Science

What is kinetic energy, and what is the formula?

Quick answer

Kinetic energy is the energy an object has because it is moving. The formula is KE = ½mv², with mass in kg and speed in m/s, and the answer is in joules (J).

Ages 13–16Secondary

In one sentence

Kinetic energy is the energy of movement: anything that moves has it, and the faster and heavier it is, the more it has.

Explained

A rolling football, a flying bird, a car on the road and even the tiny particles in the air around you all have kinetic energy (KE), because they are moving. A parked car has none. The word comes from the Greek kinesis, "movement". You can see kinetic energy at work whenever a moving thing does something: a bowling ball knocks down pins, the wind turns a turbine, a hammer drives in a nail.

How much kinetic energy something has depends on two things: its mass and its speed. The formula is:

KE = ½ × m × v² (also written E_k = ½mv²)

m is the mass in kilograms (kg)
v is the speed in metres per second (m/s)
KE is the energy in joules (J)

A moving car, a kicked ball and a cyclist all have kinetic energy.

The speed is squared, and that changes everything. Double the mass and the kinetic energy doubles. But double the speed and the kinetic energy becomes four times as big; triple the speed and it becomes nine times as big. A car at 30 m/s (about 67 mph or 108 km/h) has nine times the kinetic energy of the same car at 10 m/s. That is why speed matters so much in a crash, and why braking distances grow so quickly with speed: the brakes have to get rid of all that energy.

Where does the ½ come from? Push an object of mass m, starting at rest, with a steady force F over a distance d. The work you do is F × d, and that work becomes kinetic energy. From F = ma and the motion equation v² = 2ad, the distance is d = v² ÷ 2a, so the work is ma × v² ÷ 2a = ½mv². The ½ appears because the object starts slowly: its average speed while you push is only half its final speed.

Kinetic energy is a scalar: it has a size but no direction, and it is never negative. That makes it different from momentum (p = mv), which has a direction.

Energy is never created or destroyed, only transferred (see energy). Kinetic energy often swaps with gravitational potential energy (GPE = mgh). A ball dropped from a height loses GPE and gains the same amount of KE, if air resistance is small. On a roller coaster or a swing, energy flows back and forth between the two all the time. Gravity is what pulls the ball down and speeds it up.

Kinetic energy can also be turned into electricity: wind turbines use the kinetic energy of moving air, hydroelectric dams use falling water, a bike dynamo uses the turning wheel, and electric and hybrid cars use regenerative braking (in motor racing called kinetic energy recovery) to charge their batteries as they slow down.

Examples

Object	Mass	Speed	Kinetic energy
Person walking	60 kg	1.5 m/s	67.5 J
Football after a kick	0.45 kg	20 m/s	90 J
Cyclist with bike	80 kg	6 m/s	1,440 J
Car in town	1,200 kg	15 m/s	135,000 J (135 kJ)
Same car on a motorway	1,200 kg	30 m/s	540,000 J (540 kJ)

Worked example: the car on the motorway goes twice as fast as in town, so its kinetic energy is 2² = 4 times as big: 4 × 135 kJ = 540 kJ. Check with the formula: ½ × 1,200 × 30² = 600 × 900 = 540,000 J.

Common mistakes

Forgetting to square the speed. For 4 kg at 3 m/s: ½ × 4 × 3² = ½ × 4 × 9 = 18 J, not ½ × 4 × 3 = 6 J. Square v first, before anything else.
Squaring the wrong thing. Only v is squared, not m and not the ½. (½ × 4 × 3)² = 36 is wrong.
Forgetting the ½. m × v² gives double the right answer.
Using grams or km/h. Convert first: 50 g = 0.05 kg, and km/h ÷ 3.6 = m/s (36 km/h = 10 m/s).
Thinking double speed means double energy. It means four times the energy.

Step by step

Write down m and vCheck the units: mass in kg, speed in m/s. Convert grams (÷ 1,000) and km/h (÷ 3.6) first.
Square the speedWork out v × v.
Multiply by the massM × v².
Halve itDivide by 2 (or multiply by ½).
Give the unitThe answer is in joules (J). For big numbers, 1 kJ = 1,000 J.

To rearrange the formula: m = 2 × KE ÷ v² and v = √(2 × KE ÷ m). To find the speed, multiply the energy by 2, divide by the mass, then take the square root.

Practice questions

Kinetic energy calculations

Try each one, then check your answer.

Find the kinetic energy of a 4 kg object moving at 5 m/s.
Show answer
½ × 4 × 5² = ½ × 4 × 25 = 50 J
A 60 kg runner moves at 5 m/s. What is her kinetic energy?
Show answer
½ × 60 × 25 = 750 J
Find the kinetic energy of a 0.5 kg ball moving at 10 m/s.
Show answer
½ × 0.5 × 100 = 25 J
Find the kinetic energy of a 1,000 kg car moving at 20 m/s.
Show answer
½ × 1,000 × 400 = 200,000 J (200 kJ)
A 50 g ball moves at 20 m/s. Find its kinetic energy.
Show answer
50 g = 0.05 kg; ½ × 0.05 × 400 = 10 J
A 1,200 kg car moves at 36 km/h. Find its kinetic energy.
Show answer
36 ÷ 3.6 = 10 m/s; ½ × 1,200 × 100 = 60,000 J (60 kJ)
An object moving at 2 m/s has 100 J of kinetic energy. What is its mass?
Show answer
m = 2 × 100 ÷ 2² = 200 ÷ 4 = 50 kg
An 8 kg object has 400 J of kinetic energy. How fast is it moving?
Show answer
v = √(2 × 400 ÷ 8) = √100 = 10 m/s
A 0.2 kg ball has 40 J of kinetic energy. Find its speed.
Show answer
v = √(2 × 40 ÷ 0.2) = √400 = 20 m/s
An object has 90 J of kinetic energy. What will it have if its speed triples?
Show answer
3² = 9, so 9 × 90 = 810 J
A car has 400 kJ of kinetic energy at 20 m/s. What does it have at 10 m/s?
Show answer
Half the speed gives ¼ of the energy: 400 ÷ 4 = 100 kJ
Which has more kinetic energy: a 0.01 kg bullet at 400 m/s or a 70 kg person running at 5 m/s?
Show answer
Bullet: ½ × 0.01 × 160,000 = 800 J. Person: ½ × 70 × 25 = 875 J. The person, just.
A 4 kg object has a momentum of 20 kg m/s. Find its kinetic energy without working out the speed first.
Show answer
KE = p² ÷ 2m = 400 ÷ 8 = 50 J
A 0.5 kg ball is dropped from 5 m. Ignoring air resistance (g = 10 N/kg), how fast is it going just before it lands?
Show answer
GPE = 0.5 × 10 × 5 = 25 J = KE; v = √(2 × 25 ÷ 0.5) = √100 = 10 m/s
A 1,500 kg car brakes from 20 m/s to 10 m/s. How much kinetic energy do the brakes transfer?
Show answer
½ × 1,500 × (20² − 10²) = 750 × 300 = 225,000 J (225 kJ), mostly as heat in the brakes

Word problems

Kinetic energy in real life

Try each one, then check your answer.

A cyclist and her bike have a total mass of 75 kg and move at 8 m/s. Find their kinetic energy.
Show answer
½ × 75 × 64 = 2,400 J
A 0.058 kg tennis ball is served at 50 m/s. How much kinetic energy does it have?
Show answer
½ × 0.058 × 2,500 = 72.5 J
A 50 kg skateboarder has 900 J of kinetic energy. How fast is she going?
Show answer
v = √(2 × 900 ÷ 50) = √36 = 6 m/s
How many times more kinetic energy does a 1,000 kg car have at 30 m/s than at 10 m/s?
Show answer
At 10 m/s: 50,000 J. At 30 m/s: 450,000 J. 9 times as much (3² = 9)
A 500 kg roller coaster car starts at rest and drops 20 m. Ignoring friction (g = 10 N/kg), how fast is it at the bottom?
Show answer
GPE = 500 × 10 × 20 = 100,000 J = KE; v = √(2 × 100,000 ÷ 500) = √400 = 20 m/s
About 1.2 kg of air (roughly one cubic metre) blows past a wind turbine at 10 m/s. How much kinetic energy does it carry?
Show answer
½ × 1.2 × 100 = 60 J

Questions people ask

What is the formula for kinetic energy?

KE = ½mv²: half the mass times the speed squared. With mass in kilograms and speed in metres per second, the answer is in joules. In UK exams it is written E_k = ½mv².

What is kinetic energy measured in?

In joules (J), like every kind of energy. One joule is 1 kg m²/s², which is exactly what you get from kg × (m/s)². Large amounts are given in kilojoules (kJ).

What is the difference between kinetic and potential energy?

Kinetic energy is energy because something is moving. Potential energy is stored energy because of position or shape, such as a book on a high shelf (gravitational) or a stretched spring (elastic). A falling object turns potential energy into kinetic energy.

Why is there a ½ in the kinetic energy formula?

It comes from the work needed to speed an object up from rest. While it speeds up steadily, its average speed is only half its final speed, so the force acts over a shorter distance than if it had been at full speed all along. The maths gives exactly ½mv².

How do you find kinetic energy without the velocity?

Use another quantity you know. From momentum p: KE = p² ÷ 2m. For something falling from height h with no air resistance: KE gained = mgh. Or from the work done on it: KE gained = force × distance.

How is kinetic energy turned into electricity?

A moving thing turns a generator: the wind turns turbine blades, falling water turns a hydroelectric turbine, a turning wheel spins a dynamo. Inside the generator a magnet and a coil of wire move past each other, which makes a current flow.

What is kinetic energy in chemistry?

The particles of every substance are always moving, so they have kinetic energy. Temperature is a measure of the average kinetic energy of the particles: hotter means faster particles. Reactions also happen faster when particles collide with more energy.

Does kinetic energy have a direction?

No. Kinetic energy is a scalar and is never negative, because v² is always positive. A ball moving left at 3 m/s has exactly the same kinetic energy as one moving right at 3 m/s.

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