A free-fall calculator that solves any one of five variables — distance, time, final velocity, initial velocity or gravitational acceleration — given the others. It covers every planet in the solar system plus a custom-g mode, shows full step-by-step working, and reports kinetic energy per unit mass alongside the primary answer.
How it works
Free fall is constant-acceleration motion under gravity alone, with no air resistance. The motion is governed by the classic SUVAT equations, where u is the initial velocity (positive = downward), g is the local gravitational acceleration, t is the time elapsed and d is the distance fallen:
d = u·t + ½·g·t²
v = u + g·t
v² = u² + 2·g·d
The calculator picks the correct rearrangement for whichever variable you choose to solve for:
- Distance: direct substitution into d = u·t + ½·g·t².
- Time (u = 0): t = √(2d / g). When u is non-zero, the quadratic ½g·t² + u·t − d = 0 is solved via the positive root of the discriminant.
- Final velocity: v = √(u² + 2·g·d), derived from the energy equation v² = u² + 2·g·d.
- Initial velocity: rearranged as u = d/t − ½·g·t.
- Gravitational acceleration: rearranged as g = 2(d − u·t) / t², useful for deducing g from experiment. The result is matched to the nearest known body as a cross-check.
The tool also computes kinetic energy per unit mass (½v²) and potential energy lost per unit mass (g·d) — in a vacuum they are equal, confirming energy conservation and providing a useful sanity check.
Worked example — skydiver exit
A skydiver exits an aircraft at 4,000 m altitude with zero vertical speed. Ignoring air resistance (so this gives the theoretical maximum):
- Time to fall 4,000 m from rest: t = √(2 × 4000 / 9.80665) = √815.7 ≈ 28.56 s
- Impact speed: v = g·t = 9.80665 × 28.56 ≈ 280 m/s (about 1,008 km/h)
- KE per kg at impact: ½ × 280² = 39,200 J/kg
In reality a skydiver reaches terminal velocity around 55 m/s in the spread-eagle position because of air drag — emphasising that this calculator models ideal (vacuum) free fall.
Planet comparison — 100 m drop from rest
| Body | g (m/s²) | Fall time (s) | Impact speed (m/s) |
|---|---|---|---|
| Earth | 9.807 | 4.52 | 44.3 |
| Mars | 3.72 | 7.33 | 27.3 |
| Moon | 1.62 | 11.1 | 18.0 |
| Venus | 8.87 | 4.75 | 42.1 |
| Mercury | 3.7 | 7.35 | 27.2 |
| Jupiter | 24.79 | 2.84 | 70.4 |
Formula note
The equations assume a uniform gravitational field — valid near a planet’s surface where altitude is small relative to the planet’s radius. For falls that cover a significant fraction of the planet’s radius, the inverse-square law F = GMm/r² gives a larger correction. For all practical physics problems, exam questions and engineering estimates at ordinary altitudes, the constant-g SUVAT model is exact.
The standard value of g on Earth is defined as exactly 9.80665 m/s² by the International Bureau of Weights and Measures (BIPM), though local values vary from about 9.764 m/s² (high altitude equator) to 9.834 m/s² (polar sea level).