The kinetic energy equation

The kinetic energy equation is Ek = ½mv²: kinetic energy (J) = 0.5 × mass (kg) × speed² (m/s)². It is the equation that links kinetic energy, mass and speed. Because the speed is squared, doubling the speed of an object gives it four times the kinetic energy, while doubling its mass only doubles it. A 1000 kg car at 20 m/s has ½ × 1000 × 20² = 200 000 J of kinetic energy.

The equation for kinetic energy

Only v is squared. The ½ and the mass are not, so work out v² first, then multiply by m, then halve. Kinetic energy is never negative: it does not matter which way the object is moving, only how fast.

Rearranging the kinetic energy equation

You can rearrange Ek = ½mv² to find the mass or the speed.

To get the speed, start from Ek = ½mv². Multiply both sides by 2 to get 2Ek = mv². Divide by m to get v² = 2Ek ÷ m. Then take the square root:

Is there a kinetic energy equation triangle?

Formula triangles work for equations with three letters multiplied together, like V = IR. The kinetic energy equation has a ½ and a square in it, so a triangle does not work cleanly: you would still have to deal with the 2 and the square root yourself. It is quicker and safer to rearrange, as above.

The Kinetic Energy Calculator does any of these rearrangements for you, converts units such as grams and km/h, and shows each step.

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Kinetic Energy Calculator

Find kinetic energy, mass or speed with full working

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Why speed matters more than mass

The graph shows the kinetic energy of a 1000 kg car at different speeds. Here x is the speed in m/s and y is the kinetic energy in kJ.

The curve gets steeper as the speed rises. From 10 m/s to 20 m/s the kinetic energy goes up 4 times; from 10 m/s to 30 m/s it goes up 9 times. Kinetic energy is proportional to the square of the speed.

This is why speed matters so much on roads. The brakes have to take away all of a car's kinetic energy to stop it, so a car going twice as fast needs about four times the braking distance, if the braking force is the same.

Where the kinetic energy equation comes from

Kinetic energy is the work done to get an object up to its speed. Imagine a resultant force F pushing an object of mass m from rest, over a distance s, until it reaches speed v.

• The work done is W = Fs.
• By Newton's second law, F = ma.
• From the equation of motion v² = u² + 2as with u = 0, the distance is s = v² ÷ 2a.

Put them together:

The a cancels, which is why the kinetic energy depends only on the mass and the final speed, not on how quickly the object got there. The same steps with a starting speed u give the change in kinetic energy, ½mv² − ½mu².

Here is a quick check before the examples.

Kinetic energy equation worked examples

Example 1: finding the kinetic energy

A cyclist and bike have a total mass of 85 kg and travel at 6 m/s. Find their kinetic energy.

Example 2: converting units first

A 650 g football is kicked at 72 km/h. Find its kinetic energy.

Convert the mass: 650 g = 0.65 kg. Convert the speed: 72 ÷ 3.6 = 20 m/s.

Example 3: finding the speed

A 0.16 kg cricket ball has 72 J of kinetic energy. How fast is it moving?

Example 4: finding the mass

An object moving at 8 m/s has 1600 J of kinetic energy. Find its mass.

Example 5: kinetic energy into height

A 0.2 kg ball is thrown straight up at 14 m/s. Ignoring air resistance, how high does it go? Use g = 9.8 N/kg.

Its kinetic energy at the start is ½ × 0.2 × 14² = 19.6 J. At the top, all of this has been transferred to the gravitational potential energy store, so mgh = 19.6:

Example 6: braking distance

A 900 kg car travels at 20 m/s. Its brakes provide a constant force of 7500 N. How far does it travel while braking? What if it were going at 40 m/s?

The brakes do work equal to the kinetic energy they remove: work done = force × distance.

At 40 m/s the kinetic energy is ½ × 900 × 40² = 720 000 J, four times as much, so the braking distance is 720 000 ÷ 7500 = 96 m, four times as far.

Example 7: from momentum (A-level)

A 0.5 kg trolley has a momentum of 3 kg m/s. Find its kinetic energy.

Since p = mv, v = p ÷ m. Substituting into Ek = ½mv² gives Ek = p² ÷ 2m:

Kinetic energy and other energy stores

Kinetic energy is one of the energy stores you meet at GCSE. Energy is transferred into or out of the kinetic store all the time:

• a falling object gains kinetic energy as it loses gravitational potential energy, mgh;
• a stretched catapult transfers elastic potential energy, ½ke², to the kinetic store of the stone;
• friction and air resistance transfer kinetic energy to the thermal store of the surroundings.

Our guides to gravitational potential energy, elastic potential energy and energy stores cover the other stores.

The relativistic kinetic energy equation

Ek = ½mv² works perfectly for everyday speeds, but it fails as an object gets close to the speed of light, c. Einstein's special relativity replaces it. The total energy of an object of rest mass m₀ moving at speed v is:

and its kinetic energy is this total energy minus its rest energy, Ek = E − m₀c². At low speeds the two equations give the same answer. AQA's A-level data booklet gives the relativistic formula in the Turning points in physics option.

Common mistakes

• Forgetting to square the speed. ½ × 2 × 3 is not the kinetic energy of a 2 kg mass at 3 m/s; ½ × 2 × 3² = 9 J is.
• Squaring everything. Only v is squared, not ½mv as a whole.
• Mass in grams. 650 g is 0.65 kg.
• Speed in km/h. Divide by 3.6 to get m/s.
• Forgetting the square root when finding speed. v² = 900 means v = 30 m/s, not 900 m/s.
• Using the change in speed instead of the change in kinetic energy. Going from 10 m/s to 20 m/s changes the kinetic energy by ½m(20² − 10²), not ½m(10²).

Is the kinetic energy equation on the equation sheet?

Yes, on both AQA sheets. This was checked against AQA's documents in November 2026; other boards publish their own sheets, so check yours.

AQA's GCSE specification also lists it as an equation students should be able to recall and apply, so learn it anyway: questions often ask you to choose the right equation before you can use it.

Practise energy questions in a short timed paper

GCSE Physics Higher and A-level Physics: Year 1 each give you a short timed paper every day, with 8 questions worth 25 to 35 marks across the course, so energy questions turn up regularly but not in every paper.

Read next

Physics · EnergyKinetic energy calculator: KE = 1/2 mv² with stepsFind kinetic energy, mass or speed from KE = ½mv², or the change in kinetic energy, with unit conversions and worked examples you can check.