Kinetic energy formula: E = ½mv²

Kinetic energy = ½ × mass × speed². In symbols, E_k = ½ m v². E_k is the kinetic energy in joules (J), m is the mass in kilograms (kg) and v is the speed in metres per second (m/s). AQA, Edexcel and OCR all put this one on the equation sheet you get in your GCSE physics exam.

Equation

E_k = ½ m v²

Symbol	Quantity	Unit
`E_k`	kinetic energy	J
`m`	mass	kg
`v`	speed	m/s

What each specification says

Exam board	Status	Equation
AQA Physics (8463)	Recall	`E_k = ½ m v²`
AQA Combined Science (8464)	Recall	`E_k = ½ m v²`
Edexcel Physics (1PH0)	Recall	`KE = ½ m v²`
Edexcel Combined Science (1SC0)	Recall	`KE = ½ m v²`
OCR Gateway Physics A (J249)	Recall	`E = ½ m v²`
OCR Twenty First Century Physics B (J259)	Recall	`E = ½ m v²`

Calculator: E_k = ½ m v²

What kinetic energy means

Anything that moves has energy because of its motion. Physicists call this its kinetic energy store. A parked car has none. The same car on a motorway has a lot, and the brakes must turn all of it into heat to stop it.

Two things decide the size of the store: how heavy the object is, and how fast it goes. Mass counts once. Speed counts twice, because the equation squares it. So if you double the speed, the kinetic energy becomes four times bigger. That is why a crash at 60 mph does far more damage than one at 30 mph. Your momentum also depends on mass and speed, but there the speed is not squared.

How each exam board prints it

All three boards use the same maths. The letters change a little, and you should read the version your own board gives you.

AQA (Physics 8463 and Combined Science 8464): E_k = ½ m v².
Edexcel (Physics 1PH0 and Combined Science 1SC0): KE = ½ m v².
OCR Gateway (J249) and OCR Twenty First Century (J259): E = ½ m v².

Each specification puts kinetic energy on its recall list. That label comes from the specification itself. For exams in 2025, 2026 and 2027, every board prints the whole list on a sheet inside the paper, and the sheets carry on from 2028. You can see every board's list on the physics equation sheet page, or go straight to the AQA equation list.

Rearranging for mass or speed

Exam questions often give you the energy and ask for something else. These are the two rearranged forms:

m = 2E_k ÷ v²
v = √(2E_k ÷ m)

For speed, do the division first and take the square root last. Many students take the square root too early, or forget it.

Kinetic energy often turns up with gravitational potential energy. When a ball falls, energy moves from the gravity store into the kinetic store. Set m g h equal to ½ m v² and you can find the speed at the bottom of a drop.

Worked examples

E_k

Change the mass to kilograms: 60 g = 0.06 kg
E_k = ½ m v²
E_k = 0.5 × 0.06 × 50²
E_k = 0.5 × 0.06 × 2500
E_k = 75 J

75 J

v

Change the energy to joules: 240 kJ = 240 000 J
v = √(2E_k ÷ m)
v = √(2 × 240 000 ÷ 1200)
v = √400
v = 20 m/s

20 m/s

m

m = 2E_k ÷ v²
m = 2 × 2560 ÷ 8²
m = 5120 ÷ 64
m = 80 kg

80 kg

Common mistakes

Squaring the half as well as the speed. Only v is squared, so work out v² first and then multiply by ½ and by m.
Leaving the mass in grams. A 60 g ball put in as 60 gives an answer 1000 times too big.
Forgetting the square root when you solve for speed. If your car is doing 400 m/s, you have stopped at v².
Using a speed in km/h. Convert to m/s first by dividing by 3.6, or the energy comes out far too large.

Practise making m or v the subject in the Equation Rearranger game.

Questions people ask

Why is the speed squared?

It takes work to bring an object up to speed. Going faster needs a bigger force over a longer distance, and the two together make the energy grow with the square of the speed.

Can kinetic energy be negative?

No. Mass is always positive, and a squared speed is never negative, so the answer is zero or more. A negative answer means a slip in your working.

Is kinetic energy on the equation sheet?

Yes. Every board lists it as a recall equation in its specification, and for exams from 2025 onwards the sheet in the paper prints it too.