# Potential Energy Calculator

> Computes the gravitational potential energy of a mass raised to a height near a planet's surface with PE = m·g·h (standard gravity by default), and the speed the mass would reach if it fell that height freely.

- Calculator id: `potential-energy` · Category: Physics (`physics`) · Tool name: `calculate_potential_energy`
- Canonical page: https://tttkmbb.com/physics/potential-energy · This document: https://tttkmbb.com/physics/potential-energy.md · JSON definition: https://tttkmbb.com/physics/potential-energy.json

## Purpose

Computes the gravitational potential energy of a mass raised to a height near a planet's surface with PE = m·g·h (standard gravity by default), and the speed the mass would reach if it fell that height freely.

**Use when:** You need the energy stored by lifting a mass to a height, the minimum work to lift it, or the energy released when it falls (lifted loads, pumped hydro, dropped objects).

**Do not use when:** Heights are a significant fraction of the planet's radius (g is not constant; use gravitational-force), or you need elastic energy (use hookes-law) or energy of motion (use kinetic-energy).

## Input

| Parameter | Type | Unit | Required | Description |
| --- | --- | --- | --- | --- |
| `mass_kg` | number | kg | required | Mass of the object in kilograms. (> 0) |
| `height_m` | number | m | required | Height above the reference level (ground) in metres. (min 0) |
| `gravity_m_s2` | number | m/s² | optional, default 9.80665 | Local gravitational acceleration. Default is standard gravity g_n = 9.80665 m/s² (Moon ≈ 1.62, Mars ≈ 3.72, Jupiter ≈ 24.8). (> 0, max 10000) |

## Output

| Field | Type | Unit | Description |
| --- | --- | --- | --- |
| `potential_energy_j` | number | J | PE = m × g × h. |
| `potential_energy_kj` | number | kJ | Potential energy in kilojoules. |
| `potential_energy_kwh` | number | kWh | Potential energy in kilowatt-hours (J ÷ 3,600,000). |
| `fall_speed_m_s` | number | m/s | √(2·g·h): speed reached if all the potential energy became kinetic energy (no air resistance). |

## Formula

`PE = mass_kg × gravity_m_s2 × height_m; fall_speed_m_s = √(2 × gravity_m_s2 × height_m)`

Valid where g can be treated as constant (heights small compared with the planet's radius, i.e. within a few tens of kilometres of Earth's surface).

## Data Sources

- HyperPhysics – Gravitational potential energy — http://hyperphysics.phy-astr.gsu.edu/hbase/pegrav.html (reference, retrieved 2026-09-23)
- NIST CODATA – standard acceleration of gravity g_n = 9.80665 m/s² — https://physics.nist.gov/cgi-bin/cuu/Value?gn (standard, retrieved 2026-09-23)

Data freshness: `static`. Deterministic formula with fixed constants; results never go stale. Inputs supplied by the caller determine the output.

## API

- `GET https://tttkmbb.com/api/v1/calculate/potential-energy?mass_kg=…&height_m=…`
- `POST https://tttkmbb.com/api/v1/calculate/potential-energy` with JSON body `{"inputs": {…}}`
- Response: unified envelope (`success`, `request`, `result.values`, `result.units`, `sources`, `freshness`, `timestamp`, `next_actions`, `links`); see https://tttkmbb.com/docs/response-format.md
- Schema: https://tttkmbb.com/api/v1/calculators/potential-energy · OpenAPI operationId `calculate_potential_energy` in https://tttkmbb.com/openapi.json
- Authentication: none. Rate limit: fair use, see https://tttkmbb.com/docs/rate-limits.md.

## MCP

- Server: `https://tttkmbb.com/mcp` (Streamable HTTP, JSON-RPC 2.0, no auth)
- Tool:  `run_calculator` with `{"calculator_id": "potential-energy", "inputs": {…}}`

## Example

- 70 kg lifted 10 m: inputs `{"mass_kg":70,"height_m":10}` → `{"potential_energy_j":6864.655,"potential_energy_kj":6.864655,"potential_energy_kwh":0.001907,"fall_speed_m_s":14.0047}`
- 2 kg at 1.5 m with g = 9.81: inputs `{"mass_kg":2,"height_m":1.5,"gravity_m_s2":9.81}` → `{"potential_energy_j":29.43,"fall_speed_m_s":5.4249}`

```
GET https://tttkmbb.com/api/v1/calculate/potential-energy?mass_kg=70&height_m=10
```

## Limitations

Heights are a significant fraction of the planet's radius (g is not constant; use gravitational-force), or you need elastic energy (use hookes-law) or energy of motion (use kinetic-energy). Valid where g can be treated as constant (heights small compared with the planet's radius, i.e. within a few tens of kilometres of Earth's surface). All values are computed from the formula above; no measurement or live data is involved.

## FAQ

**Which reference level should I use?**

Any level you choose; only differences in potential energy matter. Height is measured from that reference, and objects below it have negative potential energy.

**Should I use 9.81 or 9.80665?**

9.80665 m/s² is the defined standard gravity; the actual value varies from about 9.78 at the equator to 9.83 at the poles. The difference is under 0.3 %.

## Related

- [Kinetic Energy Calculator](https://tttkmbb.com/physics/kinetic-energy.md) — Energy of motion gained when the object falls.
- [Free Fall Calculator](https://tttkmbb.com/physics/free-fall.md) — Fall time and impact speed from the same height.
- [Work & Power Calculator](https://tttkmbb.com/physics/work-power.md) — Power needed to lift the mass in a given time.