Kinetic Energy

Anything that is moving has energy because of its motion — this is its kinetic energy. A sprinter, a thrown cricket ball, a speeding car and a falling raindrop all carry kinetic energy, and the faster and heavier they are, the more they have. Kinetic energy explains why a fast car needs a far greater distance to stop than a slow one, why a heavy lorry is so much more dangerous in a crash than a bicycle, and why even a small object moving very fast (like a bullet) can do enormous damage. This lesson, part of Topic P7 (Energy) of OCR Gateway Science A, defines kinetic energy, introduces and rearranges the equation $E_k = \dfrac{1}{2}mv^2$ , and explores the all-important effect of doubling the speed.

By the end of this lesson you should be able to define kinetic energy, use and rearrange $E_k = \dfrac{1}{2}mv^2$ , carry out calculations including finding $v$ or $m$ , and explain why doubling the speed of an object quadruples its kinetic energy.

What Is Kinetic Energy?

Kinetic energy is the energy an object has because it is moving. Any moving object — whatever its direction — has energy stored in its kinetic store. The amount of kinetic energy depends on two things:

the mass of the object — a heavier object moving at the same speed has more kinetic energy;
the speed of the object — a faster object has more kinetic energy, and (as we will see) speed has a much bigger effect than mass.

When an object speeds up, energy is transferred into its kinetic store; when it slows down, energy is transferred out of its kinetic store (often to the thermal store, by friction, when braking). Kinetic energy, like all energy, is measured in joules (J).

Exam Tip: Kinetic energy depends on mass and speed. Because the speed is squared in the equation, speed matters far more than mass — a small increase in speed makes a big increase in kinetic energy.

The Kinetic Energy Equation

The kinetic energy of a moving object is given by:

$E_k = \dfrac{1}{2}mv^2$

where $E_k$ is the kinetic energy (in joules, $\text{J}$ ), $m$ is the mass (in kilograms, $\text{kg}$ ) and $v$ is the speed (in metres per second, $\text{m/s}$ ).

Two things in this equation catch people out, so note them carefully:

the $\dfrac{1}{2}$ at the front — you must halve the result; forgetting it gives an answer twice too big;
the $v^2$ — you square the speed only, not the mass. Work out $v^2$ first, then multiply by the mass, then halve.

Worked Example 1 — Kinetic energy of a car

Calculate the kinetic energy of a $1200 \text{ kg}$ car travelling at $20 \text{ m/s}$ .

Step 1 — write the equation: $E_k = \dfrac{1}{2}mv^2$ .

Step 2 — square the speed first: $v^2 = 20^2 = 400$ .

Step 3 — substitute: $E_k = \dfrac{1}{2} \times 1200 \times 400$ .

Step 4 — calculate: $E_k = 0.5 \times 1200 \times 400 = 240\,000 \text{ J}$ .

Answer: the car has $240\,000 \text{ J}$ (= $240 \text{ kJ}$ ) of kinetic energy.

Worked Example 2 — Kinetic energy of a runner

A runner of mass $60 \text{ kg}$ sprints at $8 \text{ m/s}$ . Calculate their kinetic energy.

Step 1 — write the equation: $E_k = \dfrac{1}{2}mv^2$ .

Step 2 — square the speed: $v^2 = 8^2 = 64$ .

Step 3 — substitute: $E_k = \dfrac{1}{2} \times 60 \times 64$ .

Step 4 — calculate: $E_k = 0.5 \times 60 \times 64 = 1920 \text{ J}$ .

Answer: the runner has $1920 \text{ J}$ of kinetic energy.

Exam Tip: Always square the speed first, then multiply by the mass, then halve. A common error is to halve the mass and forget to square the speed, or to square the whole of $mv$ — neither is correct. Lay the working out step by step.

Rearranging the Kinetic Energy Equation

Sometimes you are given the kinetic energy and asked to find the speed or the mass. Rearranging takes a little care because of the square.

To find the speed $v$ :

$v = \sqrt{\dfrac{2E_k}{m}}$

(Multiply $E_k$ by 2, divide by the mass, then take the square root.)

To find the mass $m$ :

$m = \dfrac{2E_k}{v^2}$

(Multiply $E_k$ by 2, then divide by the speed squared.)

Worked Example 3 — Finding the speed

A ball of mass $0.5 \text{ kg}$ has $25 \text{ J}$ of kinetic energy. Calculate its speed.

Step 1 — rearrange for speed: $v = \sqrt{\dfrac{2E_k}{m}}$ .

Step 2 — substitute: $v = \sqrt{\dfrac{2 \times 25}{0.5}} = \sqrt{\dfrac{50}{0.5}}$ .

Step 3 — simplify inside the root: $\dfrac{50}{0.5} = 100$ .

Step 4 — take the square root: $v = \sqrt{100} = 10 \text{ m/s}$ .

Answer: the ball is moving at $10 \text{ m/s}$ .

Worked Example 4 — Finding the mass

A moving object has $400 \text{ J}$ of kinetic energy and a speed of $4 \text{ m/s}$ . Calculate its mass.

Step 1 — rearrange for mass: $m = \dfrac{2E_k}{v^2}$ .

Step 2 — square the speed: $v^2 = 4^2 = 16$ .

Step 3 — substitute: $m = \dfrac{2 \times 400}{16} = \dfrac{800}{16}$ .

Step 4 — calculate: $m = 50 \text{ kg}$ .

Answer: the object has a mass of $50 \text{ kg}$ .

Exam Tip: To find a speed from kinetic energy, don't forget the final square root — it is the step most often missed. A quick check: substitute your answer back into $E_k = \dfrac{1}{2}mv^2$ and see if you get the original energy.

The Effect of Doubling the Speed

Because the speed is squared in the equation, changing the speed has a dramatic effect on the kinetic energy. This is one of the most important ideas in the whole topic.

If you double the speed (multiply $v$ by 2), the kinetic energy is multiplied by $2^2 = 4$ — it becomes four times as large. If you treble the speed (×3), the kinetic energy becomes $3^2 = 9$ times as large. In general, multiplying the speed by a factor multiplies the kinetic energy by the square of that factor.

The table below shows the kinetic energy of a $1000 \text{ kg}$ car at different speeds. Watch how the kinetic energy grows far faster than the speed.

Speed / $\text{m/s}$	$v^2$	$E_k = \dfrac{1}{2}mv^2$ / $\text{J}$
10	100	50 000
20	400	200 000
30	900	450 000
40	1600	800 000

Kinetic Energy

Kinetic Energy

What Is Kinetic Energy?

The Kinetic Energy Equation

Worked Example 1 — Kinetic energy of a car

Worked Example 2 — Kinetic energy of a runner

Rearranging the Kinetic Energy Equation

Worked Example 3 — Finding the speed

Worked Example 4 — Finding the mass

The Effect of Doubling the Speed

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