IB Physics HLTopic 1 — Motion, Forces & EnergyPaper 1 & 2Work, Energy & Power~9 min read
Elastic Potential Energy
Stretch a spring, pull back a bow, or squash a trampoline, and you’re storing energy in the material itself — energy that springs straight back the instant you release it. That’s elastic potential energy (sometimes called strain energy). It’s the third member of the mechanical-energy family, alongside kinetic and gravitational potential energy, and for anything that obeys Hooke’s law there’s a neat equation to work out exactly how much is stored.
📘 What you need to know
Elastic potential energy is the energy stored in a material when it’s stretched or compressed
For a material obeying Hooke’s law: EH = ½kΔx2
It can also be written EH = ½FΔx, using the restoring force F
k is the spring constant (N m−1) and Δx is the extension or compression (m)
The stored energy is the area under a force–extension graph
Both stretching and compressing store energy — either way Δx is squared, so the energy is positive
When a stretched material is released, its elastic PE converts to kinetic energy
The elastic potential energy equation
For a spring (or any material) that obeys Hooke’s law, the elastic potential energy stored is:
Elastic potential energyEH = ½kΔx2
where EH is the elastic PE in joules (J), k is the spring constant in newtons per metre (N m−1), and Δx is the extension or compression in metres (m). Just like kinetic energy, the Δx is squared — so stretching a spring twice as far stores four times the energy.
Whether a spring is stretched or compressed by Δx, it stores elastic potential energy ½kΔx2. Because Δx is squared, the energy is positive either way.
WE 1
A spring with a spring constant of 200 N m−1 is stretched by 0.15 m. Calculate the elastic potential energy stored.
Step 1 — write the equation
EH = ½kΔx²
Step 2 — substitute (square the extension only)EH = ½ × 200 × 0.15²EH = ½ × 200 × 0.0225EH = 2.25 JOnly the 0.15 gets squared — the spring constant and the half stay as they are.
The force form of the equation
There’s a second version that’s sometimes more convenient. Since Hooke’s law says the restoring force is F = kΔx, we can swap out one of the kΔx factors for F and write:
Elastic potential energy — force formEH = ½FΔx
where F is the restoring force in newtons (N). This is handy when a question gives you the force needed to stretch the spring rather than its spring constant. Both forms give exactly the same answer — they’re the same equation with F = kΔx substituted in.
EH = ½kΔx²
put F = kΔx →
½(kΔx)Δx
so →
½FΔx
WE 2
A force of 24 N is needed to stretch a spring by 0.08 m. Calculate the elastic potential energy stored, using the force form of the equation.
Step 1 — use the force form
EH = ½FΔx
Step 2 — substituteEH = ½ × 24 × 0.08EH = 0.96 JNo spring constant needed — the force and extension are enough. (If you wanted k, it would be 24 ÷ 0.08 = 300 N m⁻¹.)
Energy as the area under a graph
Both forms of the equation come from the same picture. Plot the restoring force against extension for a Hooke’s-law material and you get a straight line through the origin (since F = kΔx). The area under that line is the work done stretching the spring — which is exactly the elastic PE stored. That area is a triangle: ½ × base × height = ½ × Δx × F, giving EH = ½FΔx straight away.
For a Hooke’s-law spring, force against extension is a straight line. The shaded triangular area, ½FΔx, is the elastic potential energy stored.
Releasing the stored energy
Stored elastic PE doesn’t stay put — release the spring and it transfers to kinetic energy. This is exactly how a catapult, a bow, or a spring-loaded launcher works: energy you put in by stretching comes back out as motion. Setting the elastic PE equal to the kinetic energy gained lets you find the launch speed:
Elastic PE converts to kinetic energy
½kΔx2 = ½mv2
WE 3
A toy launcher uses a spring of spring constant 800 N m−1, pulled back 0.25 m. It fires a 0.050 kg ball. Assuming all the stored energy becomes kinetic energy, find the launch speed.
Step 1 — find the elastic PE storedEH = ½ × 800 × 0.25² = 25 JStep 2 — all of it becomes kinetic energy
½mv² = 25 J
Step 3 — rearrange for v and substitutev = √(2 × 25 ÷ 0.050) = √1000v = 31.6 m s⁻¹ (3 s.f.)The 25 J stored in the spring is handed straight to the ball as motion.
This energy transfer is also why a stretched wire snapping is dangerous. All that elastic PE built up in the strained wire is released in an instant as kinetic energy — the greater the extension before it breaks, the faster the ends whip back. Same physics as the catapult, just less friendly.
🛠️ Solving an elastic PE problem
Check what you’re given — spring constant k, or the force F?
Pick the matching form: ½kΔx2 if you have k, or ½FΔx if you have F.
Convert units — extensions in cm or mm must become metres first.
Square the extension only in the k form — never the k or the half.
For a launch or snap, set the elastic PE equal to the kinetic energy gained and solve for speed.
💡 Top tips
Watch the units. Extensions given in cm or mm must be converted to metres before squaring — a common source of factor-of-100 errors.
Only Δx is squared in the ½kΔx2 form, just like speed in kinetic energy.
Two forms, same answer. Use whichever suits the data — F = kΔx links them.
Compression counts too. Squashing a spring stores just as much energy as stretching it by the same amount.
Quick recap: Elastic potential energy is the energy stored when a material is stretched or compressed. For a Hooke’s-law spring, EH = ½kΔx2, or equivalently EH = ½FΔx — the triangular area under a force–extension graph. When released, that stored energy transfers to kinetic energy.
⚠ Common mistakes
Forgetting to convert cm or mm to metres before squaring the extension
Using EH = FΔxwithout the ½ — the force builds up from zero, so it’s an average of ½F
Squaring the spring constant or the half — only the extension is squared
Assuming the equation works beyond the elastic limit — it only holds while Hooke’s law does
Thinking compression stores no energy — it stores just as much as stretching
That completes the three stores of mechanical energy: kinetic, gravitational potential, and now elastic potential. The exciting part comes next — putting them together and seeing how, with no friction, mechanical energy stays perfectly conserved as it swaps between all three. That’s conservation of mechanical energy, and it’s where all this groundwork pays off.
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