# Free Fall Calculator

> Computes the time to fall a given height from rest, or the height fallen in a given time, together with the final (impact) velocity, using h = ½·g·t² with no air resistance.

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

## Purpose

Computes the time to fall a given height from rest, or the height fallen in a given time, together with the final (impact) velocity, using h = ½·g·t² with no air resistance.

**Use when:** An object is dropped from rest near a planet's surface and you need how long it falls, how far, or how fast it lands (a dropped tool, a stone in a well).

**Do not use when:** Air drag matters (light or fast objects, skydivers, long falls near terminal velocity), the object has an initial velocity or launch angle (use projectile-motion), or heights are comparable to the planet's radius.

## Input

| Parameter | Type | Unit | Required | Description |
| --- | --- | --- | --- | --- |
| `height_m` | number | m | optional | Height fallen in metres. Give either height_m or time_s. (min 0) |
| `time_s` | number | s | optional | Fall duration in seconds. Give either height_m or time_s. (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 |
| --- | --- | --- | --- |
| `fall_time_s` | number | s | Time to fall the height from rest: √(2h / g). |
| `height_m` | number | m | Distance fallen: ½ × g × t². |
| `final_velocity_m_s` | number | m/s | Velocity at the end of the fall: g × t = √(2gh). |
| `final_velocity_km_h` | number | km/h | Impact velocity in kilometres per hour. |
| `average_velocity_m_s` | number | m/s | Mean velocity over the fall, half the final velocity. |

## Formula

`fall_time_s = √(2 × height_m / g); height_m = ½ × g × time_s²; final_velocity_m_s = g × fall_time_s = √(2 × g × height_m)`

Object released from rest in a vacuum (no drag), constant g. In air, drag becomes noticeable for a human after roughly 2–3 s of fall and dominates for light objects.

## Data Sources

- Wikipedia – Equations for a falling body — https://en.wikipedia.org/wiki/Equations_for_a_falling_body (reference, retrieved 2026-09-23)
- HyperPhysics – Trajectories — http://hyperphysics.phy-astr.gsu.edu/hbase/traj.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/free-fall?`
- `POST https://tttkmbb.com/api/v1/calculate/free-fall` 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/free-fall · OpenAPI operationId `calculate_free_fall` 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": "free-fall", "inputs": {…}}`

## Example

- Dropped from 20 m: inputs `{"height_m":20}` → `{"fall_time_s":2.0196,"height_m":20,"final_velocity_m_s":19.8057,"final_velocity_km_h":71.3005,"average_velocity_m_s":9.9029}`
- Falling for 3 s: inputs `{"time_s":3}` → `{"height_m":44.1299,"fall_time_s":3,"final_velocity_m_s":29.42}`

```
GET https://tttkmbb.com/api/v1/calculate/free-fall?height_m=20
```

## Limitations

Air drag matters (light or fast objects, skydivers, long falls near terminal velocity), the object has an initial velocity or launch angle (use projectile-motion), or heights are comparable to the planet's radius. Object released from rest in a vacuum (no drag), constant g. In air, drag becomes noticeable for a human after roughly 2–3 s of fall and dominates for light objects. All values are computed from the formula above; no measurement or live data is involved.

## FAQ

**Does the object's mass matter?**

Not without air resistance: all masses fall with the same acceleration g. With air, heavier and denser objects are affected less by drag.

**What if I know both the height and the time?**

Provide only one; the other is computed. Giving both returns an INVALID_PARAMETER error because they cannot be independent.

## Related

- [Projectile Motion Calculator](https://tttkmbb.com/physics/projectile-motion.md) — Launched objects with initial speed and angle.
- [Potential Energy Calculator](https://tttkmbb.com/physics/potential-energy.md) — Energy released by the fall.
- [Acceleration Calculator](https://tttkmbb.com/physics/acceleration.md) — General constant-acceleration kinematics.