Potential Energy Calculator
Work out gravitational potential energy from mass, gravity and height, or the energy stored in a stretched or compressed spring. Then use energy conservation to find how fast an object is moving after a fall or slide, or how fast a spring launches a mass. Choose any units, solve for any variable, and see the working. Runs in your browser; free, no sign-up.
Gravitational potential energy is measured from a reference height you choose — usually the ground or the lowest point. Heights below it are negative.
Calculated
Results
490.3 J
Potential energy PE
The working
- Start from PE = m × g × h.
- PE = 5 × 9.80665 × 10 = 490.3 J
The same energy in other units
| Unit | Energy |
|---|---|
| J | 490.3 |
| kJ | 0.4903 |
| kcal | 0.1172 |
| Wh | 0.1362 |
| kWh | 1.362 × 10⁻⁴ |
| ft·lbf | 361.7 |
Assumes a uniform gravitational field (valid near a planet's surface), an ideal spring that obeys Hooke's law, and no friction or air resistance.
How to Use the Potential Energy Calculator
- 1. Choose the calculation. Gravitational energy for an object above a reference height, Spring energy for a stretched or compressed spring, Speed after a fall or slide for energy conservation under gravity, or Speed of a mass released by a spring.
- 2. Pick the unknown and enter the rest. The box you are solving for becomes the answer. Choose Earth, the Moon, Mars or your own value of g where gravity appears, and any unit for each quantity.
- 3. Read the working. The equation, its rearrangement and your numbers are shown step by step.
- 4. Compare units. Energy results come with the same value in joules, kilojoules, kilocalories, watt-hours, kilowatt-hours and foot-pounds.
The Formulas
| Situation | Formula | Notes |
|---|---|---|
| Gravitational | PE = m g h | h is measured from a reference level you choose |
| Elastic (spring) | PE = ½ k x² | x is the stretch or compression from the relaxed length |
| Fall or slide | v² = u² + 2 g h | h is the height lost; the mass cancels |
| Spring release | ½ k x² = ½ m v² | v = x √(k ÷ m) on a frictionless surface |
Reference: OpenStax University Physics, Volume 1, sections 7.3 and 8.2. Only changes in potential energy have physical meaning, so the zero level is a matter of convenience.
Worked Examples
Height. A 5 kg object held 10 m above the ground has PE = 5 × 9.80665 × 10 = 490.3 J (the calculator's default). That is 0.4903 kJ, or about 0.1172 kcal. On the Moon (1.62 m/s²) the same object at the same height has 81 J; on Mars (3.73 m/s²), 186.5 J.
Spring. A spring with k = 200 N/m stretched 0.25 m stores ½ × 200 × 0.25² = 6.25 J. Stretch it twice as far and the energy quadruples to 25 J, because x is squared.
A fall. Drop an object from rest through 20 m. Its potential energy mgh turns into kinetic energy ½mv², so v = √(2 × 9.80665 × 20) = 19.81 m/s, which is 71.3 km/h. The mass cancels: a feather and a hammer would land at the same speed in a vacuum. A skier or roller-coaster car starting down a 30 m hill at 2 m/s reaches v = √(2² + 2 × 9.80665 × 30) = 24.34 m/s at the bottom, whatever the shape of the track, if friction is ignored.
A spring launch. The 200 N/m spring compressed 0.25 m holds 6.25 J. Released against a 0.5 kg mass on a smooth surface, all of it becomes kinetic energy: v = 0.25 × √(200 ÷ 0.5) = 5 m/s.
Choosing the Reference Level
Potential energy depends on where you call h = 0. The floor, a tabletop or the bottom of a hill are all fine, as long as you use the same level throughout a problem. A height below the reference gives negative energy, which is allowed. What never depends on the choice is the change in potential energy between two points, and that is what converts into kinetic energy.
Assumptions and Limits
- Uniform gravity. PE = mgh assumes g is constant, which holds for heights small compared with the planet's radius. For satellites and rockets use −GMm ÷ r instead; the Gravitational Force Calculator covers the underlying law.
- No friction or drag. Energy conservation here treats the fall or slide as lossless. Real objects lose some energy to heat and sound, so they end up slower than calculated.
- Ideal springs. ½kx² holds while the spring obeys Hooke's law. Beyond the elastic limit the spring deforms permanently and the formula no longer applies.
Frequently Asked Questions
- What is the formula for potential energy?
- Gravitational potential energy is PE = mgh: mass times the gravitational acceleration times the height above a reference level. Elastic potential energy of a spring is PE = ½kx², with spring constant k and stretch x. Both are in joules when SI units are used.
- Can potential energy be negative?
- Yes. Potential energy is measured from a reference level of your choice, and anything below that level has negative energy. Only differences in potential energy affect the physics, so the sign of a single value means nothing by itself.
- How is potential energy converted to kinetic energy?
- As an object falls, it loses potential energy mgh and gains the same amount of kinetic energy ½mv² when friction is ignored. Setting them equal gives v = √(2gh) for a fall from rest. The mass cancels, so heavy and light objects reach the same speed.
- Why is spring energy proportional to the stretch squared?
- The force grows in proportion to the stretch (F = kx), so the work done stretching a spring is the area under a straight line, a triangle of area ½ × x × kx = ½kx². Doubling the stretch doubles the force and the distance, so it quadruples the energy.
- Does the path matter for gravitational potential energy?
- No. Gravity is a conservative force, so the energy depends only on the starting and ending heights, not the route. A ball rolled down a ramp and one dropped straight down from the same height reach the bottom at the same speed if friction is ignored.
- What is the unit of potential energy?
- The joule (J), which is 1 kg·m²/s², the same as a newton-metre. Other common units are kilojoules, kilocalories, watt-hours and foot-pounds; the results table shows several of them.