Kinetic Energy — KE = ½mv²

Kinetic energy is the energy an object has because of its motion. It depends on the object's mass and the square of its speed. Doubling the speed quadruples the kinetic energy. The kinetic energy of an object can be transferred to other forms when it stops.

What is Kinetic Energy?

Kinetic energy (KE) is the energy an object has due to its motion. Any moving object has kinetic energy.

$KE = \frac{1}{2}mv^2$

Symbol	Quantity	Unit
$KE$	kinetic energy	Joules (J)
$m$	mass	kg
$v$	speed	m/s

Warning

Notice that $v$ is squared. Doubling the speed makes the KE four times greater. Tripling the speed makes it nine times greater.

Calculating Kinetic Energy

A 1,200 kg car travels at 10 m/s. Calculate its kinetic energy.

$KE = \frac{1}{2} \times 1200 \times 10^2 = \frac{1}{2} \times 1200 \times 100 = 60{,}000 \text{ J} = 60 \text{ kJ}$

Finding Speed from KE

Rearranging the formula:

$v = \sqrt{\frac{2 \times KE}{m}}$

Finding Speed

A 2 kg ball has 100 J of kinetic energy. Find its speed.

$v = \sqrt{\frac{2 \times 100}{2}} = \sqrt{100} = 10 \text{ m/s}$

Energy Transfers Involving KE

A falling object converts gravitational potential energy → kinetic energy.
A ball rolling to a stop converts kinetic energy → thermal energy (due to friction).
A compressed spring releasing converts elastic potential energy → kinetic energy.

Real world: Car crash energy

A 1,500 kg car travelling at 20 m/s (72 km/h) collides and is brought to rest.

KE to be absorbed by the crash: $KE = \tfrac{1}{2} \times 1500 \times 20^2 = 300{,}000 \text{ J} = 300 \text{ kJ}$

At 40 m/s (144 km/h) the KE would be 1,200 kJ — four times greater. This is why motorway crashes cause so much more damage than urban ones.

Real world: Cyclist on a flat road

A 75 kg cyclist plus 10 kg bike are riding at 8 m/s.

$KE = \tfrac{1}{2} \times 85 \times 8^2 = \tfrac{1}{2} \times 85 \times 64 = 2{,}720 \text{ J}$

To stop from this speed, the brakes must remove 2,720 J of kinetic energy, converting it to heat.

Real world: Football penalty kick

A 0.43 kg football is struck and leaves the boot at 28 m/s.

$KE = \tfrac{1}{2} \times 0.43 \times 28^2 = \tfrac{1}{2} \times 0.43 \times 784 \approx 169 \text{ J}$

All of that energy came from the footballer's muscles doing work on the ball during the kick.

Tip

Where do you see this in real life? Road safety speed limits, crumple zones in cars, and helmets for cyclists all exist to manage kinetic energy — the faster or heavier the object, the more energy must be safely absorbed in a collision.

Practice Problems

Calculate the kinetic energy of a 0.5 kg ball moving at 6 m/s.

Show answer

KE = ½ × 0.5 × 6² = ½ × 0.5 × 36 = 9 J

A car's speed doubles from 15 m/s to 30 m/s. How does its kinetic energy change?

Show answer

KE is proportional to v². Doubling the speed quadruples the kinetic energy (KE becomes 4 times greater).

A 4 kg object has 200 J of KE. What is its speed?

Show answer

v = √(2 × 200 / 4) = √100 = 10 m/s