Energy is stored in different ways, including the movement of an object, its position in a gravitational field and the stretching of an elastic object. Three equations allow us to calculate these energy stores in joules (J). Choosing the right equation and using the correct units makes energy calculations much easier.
On this page: Choosing an equation · Kinetic energy · Gravitational potential energy · Elastic potential energy · Calculation checklist · Summary · Quick questions
Key vocabulary
- Energy store: A way energy is held within a system, such as the kinetic store of a moving object.
- Kinetic energy: Energy in the store of a moving object.
- Gravitational potential energy: Energy in a store associated with an object’s position in a gravitational field.
- Elastic potential energy: Energy in a store when an elastic object is stretched or compressed.
- Gravitational field strength (g): Force per kilogram of mass, measured in newtons per kilogram (N/kg).
- Spring constant (k): A measure of a spring’s stiffness, measured in newtons per metre (N/m).
- Extension (e): The increase in a spring’s length from its original, unstretched length.
Which energy store and equation?
Identify what is changing. Movement points to kinetic energy; a change in height points to gravitational potential energy; and a change in a spring’s extension points to elastic potential energy.
Kinetic energy
A moving object has energy in its kinetic energy store. A greater mass or speed gives it more kinetic energy.
Kinetic energy equation
Equation: Ek = ½mv2
Symbols: Ek = kinetic energy (J); m = mass (kg); v = speed (m/s).
Worked example: A 2.0 kg ball moves at 5.0 m/s.
Ek = ½ × 2.0 × 5.02 = ½ × 2.0 × 25 = 25 J.
Important: The speed is squared. If speed doubles, kinetic energy becomes four times greater, provided mass stays the same.
Gravitational potential energy
An object’s gravitational potential energy store changes when its height changes. Lifting a heavier object, or lifting it higher, increases this store.
Gravitational potential energy equation
Equation: Ep = mgh
Symbols: Ep = change in gravitational potential energy (J); m = mass (kg); g = gravitational field strength (N/kg); h = vertical height change (m).
Worked example: A 3.0 kg bag is lifted by 2.0 m. Take g = 10 N/kg, as given in this example.
Ep = 3.0 × 10 × 2.0 = 60 J.
Important: Use the value of g supplied in the question. Height means the vertical change in height relative to a chosen starting level, not the distance travelled along a slope.
Elastic potential energy
A spring stores elastic potential energy when it is stretched or compressed. The extension of a stretched spring is the difference between its stretched length and its original length.
Elastic potential energy equation
Equation: Ee = ½ke2
Symbols: Ee = elastic potential energy (J); k = spring constant (N/m); e = extension (m).
Condition: Use this GCSE equation when the limit of proportionality has not been exceeded.
Worked example: A spring has k = 200 N/m and an extension of 10 cm.
Step 1: Convert the extension: 10 cm ÷ 100 = 0.10 m.
Step 2: Ee = ½ × 200 × (0.10)2 = ½ × 200 × 0.01 = 1.0 J.
Important: A spring’s full length is not its extension. Squaring 10 instead of 0.10 would give a very large, incorrect answer.
Rearranging and checking calculations
Sometimes the energy is known and you need to find another quantity. Rearrange the equation before substituting numbers.
Kinetic speed: v = √(2Ek ÷ m). For Ek = 36 J and m = 2.0 kg: v = √(72 ÷ 2.0) = √36 = 6.0 m/s.
Gravitational height: h = Ep ÷ (mg). For Ep = 100 J, m = 2.0 kg and g = 10 N/kg: h = 100 ÷ 20 = 5.0 m.
Spring constant: k = 2Ee ÷ e2. For Ee = 1.0 J and e = 0.10 m: k = 2.0 ÷ 0.01 = 200 N/m.
Calculation checklist: Select the correct store → write the equation → convert to kg, m and m/s where needed → substitute → square or square-root carefully → give a unit.
How energy moves between stores
When an object falls, its gravitational potential energy store decreases as its kinetic energy store increases. If air resistance and friction transfer energy to the surroundings, not all the decrease in gravitational potential energy becomes kinetic energy. Energy is transferred, not destroyed.
Key summary
- Kinetic energy is stored by moving objects: Ek = ½mv2. Speed must be in m/s.
- Gravitational potential energy changes when an object changes height: Ep = mgh. Use height change in metres.
- Elastic potential energy is stored when a spring stretches or compresses: Ee = ½ke2, within its limit of proportionality.
- Squared quantities matter: doubling speed makes kinetic energy four times greater; doubling extension makes elastic energy four times greater when k stays constant.
- Calculation accuracy depends on converting units, substituting values carefully, showing working and giving energy in joules (J).
Quick questions
Write the equation for kinetic energy.
Ek = ½mv2.
A 2 kg ball travels at 3 m/s. Calculate its kinetic energy.
Ek = ½ × 2 × 32 = 9 J.
Write the equation for gravitational potential energy.
Ep = mgh.
What does extension mean in the elastic potential energy equation?
Extension is the increase in length from the spring’s original, unstretched length.
A spring extends by 5 cm. What value of e should you use?
0.05 m, because 5 ÷ 100 = 0.05.