4.1.1.2a - Kinetic Energy

4.1.1.2a - Kinetic Energy

Kinetic energy is the energy associated with a moving object. This lesson is about one equation: how the mass and speed of an object determine its kinetic energy. For this specification point, the kinetic-energy equation is recall and apply, so you need to know it and use it accurately.

The kinetic energy store

Any object that is moving has kinetic energy. A moving football, cyclist, train, skateboarder or thrown ball all have kinetic energy because they are moving.

Kinetic energy

Kinetic energy is the energy associated with a moving object.

If the object is not moving, its speed is zero, so its kinetic energy is zero. If the same object moves faster, it has more kinetic energy. If two objects move at the same speed, the object with the greater mass has more kinetic energy.

In this lesson, use speed as a positive value in metres per second. Direction is not part of the kinetic-energy calculation at GCSE: the equation uses speed, not displacement or direction.

The equation and units

The kinetic-energy equation is:

Kinetic energy

Ek=12mv2E_k = \frac{1}{2}mv^2

In words:

kinetic energy = 0.5 x mass x speed^2

The symbols and units are:

QuantitySymbolUnit
kinetic energyE_kjoules, J
massmkilograms, kg
speedvmetres per second, m/s

The speed is squared. That means v^2 = v x v. Only the speed is squared, not the mass and not the whole answer.

Because the equation contains speed^2, changing speed has a bigger effect than changing mass by the same factor. If mass stays the same, doubling speed makes kinetic energy four times as large because (2v)^2 = 4v^2. Doubling the mass only doubles the kinetic energy.

Calculating kinetic energy

A good calculation answer shows the equation, substitution, answer and unit. Before substituting, check that mass is in kilograms and speed is in metres per second.

Worked example: a ball has mass 250 g and moves at 12 m/s. Calculate its kinetic energy.

First convert the mass:

250 g = 0.250 kg

Then substitute into the equation:

Ek=12mv2E_k = \frac{1}{2}mv^2 Ek=0.5×0.250×122E_k = 0.5 \times 0.250 \times 12^2 Ek=0.5×0.250×144E_k = 0.5 \times 0.250 \times 144 Ek=18 JE_k = 18\text{ J}

The answer is 18 J. It is not 18 kg or 18 m/s, because the quantity calculated is energy.

When a question gives a value in kilojoules, remember that 1 kJ = 1000 J. When a question gives mass in grams, remember that 1000 g = 1 kg.

Rearranging for mass or speed

Sometimes the question gives kinetic energy and asks for mass or speed instead. Start from the same equation, then rearrange it carefully.

To find mass:

Ek=12mv2E_k = \frac{1}{2}mv^2 2Ek=mv22E_k = mv^2 m=2Ekv2m = \frac{2E_k}{v^2}

Worked example: an object has kinetic energy 72 J and speed 6.0 m/s. Calculate its mass.

m=2Ekv2m = \frac{2E_k}{v^2} m=2×726.02m = \frac{2 \times 72}{6.0^2} m=14436m = \frac{144}{36} m=4.0 kgm = 4.0\text{ kg}

To find speed:

Ek=12mv2E_k = \frac{1}{2}mv^2 2Ek=mv22E_k = mv^2 v2=2Ekmv^2 = \frac{2E_k}{m} v=2Ekmv = \sqrt{\frac{2E_k}{m}}

The square root is needed because the original equation contains v^2. A common error is to stop at v^2 and forget that speed is v.

Worked example: a trolley has kinetic energy 180 J and mass 10 kg. Calculate its speed.

v=2Ekmv = \sqrt{\frac{2E_k}{m}} v=2×18010v = \sqrt{\frac{2 \times 180}{10}} v=36v = \sqrt{36} v=6.0 m/sv = 6.0\text{ m/s}

Checking answers

The kinetic-energy equation is simple, but it is easy to lose marks through small mistakes. Use this checklist whenever you calculate kinetic energy:

  • mass must be in kg
  • speed must be in m/s
  • speed must be squared before multiplying by mass and 0.5
  • kinetic energy must be in J
  • if finding speed, take the square root at the end
  • round to a sensible number of significant figures when needed

You can also use the equation as a quick sense-check. For a 2.0 kg object:

Speed in m/sv^2E_k = 0.5 x 2.0 x v^2
1.01.01.0 J
2.04.04.0 J
3.09.09.0 J

The pattern is not 1, 2, 3 J because kinetic energy depends on speed squared.

One final exam habit: if your answer has the wrong unit, it is probably the wrong quantity. Kinetic energy is an energy, so the unit is joules.