How to calculate free fall
Updated on June 30, 2026 · by Rafael Rossi
Free fall is the motion of a body acted on only by gravity, with no air resistance. It is the simplest example of accelerated motion — and the one that most contradicts intuition, because in it the object's mass does not matter.
The formulas
- Distance fallen:
h = g·t² ÷ 2 - Speed after t seconds:
v = g·t - Speed after falling h metres:
v = √(2·g·h) - Time to fall h metres:
t = √(2h ÷ g)
Where g ≈ 9.8 m/s² (32.2 ft/s²) at Earth's surface. Note that mass appears in none of them.
Worked example
An object falls 33 feet (10 m) — about a third-floor window. How long does it take, and how fast does it land?
- t = √(2 × 10 ÷ 9.8) = √2.04 = 1.43 seconds
- v = √(2 × 9.8 × 10) = √196 = 14 m/s, or 31 mph
A second and a half, and the object reaches the ground at residential-street speed.
Fall table
| Height | Time | Impact speed |
|---|---|---|
| 3 ft | 0.43 s | 9 mph |
| 16 ft | 1.00 s | 22 mph |
| 33 ft | 1.43 s | 31 mph |
| 66 ft | 2.02 s | 44 mph |
| 148 ft | 3.03 s | 67 mph |
The table shows why work at height is so heavily regulated: a fall of just 16 feet — a storey and a half — already lands at over 20 mph.
Why mass does not matter
This is the part almost everyone gets wrong when guessing. A hammer and a feather, in a vacuum, fall exactly together.
The reason is two laws cancelling. The force of gravity on a body is proportional to its mass (F = m·g), but its resistance to being accelerated — inertia — is also proportional to mass (F = m·a). Set them equal: m·g = m·a, and mass drops out of both sides. What remains is a = g, identical for every body.
The most famous demonstration was performed on the Moon in 1971 by astronaut David Scott: a hammer and a falcon feather released together hit the surface at the same instant, because there is no air there.
What air changes
On Earth, air changes everything for light or wide objects. Air resistance grows with the square of speed until it equals the object's weight — at that point acceleration stops and it falls at a constant terminal velocity.
That is why a sheet of paper drifts down while the same sheet crumpled drops fast: the mass is identical, but the area exposed to air changed. Terminal velocity for a human in a belly-down position is roughly 120 mph, and considerably higher head-down — which is also why a skydiver's body position matters before the canopy opens.
For dense, compact objects falling a few metres, the difference is negligible and the formulas above hold well.
A practical use: measuring the depth of a well
The time formula inverts easily and becomes a field tool. Drop a stone into a well, time it until you hear the impact, and calculate:
h = 9.8 × t² ÷ 2
Three seconds would give 145 feet — but there is a correction almost every version of this tip forgets: the sound also takes time to travel back up. At 1,125 ft/s, your stopwatch is measuring the fall plus the sound's return. Solving both stages together, the real depth is about 133 feet.
The error grows with depth, because the fall accelerates and the sound does not: in a 330-foot well, ignoring the sound would have you estimate 374 feet — 13% too deep.
Frequently asked questions
Can a penny dropped from a skyscraper kill someone? No. It is the most widespread free-fall myth, and it ignores terminal velocity precisely: a penny is light, flat and tumbles as it falls, reaching a terminal velocity somewhere around 25 to 40 mph. It stings; it does not kill. A bolt or a hand tool, dense and aerodynamic, is a genuine hazard.
What if the object is thrown downward? Add the initial speed: v = v₀ + g·t and h = v₀·t + g·t² ÷ 2. The formulas above are the special case where v₀ = 0.
Why is g 9.8 and not 10? 9.8 m/s² is the average at Earth's surface; the real value ranges from about 9.78 at the equator to 9.83 at the poles, because of the planet's oblateness and rotation. Using 10 simplifies the arithmetic with a 2% error, acceptable for an estimate.
Does a heavier object really land first in practice? In air, yes, when both have similar shapes — because air resistance weighs proportionally less on the denser one. It is not gravity pulling harder: it is air holding back less.