Gravitational Potential Energy

Gravitational Potential Energy is the energy an object has as a result of its height above the surface of the Earth. In practice, objects change their relative position with respect to the surface all the time, as such, it is often more meaningful to look at the change in Gravitational Potential Energy as the object’s height changes

_Change in Gravitational Potential Energy = mass x gravity x change in height _

Δ_GPE = mg_Δ_h_

[note the symbol ‘Δ’ before any symbol means ‘a change in’ that quantity. It is pronounced delta.]

In common with all forms of energy GPE is measuring in Joules (J).


A roller coaster carriage with a mass of 450 kg climbs from 2 m to 15m, what is the change in GPE?

Energy, figure 1

Δ_GPE = mgh_ - recall g = 10 N/kg

Δ_h_ = 15 - 2 = 13 m

Δ_GPE =_ 450 x 10 x 13

Δ_GPE = 58,500 _J or _ 5.85 x104 _J

The value of Δ_GPE_ would have the same magnitude but would be negative when the roller coaster descends from 15 m to 2 m. That is, it is losing GPE rather than gaining it.

Example 2

A ball of mass 0.5 kg has gained 20 J of GPE, having been kicked from the ground in a game of football, how high did the ball reach?

Δ_GPE_ = mg_Δ_h

∴ Δ_h_ = Δ_GPE/mg_ Δ_h_ = 20 / (0.5 x_10) = 4_m As the starting height was 0m the ball has reached a maximum height above ground of 4 m.

Kinetic Energy

Kinetic Energy is the energy an object has because of its movement. As with all forms of energy it is measured in Joules (J).

Kinetic Energy = 1/2 x mass x velocity2

KE _= 1/2_mv2

The equation tells us that the kinetic energy increases rapidly with velocity, it is proportional to the square of the velocity of the object. This is the reason that even small objects moving fast can cause a lot of manage, they have a lot of energy.

Example 1: How much kinetic energy does a hockey ball of 160g have at 1 m/s (a gentle tap from a hockey stick).

KE = 1/2 m v2

KE = 1/2 x 0.16 x (1)2 =0.08 J -160g = 0.16 kg mass must always be in kg.

Energy, figure 1

What is the kinetic energy of the same ball at 15 m/s (a fast pass or energetic aim at goal)?

KE =1/2mv2 = 1/2 x 0.16 x (15)2 = 1/2 x 0.16 x 225 = 18_ J _

Example 3

A roller coaster at the base of a drop has 58,500 J of kinetic energy, how fast is it moving, if it has a mass of 450 kg?

Energy, figure 2

Energy, figure 3

Remember in the exam you do not have to memorise all these equations. It is more important to be able to choose the correct equation and be able to rearrange them to change the subject of the equations.

Make m the subject of the kinetic energy equation.

KE = 1/2 mv2 Move the 1/2 first 2KE = mv2

  • When you move a variable or value across the = sign it does the opposite function so ½ (divide by 2) becomes multiply by 2.

Move the v2 next - this is multiplied on the right of the = so it is divide by on the left.

2KE / v2 = m it normal to put the subject of the equation on the left we write its as;

m = 2KE / v2

Energy Transfer

Energy cannot be destroyed. When people use the expression they have run out of energy, they really mean all the useful store of energy they had has been changed into other forms of energy.

Energy can exist in a number of states or forms.

Heat (Thermal)

Light (including all forms of Electromagnetic radiation)


Kinetic (movement)

Gravitational Potential (height above the ground)

Elastic or stretch (think elastic bands or springs)



Energy, figure 1

It is possible to convert one type of energy into others.

Light bulbs take in Electrical Energy and convert it into Light and Heat Energies.

The more efficient the bulb is the more light is produced and less is wasted as heat.

Energy, figure 2

Energy Transfer Diagrams

These are diagrammatic representations of the amounts or percentages of energy changes in a system. They are known as Sankey diagrams.

Energy, figure 3

A typical Sankey diagram for a conventional filament light bulb. (No longer on sale in the EU or UK, due to the fact that they waste 90% of the electrical energy but convert it into heat, not light).

The left hand side represents the input energy type and the right hand arrows the energy transfers. The width of the arrow in each section is proportional to the energy or the percentage division.

Energy, figure 4

What sort of device might this be a Sankey diagram for?

Answer: A typical plasma TV.

Example of Energy Transfer:

When a roller coaster is moving up to the start of the ride;
Electrical to Kinetic and Heat, (some sound too)
Kinetic to Gravitational Potential Energy.

As the roller coaster runs down from the top of the ride, the Gravitational Potential energy is converted to Kinetic energy, as well as some waste heat (from friction) and sound, (from the carriage not the people screaming - that’s a different energy change).

Importantly - The total energy before the transfer must equal the total energy after the transfer. (Law of conservation of Energy).

What was the velocity of the bike and rider in question 1 above, at the base of the second slope?
Your answer should include: 54.77m/s / 54.77
Explanation: v =√2Ke / m = √2 x 1.575 x102 / 10⁵ = 54.77 m/s