries four times the wrecking energy it had at 20 mph — not twice — which is why a small rise in speed makes a crash so much more violent, and why the braking distance grows so alarmingly (the brakes have four times as much energy to soak up).

Worked examples

1. A family car. Mass m = 1000\ \text{kg}, speed v = 20\ \text{m/s} (about 45 mph).

2. The same car, twice as fast. Now v = 40\ \text{m/s}. Nothing else changed.

Doubling the speed from 20 to 40 m/s took the energy from 200 kJ to 800 kJ — exactly four times as much, just as the squaring promised.

3. A kicked football. Mass m = 0.4\ \text{kg}, speed v = 25\ \text{m/s}.

4. Rearranging to find the speed. A 2\ \text{kg} ball carries E_k = 400\ \text{J}. How fast is it going? Start from E_k = \tfrac12 m v^2 and unwrap v: multiply both sides by 2, divide by m, then square-root.

Mass matters too: a lorry versus a car

Speed is squared, but mass counts as well — straight, not squared. Put a heavy lorry and a light car side by side at the same speed and the lorry's extra mass alone gives it a huge kinetic-energy advantage.

A loaded lorry (m = 20\,000\ \text{kg}) at v = 20\ \text{m/s}:

That is 20 times the 200 kJ of our 1000 kg car at the same speed — because it has 20 times the mass. A slow-moving lorry is anything but harmless. And a fast marble is not nothing either: a tiny mass at a huge speed (a bullet, a hailstone, space debris) can carry plenty of energy, because that squared speed does the heavy lifting. To have a lot of kinetic energy you need either a big mass or a big speed — and a big speed is worth much more.

Try it: watch the curve steepen

Below is a graph of kinetic energy against speed for a car. Drag the speed slider and the marker climbs the curve; the readout works out E_k = \tfrac12 m v^2 live. Notice the shape — it is not a straight line but a curve that steepens, because of the v^2. Go from 10 to 20 m/s and the energy roughly quadruples; the same jump higher up adds far more still. Then drag the mass slider to make the whole curve steeper: more mass lifts the energy at every speed.