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Unlock This CourseRemember, anything that is moving has energy in its kinetic store. The amount of energy in that store is called its kinetic energy, or KE for short. A parked car has no kinetic energy. The same car moving along a road has a lot.
Kinetic energy depends on just two things: the mass of the object and how fast it is moving.
You need to know this relationship by heart. It is not given on the formulae page of the exam paper.
kinetic energy = ½ × mass × speed2
KE = ½ × m × v2
| Quantity | Symbol | Unit |
|---|---|---|
| Kinetic energy | KE | joule (J) |
| Mass | m | kilogram (kg) |
| Speed | v | metre per second (m/s) |
Key terms:
The little 2 belongs to the speed only. It does not mean you square the whole answer. So always work in this order:
Set out every calculation in three lines: the equation, the numbers, then the answer with its unit.
A kangaroo of mass 50 kg hops along at 12 m/s. Calculate its kinetic energy.
KE = ½ × m × v2
KE = ½ × 50 × 122
KE = ½ × 50 × 144
KE = 3600 J
Notice the extra line where 122 becomes 144. Writing it down stops you forgetting to square.
A golf ball of mass 45 g leaves the club at 60 m/s. Calculate its kinetic energy.
45 g = 45 ÷ 1000 = 0.045 kg
KE = ½ × m × v2
KE = ½ × 0.045 × 602
KE = ½ × 0.045 × 3600
KE = 81 J
On a scientific calculator, type 0.5 × 50 × 12, press the x2 key, then press =. Do not press = before x2. If you do, the calculator squares the whole 300 and gives 90 000, which is wrong.
Both matter, but not equally. Look closely at the equation. The mass appears once. The speed is squared.
Kinetic energy is directly proportional to mass. At the same speed, an object with twice the mass has twice the kinetic energy. A cargo ship loaded with containers has more kinetic energy than the same ship empty at the same speed.
Kinetic energy depends on speed squared. Twice the speed gives 22 = 4 times the kinetic energy. Three times the speed gives 32 = 9 times the kinetic energy.
You can check this with numbers. Here is a car of mass 1000 kg at three different speeds.
| Speed | Speed squared | Kinetic energy | Compared with 10 m/s |
|---|---|---|---|
| 10 m/s | 100 | ½ × 1000 × 100 = 50 000 J | 1 times |
| 20 m/s | 400 | ½ × 1000 × 400 = 200 000 J | 4 times |
| 30 m/s | 900 | ½ × 1000 × 900 = 450 000 J | 9 times |
Why does this happen? When the speed doubles from 10 to 20, the speed squared goes from 100 to 400. That is four times bigger, and nothing else in the equation has changed, so the kinetic energy is four times bigger too.
It works the other way as well. Halve the speed and the kinetic energy drops to a quarter.
To stop a car, the brakes must do work to transfer all of its kinetic energy to thermal stores. Work done is braking force × distance. So a car going twice as fast has four times the kinetic energy and, with the same braking force, needs four times the braking distance. This is why braking distance rises so quickly with speed, as you saw in Stopping Distances.
Sometimes the question gives you the kinetic energy and asks for the mass or the speed. Start both rearrangements the same way: multiply the kinetic energy by 2 to get rid of the ½.
| To find | Use |
|---|---|
| Kinetic energy | KE = ½ × m × v2 |
| Mass | m = (2 × KE) ÷ v2 |
| Speed | v2 = (2 × KE) ÷ m, then take the square root |
An ostrich running at 20 m/s has 24 000 J of kinetic energy. Calculate the mass of the ostrich.
m = (2 × KE) ÷ v2
m = (2 × 24 000) ÷ 202
m = 48 000 ÷ 400
m = 120 kg
A javelin of mass 0.8 kg is thrown with 360 J of kinetic energy. Calculate the speed of the javelin.
v2 = (2 × KE) ÷ m
v2 = (2 × 360) ÷ 0.8
v2 = 720 ÷ 0.8 = 900
v = √900
v = 30 m/s
The last step is the one people forget. 900 is the speed squared. You must press the square root key to get the speed itself.
Always check an answer by putting it back into the original equation. For the javelin: ½ × 0.8 × 302 = ½ × 0.8 × 900 = 360 J. It matches, so the answer is right.
A jet ski and its rider have a total mass of 300 kg. They travel across a lake at a speed of 8 m/s.
(a) State the equation linking kinetic energy, mass and speed. [1 mark]
(b) Calculate the kinetic energy of the jet ski and rider. [2 marks]
(c) The jet ski speeds up to 16 m/s. Without doing a full calculation, state how its kinetic energy changes and explain your answer. [2 marks]
(d) A second jet ski and rider travel at 10 m/s and have 17 500 J of kinetic energy. Calculate their total mass. [3 marks]
(e) Later, the first jet ski and rider (total mass 300 kg) have 21 600 J of kinetic energy. Calculate their speed. [3 marks]
(a) kinetic energy = ½ × mass × speed2, or KE = ½ × m × v2 (1)
(b) KE = ½ × 300 × 82 (1) = ½ × 300 × 64 = 9600 J (1)
(c) The kinetic energy becomes four times bigger (1) because the speed has doubled and kinetic energy depends on speed squared, and 22 = 4 (1).
(d) m = (2 × KE) ÷ v2 (1) = (2 × 17 500) ÷ 102 = 35 000 ÷ 100 (1) = 350 kg (1)
(e) v2 = (2 × KE) ÷ m = (2 × 21 600) ÷ 300 (1) = 144 (1), so v = √144 = 12 m/s (1)
In part (b), write the line with 82 = 64 in your working. It shows the examiner you squared the speed, and it earns the substitution mark even if you slip later. In part (e), do not stop at 144. That is v2, and the question asks for v.