# What is the potential energy?

Computes gravitational potential energy U = m g h from mass and height, or the elastic potential energy of a spring U = ½ k x², in joules, kilojoules and foot-pounds.

- Page: https://www.acalculator.org/physics/potential-energy-calculator
- JSON spec: https://www.acalculator.org/physics/potential-energy-calculator.json
- Version: cf6265528265

## Default answer

Example with the default inputs (Kind Gravitational, Mass 22.0462262184878 lb, Height 16.4041994750656 ft, Gravity (m/s²) 9.80665): The potential energy is 490.333 J (361.651 ft·lbf).

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| kind | Kind | Gravitational potential energy (a raised mass) or elastic potential energy (a stretched spring). |
| m | Mass | The mass that is raised. |
| h | Height | How high the mass is above the level you measure from. |
| g | Gravity (m/s²) | The acceleration of gravity in m/s²: 9.80665 on Earth, about 1.62 on the Moon. |
| k | Spring constant | The spring constant k: the force per unit of stretch, in the unit chosen below. |
| ku | Spring constant unit | The unit of the spring constant: newtons per meter, newtons per millimeter, or pounds-force per inch. |
| x | Stretch or squeeze | How far the spring is stretched or squeezed from its rest length. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| joules | Potential energy (J) | The potential energy in joules. |
| kj | Potential energy (kJ) | The same energy in kilojoules. |
| ftlbf | Potential energy (ft·lbf) | The same energy in foot-pounds (1 ft·lbf = 1.3558179483314004 J). |
| speed | Speed if dropped (m/s) | For a raised mass: the speed it would reach falling the height with no air resistance, √(2gh). |

## Method

U = m × g × h (gravitational); U = ½ × k × x² (spring).

## Assumptions

- Gravitational: the height is measured from a chosen zero level, and g is the same over the whole height (near a planet’s surface).
- Spring: an ideal spring that follows Hooke’s law, F = k x, over the whole stretch.
- Exact unit sizes: 1 lb = 0.45359237 kg; 1 ft = 0.3048 m; 1 lbf = 4.4482216152605 N; 1 ft·lbf = 1.3558179483314004 J.

## Worked examples

1. kind = gravity, m = 10, h = 5, g = 9.80665 gives joules = 490.3325, kj = 0.490333, speed = 9.902853. Source: OpenStax, University Physics Volume 1, §8.1 Potential Energy of a System (U = m g y; U = ½ k x²), https://openstax.org/books/university-physics-volume-1/pages/8-1-potential-energy-of-a-system.
2. kind = gravity, m = 0.453592, h = 0.3048, g = 9.80665 gives ftlbf = 1, joules = 1.355818. Source: OpenStax, University Physics Volume 1, §8.1 Potential Energy of a System (U = m g y; U = ½ k x²), https://openstax.org/books/university-physics-volume-1/pages/8-1-potential-energy-of-a-system.
3. kind = spring, k = 200, ku = Npm, x = 0.1 gives joules = 1. Source: OpenStax, University Physics Volume 1, §8.1 Potential Energy of a System (U = m g y; U = ½ k x²), https://openstax.org/books/university-physics-volume-1/pages/8-1-potential-energy-of-a-system.
4. kind = gravity, m = 70, h = 100, g = 1.62 gives joules = 11,340, kj = 11.34. Source: OpenStax, University Physics Volume 1, §8.1 Potential Energy of a System (U = m g y; U = ½ k x²), https://openstax.org/books/university-physics-volume-1/pages/8-1-potential-energy-of-a-system.

## FAQ

### What is the formula for gravitational potential energy?

U = m × g × h: the mass in kilograms, times the acceleration of gravity (9.80665 m/s² on Earth), times the height in meters. A 10 kg mass raised 5 m has 10 × 9.80665 × 5 = 490.3 J.

### What is the formula for spring potential energy?

U = ½ × k × x², where k is the spring constant and x the stretch or squeeze from the rest length. A 200 N/m spring stretched 0.1 m stores ½ × 200 × 0.01 = 1 J.

### Where is the zero of potential energy?

Only changes in potential energy matter, so you choose the zero level: the floor, the ground, or a table top. Type the height above that level.

### How fast will the object be going if it falls?

With no air resistance, all the potential energy becomes kinetic energy: m g h = ½ m v², so v = √(2gh). A mass dropped 5 m reaches √(2 × 9.80665 × 5) = 9.90 m/s, whatever its mass.

### What is a foot-pound?

A foot-pound (ft·lbf) is the energy to raise 1 lb by 1 ft: 0.3048 m × 4.4482216152605 N = 1.3558 J.

### Can I use it on the Moon or Mars?

Yes. Change the gravity: about 1.62 m/s² on the Moon and 3.71 m/s² on Mars. A 70 kg astronaut 100 m up on the Moon has 70 × 1.62 × 100 = 11,340 J.

## Sources

- OpenStax, University Physics Volume 1, §8.1 Potential Energy of a System (U = m g y near Earth’s surface; U = ½ k x² for a spring). https://openstax.org/books/university-physics-volume-1/pages/8-1-potential-energy-of-a-system (retrieved 2026-10-03)
- NIST SP 811, Appendix B.8 Factors for Units Listed Alphabetically (foot-pound-force, pound-force, standard gravity 9.80665 m/s²). https://www.nist.gov/pml/special-publication-811/nist-guide-si-appendix-b-conversion-factors/nist-guide-si-appendix-b8 (retrieved 2026-10-03)