Elastic potential energy formula: E = ½ k e²
Elastic potential energy = 0.5 × spring constant × extension², written Ee = ½ k e² by AQA and E = ½ k x² by Edexcel and OCR. The energy is in joules (J), k is the spring constant in newtons per metre (N/m) and the extension is in metres (m). Each board classes it as a given equation, printed on the sheet in the exam.
Equation
Ee = ½ k e²
| Symbol | Quantity | Unit |
|---|---|---|
Ee | elastic potential energy | J |
k | spring constant | N/m |
e | extension | m |
What each specification says
| Exam board | Status | Equation |
|---|---|---|
| AQA Physics (8463) | Given on the sheet | Ee = ½ k e² |
| AQA Combined Science (8464) | Given on the sheet | Ee = ½ k e² |
| Edexcel Physics (1PH0) | Given on the sheet | E = ½ k x² |
| Edexcel Combined Science (1SC0) | Given on the sheet | E = ½ k x² |
| OCR Gateway Physics A (J249) | Given on the sheet | E = ½ k x² |
| OCR Twenty First Century Physics B (J259) | Given on the sheet | E = ½ k x² |
Calculator: Ee = ½ k e²
Energy stored in a stretch
When you stretch or squash a spring, you do work on it. The spring keeps that energy in its elastic potential energy store. Let go, and the store empties, often into kinetic energy. A catapult, a trampoline and a wind-up toy all work this way.
The extension is squared. Pull a spring twice as far and it holds four times the energy. That is why the last few centimetres of a stretch feel so much harder.
This equation only works while the spring obeys Hooke's law, up to its limit of proportionality. Beyond that, the spring bends out of shape and the numbers no longer fit.
Forms on the exam sheets
- AQA (8463 and 8464): Ee = ½ k e².
- Edexcel (1PH0 and 1SC0): E = ½ k x², for energy transferred in stretching.
- OCR Gateway (J249) and OCR Twenty First Century (J259): E = ½ k x².
In every one of these six specifications, this sits in the select and apply group. What the course asks is that you pick it out and use it with the right units. You can see where it sits among OCR's list on the OCR Gateway page.
Working backwards
- k = 2E_e ÷ e²
- e = √(2E_e ÷ k)
Often a question gives you a force and an extension first. Use F = k e to find k, then put k into this equation. A second common link is to kinetic energy: the energy from a stretched catapult band becomes the kinetic energy of the stone. The full list of stores and equations is on the physics equation sheet page.
Worked examples
Ee
- 15 cm = 0.15 m
- Ee = ½ k e²
- Ee = 0.5 × 80 × 0.15²
- Ee = 0.5 × 80 × 0.0225
- Ee = 0.9 J
0.9 J
k
- k = 2E_e ÷ e²
- k = 2 × 2 ÷ 0.1²
- k = 4 ÷ 0.01
- k = 400 N/m
400 N/m
e
- e = √(2E_e ÷ k)
- e = √(2 × 5 ÷ 1000)
- e = √0.01
- e = 0.1 m
0.1 m
Common mistakes
- Squaring the extension in centimetres. 15² is 225, but 0.15² is 0.0225, so convert to metres before you square.
- Using F = k e when the question asks for energy. That gives a force in newtons, not joules.
- Forgetting the half. Without it, the energy comes out twice as big as it should.
- Using the total length of the spring. Only the stretch beyond its natural length counts.
See how fast you can rearrange ½ k e² for k in the Equation Rearranger.
Questions people ask
Why is there a half in the equation?
The force on a spring grows from zero as you stretch it. On average it is half the final force, so the work done is half of force × extension.
Does a squashed spring store energy too?
Yes. A compressed spring stores elastic energy in the same way, with e as the amount it has been shortened.