The elastic potential energy formula

The elastic potential energy formula is Ee = ½ke²: elastic potential energy = 0.5 × spring constant × extension². It gives the energy stored in a spring, elastic band or any other elastic object when it is stretched or squashed. Energy is in joules (J), the spring constant k in newtons per metre (N/m) and the extension e in metres (m). It only works up to the limit of proportionality, while the spring obeys Hooke's law.

What is elastic potential energy?

Elastic potential energy is the energy stored in an object because it has been stretched, squashed, bent or twisted out of shape. When you let go, the object springs back and the stored energy is transferred, usually to kinetic energy. A drawn bow, a compressed car spring, a stretched trampoline and a wound-up clockwork toy all store elastic potential energy.

The work you do stretching a spring is not lost: it is stored in the spring's elastic potential energy store. If the spring is not stretched beyond its limit of proportionality, all of that work can be got back.

The elastic potential energy equation

A stiff spring has a large k and a floppy one a small k. The extension is the new length minus the original length. For a spring that is squashed instead, use the compression, how much shorter it is; the formula is the same.

Rearranging the formula

Because e is squared, doubling the extension stores four times as much energy. Stretching a spring 2 cm stores four times the energy of stretching it 1 cm, and 3 cm stores nine times as much.

Where the formula comes from: the force-extension graph

Hooke's law says the force needed to stretch a spring is proportional to its extension, F = ke. So a graph of force against extension is a straight line through the origin, up to the limit of proportionality.

Here x is the extension in metres and y is the force in newtons. The work done stretching the spring, and so the energy stored, is the area under the line. Up to an extension of 0.25 m, that area is a triangle with base 0.25 m and height 10 N, so the energy is ½ × 0.25 × 10 = 1.25 J.

In general the triangle has base e and height F = ke, so its area is ½ × e × ke = ½ke². That is the elastic potential energy formula. It also gives a second version that is handy when you know the force instead of k:

Beyond the limit of proportionality the graph curves, F = ke no longer holds, and you have to find the energy from the area under the curve instead. The Hooke's Law Calculator finds k, F and e for you.

How to calculate elastic potential energy: worked examples

Example 1: the basic formula

A spring with a spring constant of 200 N/m is stretched by 0.15 m. How much elastic potential energy does it store?

It stores 2.25 J. Square the extension first, then multiply.

Example 2: extension in centimetres

A spring with k = 50 N/m is stretched by 8 cm. Find the energy stored.

Convert first: 8 cm = 0.08 m.

Using 8 instead of 0.08 would give 1600 J, ten thousand times too big.

Example 3: finding the spring constant

A spring stores 0.6 J when stretched by 0.1 m. Find its spring constant.

Example 4: finding the extension

A spring with k = 900 N/m stores 4.5 J. How far is it stretched?

It is stretched by 0.1 m, or 10 cm.

Example 5: from force and extension

A 12 N force stretches a spring by 5 cm, within its limit of proportionality. How much energy is stored?

Use Ee = ½Fe with e = 0.05 m: ½ × 12 × 0.05 = 0.3 J. You could also find k first: k = F ÷ e = 12 ÷ 0.05 = 240 N/m, and ½ × 240 × 0.05² gives the same 0.3 J.

Example 6: a spring launcher

A toy launcher has a spring with k = 400 N/m, compressed by 5 cm. When released, it fires a 0.02 kg ball. Assuming all the elastic potential energy becomes kinetic energy, how fast does the ball leave?

Energy stored: ½ × 400 × 0.05² = 0.5 J. Set that equal to the kinetic energy, ½mv²:

The ball leaves at about 7.1 m/s. In reality it would be a little slower, as some energy is lost to friction and sound.

Here is a quick check.

Finding the spring constant in an experiment

You cannot use Ee = ½ke² until you know k, and the usual way to find it is the force and extension experiment, a GCSE required practical.

1. Hang the spring from a clamp next to a vertical ruler, and record the position of its bottom end with no load.
2. Add masses one at a time, 100 g (about 1 N) at a time, and record the new position after each.
3. Work out each extension by subtracting the unloaded position, and each force from the weight, W = mg.
4. Plot force on the y-axis against extension on the x-axis.

The gradient of the straight part of the line is the spring constant k. Where the line starts to curve is the limit of proportionality. The area under the straight part, up to any extension, is the elastic potential energy stored at that extension. Read the ruler at eye level, and remove the masses at the end to check the spring returns to its original length: if it does not, it has been stretched past its elastic limit.

Elastic potential energy and other energy stores

Elastic potential energy is one of the energy stores you meet at GCSE. In a bungee jump, the jumper's gravitational potential energy becomes kinetic energy, then elastic potential energy in the cord as it stretches. At the lowest point the jumper is momentarily still, and almost all the energy is in the cord. The Potential Energy Calculator works out both gravitational and elastic potential energy, and the guide to energy stores covers the rest.

Common mistakes

• Not squaring the extension. The formula is ½ke², not ½ke.
• Using the total length. The extension is the change in length. A 10 cm spring stretched to 14 cm has e = 4 cm.
• Leaving the extension in cm or mm. Convert to metres before substituting.
• Squaring k, or squaring ½ke. Only e is squared.
• Using the formula beyond the limit of proportionality. There, the spring no longer obeys F = ke, so ½ke² does not apply.
• Forgetting the ½. Ee = ke² gives double the right answer.

Is the elastic potential energy formula on the equation sheet?

Yes, on AQA's sheets, checked in November 2026. AQA's GCSE Physics equations sheet gives elastic potential energy = 0.5 × spring constant × (extension)², Ee = ½ke², together with Hooke's law, F = ke. AQA's A-level Physics data booklet gives the energy stored as E = ½FΔL in its materials section, alongside Hooke's law F = kΔL. Other boards publish their own sheets, so check yours. Finding the spring constant from a graph is also part of the GCSE required practical on the force and extension of a spring, where the Practical Data Lab helps you plot and analyse your results.

Practise energy questions in a short timed paper

GCSE Physics Foundation and GCSE Physics Higher each give you a short timed paper every day, with 8 questions worth 25 to 35 marks across the course, including energy stores and forces.

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