Elastic potential energy is the work you did against a spring, held in the spring, waiting to be given back. For an ideal spring it is U = ½kx², where k is the stiffness and x is the distance from the spring's relaxed length. The factor of one half is not decoration — it is there because the force you push against grows as you go, so the average force over the displacement is half the final force, and energy is average force times distance.
Arb Digital built this page to solve that relation in any direction and then to make the number mean something. Joules on their own are abstract. The grid converts the stored energy into a launch speed for a mass you specify and into the height that mass would reach, which are the two things a stored-energy figure is usually a step toward. It also shows the restoring force, so you can see the linear and the quadratic behaviour side by side.
What This Elastic Potential Energy Calculator Does
Give it a spring constant and a displacement and it returns the stored energy. Switch the solve-for dropdown and it works backwards: energy and displacement give the stiffness, energy and stiffness give the displacement. That reverse direction is the useful one when you are specifying a spring rather than measuring one, because the design requirement usually arrives as an energy or a speed rather than as a stiffness.
The supporting grid reports four things. The restoring force at that displacement, which is Hooke's law and rises linearly. The energy stored at half the displacement, which is a quarter of the headline figure and demonstrates the square law more convincingly than any sentence can. The speed a mass would reach if all the energy transferred to it. And the height that mass would rise to if fired straight up, ignoring air resistance.
Compression and extension give identical answers because x is squared. A spring compressed by 12 mm holds exactly the energy of the same spring extended by 12 mm. Enter a negative displacement and the tool returns the same positive energy, which is correct rather than a lapse in sign handling.
How to Use It
- Choose what you are solving for. The field matching your selection is ignored, so leaving a stale value in it changes nothing.
- Set the displacement unit before typing. Spring travel is quoted in millimetres in mechanical design and in metres in physics problems. The energy is proportional to the square of the displacement, so a unit mistake here is squared too.
- Measure displacement from the free length. Not from the installed position, and not from the fully compressed state. A pre-loaded spring already holds energy, and the difference between two states is what a change in stored energy means.
- Add a mass if you want the outcome rather than the energy. The launch speed and lift height only appear once there is a mass for the energy to act on.
- Compare the two energy figures. The headline value and the half-displacement value differ by a factor of four every time, which is the fastest way to internalise why elastic energy is not proportional to how far you pulled.
The Formula: Why There Is a Factor of One Half
Hooke's law says the restoring force is F = −kx: proportional to displacement and directed back toward the relaxed position. Work is force times distance, but the force is not constant here, so the work done stretching from 0 to x is the area under the force–displacement line — a triangle of base x and height kx, which has area ½kx². OpenStax University Physics Volume 1, section 8.1 on the potential energy of a system, derives it as U(x) = ½kx² plus a constant, with the constant set to zero by taking the relaxed length as the reference point.
Work the defaults. A spring of k = 250 N/m displaced by 12 cm, which is 0.12 m, stores U = 0.5 × 250 × 0.12² = 0.5 × 250 × 0.0144 = 1.80 J. The restoring force at that point is F = 250 × 0.12 = 30 N. At half the displacement, 0.06 m, the stored energy is 0.5 × 250 × 0.0036 = 0.45 J, exactly a quarter.
Release that 1.80 J into a 0.5 kg mass and, if the transfer is complete, ½mv² = 1.80 gives v = √(2 × 1.80 ÷ 0.5) = √7.2 = 2.68 m/s. Fired vertically, the same energy lifts that mass to h = U ÷ mg = 1.80 ÷ (0.5 × 9.80665) = 0.367 m. Both conversions assume no losses, which is optimistic but is the right starting point.
Where This Differs From Our Hooke's Law Calculator
Arb Digital's Hooke's law calculator and this page use the same spring model and answer different questions. Hooke's law gives the force a spring exerts at a given extension: F = kx, a linear relation, in newtons. This page gives the energy stored at that extension: U = ½kx², a quadratic relation, in joules. One is what you feel while holding the spring; the other is what the spring can do when you let go.
Use the force page when you need to know whether a mounting will hold, what preload a fastener sees, or how much force is available at a particular position. Use this page when you need to know how fast something will be launched, how far a stored-energy mechanism will drive a mechanism, or whether a compressed spring holds enough energy to matter on release. The two are linked — energy is the area under the force curve — and the restoring-force figure in the grid above is exactly the Hooke's law result, included so you can see both without switching pages.
The boundary matters because the two scale differently. A spring at twice the extension exerts twice the force and holds four times the energy. Any intuition built on one of those will mislead you about the other, which is the practical reason to keep the calculations separate.
Where the Ideal Spring Model Stops Working
U = ½kx² assumes a constant k, and real springs only oblige within a range. Push a coil spring far enough and the coils touch — solid height — and stiffness rises abruptly toward infinity. Stretch it beyond its elastic limit and it takes a permanent set, after which the free length has changed and every subsequent calculation from the old reference is wrong.
Some springs are not linear by design. Progressive-rate suspension springs have variable pitch so stiffness rises with compression, and a single k value cannot describe them. Rubber and elastomers are worse: they are viscoelastic, so the force depends on how fast you deform them as well as how far, and they dissipate a noticeable fraction of the input energy as heat on every cycle. A rubber band returns visibly less energy than it absorbed, and it warms slightly as proof.
Gas springs follow a gas law rather than Hooke's law. A torsion spring stores ½κθ² with an angular stiffness and an angle, which is the same mathematics in rotational form but different units throughout. OpenStax section 8.4 on potential energy diagrams and stability shows why the parabolic shape of ½kx² matters beyond springs: any stable equilibrium looks like a parabola close enough to the minimum, which is why this formula turns up in molecular vibration and structural analysis far from anything anyone would call a spring.
What Happens to the Energy on Release
A stored 1.8 J does not automatically become 1.8 J of kinetic energy in your projectile. The spring itself has mass and some of the energy goes into accelerating the coils, which for a light projectile and a heavy spring can be a large fraction. Friction in the guide, air drag, and the spring's own internal damping take more. Real spring-driven mechanisms typically deliver somewhere between sixty and ninety per cent of the stored energy to the load.
The launch speed figure in the grid is therefore an upper bound. It is still the right number to start from, because it tells you the ceiling: if the ideal speed is already below what you need, no amount of mechanism refinement will get you there and the spring or the displacement has to change. Once you have the ideal figure, the kinetic energy calculator works the same conversion in the opposite direction, and the impulse calculator covers the force-over-time side of the release if the duration matters.
Related Free Tools From Arb Digital
For the force a spring exerts rather than the energy it stores, use the Hooke's law calculator. For gravitational rather than elastic storage, the potential energy calculator covers mgh, and the kinetic energy calculator handles the energy once it is moving. The inductor energy calculator is the electrical analogue, storing ½LI² in a magnetic field with exactly the same quadratic form. Convert results between joules, calories and watt-hours with the energy converter, and use the impulse calculator for the momentum side of a release. Everything is listed on the free online tools hub.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Forgetting the factor of one half — the force grows as you stretch, so the work done is the area of a triangle, not a rectangle. Using kx² doubles the answer.
- Measuring displacement from the installed position — x is measured from the spring's free length. A pre-loaded spring already stores energy, and the quantity you usually want is the difference between two states.
- Mixing displacement units — energy goes as the square of displacement, so entering millimetres where the formula expects metres is wrong by a factor of a million, not a thousand.
- Assuming all stored energy reaches the load — the spring's own mass, friction and internal damping take a real share. The launch speed shown here is a ceiling, not a prediction.
- Applying a single k beyond the elastic range — coil bind, permanent set, progressive-rate designs and elastomers all break the constant-stiffness assumption the formula depends on.
Frequently Asked Questions
U equals one half times the spring constant times the displacement squared, with the spring constant in newtons per metre, the displacement in metres, and the result in joules. The displacement is measured from the spring's relaxed, unloaded length.
Because the force you work against is not constant. It starts at zero and rises linearly to kx, so the average force over the displacement is half the final force. Work is average force times distance, which gives one half times k times x squared.
Hooke's law gives the force a spring exerts at a given displacement, F equals k times x, measured in newtons and linear in displacement. This page gives the energy stored at that displacement, one half k x squared, measured in joules and quadratic in displacement. One is what you feel holding the spring, the other is what it can do on release.
For an ideal spring, yes. The displacement is squared, so the sign disappears and a compression of 12 millimetres stores exactly what an extension of 12 millimetres stores. Real springs may differ once compression approaches coil bind or extension approaches the elastic limit.
Some becomes kinetic energy in the load, some accelerates the spring's own mass, and the rest is lost to friction, air resistance and internal damping. Real mechanisms typically deliver a majority of the stored energy but not all of it, so the launch speed shown here is an upper limit.
Switch the solve-for dropdown to spring constant and enter the energy and the displacement. Rearranged, k equals two times U divided by x squared. Doubling the displacement for the same energy requires a spring four times softer.
Not accurately. Rubber is viscoelastic, so its force depends on the rate of stretching as well as the amount, and it dissipates a noticeable share of the input energy as heat each cycle. The formula gives a rough upper bound at best for elastomers.
This tool is provided for educational and study use. It models an ideal linear spring with constant stiffness and no losses, and does not account for coil bind, permanent set, progressive rates, viscoelastic behaviour or the mass of the spring itself.
