Formula of Kinetic Energy: KE = ½mv², Derivation, Examples

The formula of kinetic energy is $KE = \frac{1}{2}mv^2$ : half the mass multiplied by the square of the speed. Any moving object carries this energy, from a rolling marble to a satellite in orbit, and it is measured in joules (J).

Because speed is squared, it matters far more than mass. Double the mass and the energy doubles. Double the speed and the energy is multiplied by four. That single idea explains why braking distance grows so quickly at higher speeds and why it is one of the first results in the energy and work formulas.

Quick facts

Formula
$KE = \frac{1}{2}mv^2$
SI unit
joule (J), equal to kg m²/s²
Dimensional formula
[M¹ L² T⁻²]
Scalar or vector
Scalar, never negative
With momentum
$KE = \frac{p^2}{2m}$
Rotating body
$KE_{rot} = \frac{1}{2}I\omega^2$
Courses
Class 9, Class 11, AP Physics 1, NEET, JEE

Kinetic energy formula physics: each symbol

Symbols in KE = ½mv²
Symbol	Meaning	SI unit
KE (or K)	energy of motion	joule, J
m	mass of the object, in kilograms	kg
v	speed relative to the chosen frame	m/s

The factor one half is part of the formula, not a rounding choice. Leave it out and every answer is twice too large. The speed must be in metres per second and the mass in kilograms, or the answer will not come out in joules. Kinetic energy depends on the frame of reference: a passenger sitting in a moving train has zero kinetic energy relative to the train but plenty relative to the platform.

Derive the formula of kinetic energy

Boards often ask for this in two or three marks. Start with an object of mass m that is not moving. A constant net force F acts on it over a displacement s and speeds it up to v.

Work done by the force: $W = Fs$ .
Newton's second law: $F = ma$ .
Third equation of motion with $u = 0$ : $v^2 = 2as$ , so $s = \frac{v^2}{2a}$ .
Substitute: $W = ma \cdot \frac{v^2}{2a} = \frac{1}{2}mv^2$ .

All of that work is stored as energy of motion, so $KE = \frac{1}{2}mv^2$ . The calculus version reaches the same result by integrating $F\,dx = mv\,dv$ from 0 to v, which also shows that the result holds when the force is not constant.

Worked examples using KE = ½mv²

Example 1: a cricket ball

A cricket ball of mass 0.16 kg is bowled at 25 m/s. Find its kinetic energy.

Given: m = 0.16 kg, v = 25 m/s.

$KE = \frac{1}{2}(0.16)(25)^2 = 0.08 \times 625 = 50\ \text{J}$

Answer: the ball carries 50 J.

Example 2: a car in km/h (unit conversion)

A 1200 kg car travels at 72 km/h. What is its kinetic energy, and what does it become if the speed rises to 108 km/h?

Convert: 72 km/h ÷ 3.6 = 20 m/s and 108 km/h ÷ 3.6 = 30 m/s.

$KE_1 = \frac{1}{2}(1200)(20)^2 = 240\,000\ \text{J} = 240\ \text{kJ}$

$KE_2 = \frac{1}{2}(1200)(30)^2 = 540\,000\ \text{J} = 540\ \text{kJ}$

The speed went up by a factor of 1.5, so the energy went up by 1.5² = 2.25. Check: 240 × 2.25 = 540 kJ.

Always divide km/h by 3.6 before you square. Squaring 72 instead of 20 gives an answer almost 13 times too big.

Rearranging for mass or speed

The calculator on this page uses these forms. Pick the unknown, then rearrange before you put numbers in.

$m = \frac{2\,KE}{v^2} \qquad v = \sqrt{\frac{2\,KE}{m}}$

Example: a 0.50 kg ball has 16 J of kinetic energy. Its speed is $v = \sqrt{2(16)/0.50} = \sqrt{64} = 8.0\ \text{m/s}$ . Since $p = mv$ , you can also write $KE = \frac{p^2}{2m}$ , which is handy in collision problems and in de Broglie wavelength questions. Give two bodies equal momentum and the one with less mass carries more energy, in proportion to 1/m.

Units, dimensions and graphs

Put the SI units into the formula: kg × (m/s)² = kg m²/s², which is the joule. The dimensional formula follows the same way: [M] × [L T⁻¹]² = [M¹ L² T⁻²], which every form of energy shares. A smaller unit you will meet in modern physics is the electronvolt, where 1 eV = 1.602 × 10⁻¹⁹ J.

On a graph of KE against v for a fixed mass, you get a parabola through the origin. Plot KE against v² and the line is straight with slope m/2. At a fixed speed, KE rises in direct proportion to m, with gradient v²/2.

Common mistakes and how kinetic energy links to other formulas

Using velocity with a sign. KE uses v², so the direction never makes it negative. A negative KE in your working means an arithmetic slip.
Treating energies like vectors. Kinetic energies add as plain numbers with no directions.
Assuming kinetic energy is conserved in every collision. Only elastic collisions keep it; the coefficient of restitution tells you how much is lost.
Grams left in grams. Convert to kilograms first.

Change in kinetic energy equals the net work done, which is the work energy theorem. When a body falls, it trades potential energy for kinetic energy, and the total stays fixed if there is no friction, as described on the conservation of energy page. Compare also with momentum, which is a vector.

Frequently asked questions

What is the formula of kinetic energy?

Kinetic energy equals one half of mass times speed squared, KE = ½mv². With m in kilograms and v in metres per second, the answer is in joules.

What is the SI unit of kinetic energy?

The joule (J). One joule is the kinetic energy of a 2 kg mass moving at 1 m/s, since ½ × 2 × 1² = 1.

Can kinetic energy be negative?

No. Mass is positive and the speed is squared, so kinetic energy is zero for a body at rest and positive otherwise. Its change, ΔKE, can be negative when a body slows down.

How does kinetic energy change if speed is doubled?

It becomes four times as large, because the speed is squared. If the speed is halved, the energy falls to a quarter of its old value.

What is the difference between kinetic energy and momentum?

Momentum, p = mv, is a vector measured in kg m/s. Kinetic energy, ½mv², is a scalar measured in joules. They are linked by KE = p²/(2m).