Free Fall Calculator

This calculator links the three quantities of a free fall: h = ½gt², v = gt and v² = 2gh. Enter whichever you know above and it works out the other two, with optional impact energy if you supply a mass.

Everything falls at the same rate

Ignoring air, a feather and a hammer released together land together. Gravity accelerates every object equally regardless of mass, because a heavier object needs proportionally more force to accelerate and gravity supplies exactly that much more.

David Scott demonstrated this on the Moon in 1971 during Apollo 15, dropping a hammer and a falcon feather in vacuum. They struck the surface at the same moment. On Earth the same experiment fails only because air resistance affects the feather far more than the hammer.

A worked example

Drop something from 10 metres on Earth.

• Time: t = √(2 × 10 ÷ 9.80665) = 1.428 seconds
• Impact speed: v = 9.80665 × 1.428 = 14.00 m/s, which is 50.4 km/h

Ten metres is roughly a three storey building, and the impact speed is comparable to being hit by a car in a residential street. That comparison is worth keeping in mind around any working height.

How the numbers grow

Drop height	Fall time	Impact speed
1 m	0.452 s	4.43 m/s (15.9 km/h)
10 m	1.428 s	14.00 m/s (50.4 km/h)
50 m	3.193 s	31.32 m/s (112.7 km/h)
100 m	4.516 s	44.29 m/s (159.4 km/h)

Notice that a hundredfold increase in height produces only a tenfold increase in speed, because speed rises with the square root of the height. It also means the time to fall a long way is shorter than most people expect: a 100 m drop takes under five seconds.

Air resistance, and when this stops being true

These figures assume a vacuum. In air, drag rises with the square of speed until it balances weight, after which the object stops accelerating and falls at a constant terminal velocity.

A skydiver in a spread position reaches about 53 m/s, near 190 km/h, after roughly 12 seconds and 450 m of fall. Beyond that point the calculations here overstate the speed, and increasingly so. For a dense object over a short drop, which covers most practical questions, the error is small.

Light objects diverge much sooner. A raindrop reaches terminal velocity at around 9 m/s within a few metres, which is why rain falling from a kilometre up does no damage.

Gravity is not the same everywhere

Body	Gravity (m/s²)	Fall time from 10 m
Earth	9.80665	1.43 s
Moon	1.62	3.51 s
Mars	3.72	2.32 s
Jupiter	24.79	0.90 s

Earth's own gravity varies slightly with latitude and altitude, from about 9.78 at the equator to 9.83 at the poles, because the planet bulges and rotates. The standard value of 9.80665 is a defined average rather than a measurement of any particular place.

Reaction time, and the ruler test

A common use of these equations is measuring reaction time. Have someone hold a ruler vertically, release it without warning, and catch it. The distance it fell gives the time from t = √(2h/g).

Catching it after 20 cm corresponds to 0.202 seconds, and after 30 cm to 0.247 seconds. Typical human reaction times fall between those figures, which is why a 30 cm ruler is about the right length for the test.

Impact energy

Supply a mass and the calculator also returns the kinetic energy at impact, from ½mv². A 5 kg object dropped 10 m arrives with ½ × 5 × 196.13 = 490 J.

That figure matters more than speed for anything to do with damage or safety, because energy is what has to be absorbed. It is also why dropped tools are treated seriously on site: a small spanner falling from a few floors up carries enough energy to be lethal.

Falling sideways: projectile motion

Horizontal and vertical motion are independent. An object thrown horizontally falls at exactly the same rate as one dropped from the same height, and reaches the ground at the same moment despite travelling a long way sideways.

A ball thrown horizontally at 20 m/s from a 10 m height falls for the same 1.428 seconds as a dropped one, and in that time travels 20 × 1.428 = 28.6 m across the ground. Nothing about the horizontal speed changes the fall time. This is the basis of every trajectory calculation, and it is why a bullet fired level and one dropped from the same height land together.

Working at height

These figures explain why fall protection rules are as strict as they are. A fall from 2 m, barely above head height, reaches 6.26 m/s in 0.64 seconds. That is faster than most people can run, achieved before there is any chance to react.

The time available is the crucial part. A fall from 4 m gives 0.90 seconds from start to impact, which is roughly the length of a human reaction. There is no opportunity to correct or brace, which is why the protection has to be in place beforehand rather than depending on the person.

Timing a fall to measure depth

Drop a stone into a well, time the fall, and the depth follows from h = ½gt². Three seconds gives ½ × 9.80665 × 9 = 44.1 m.

Two things spoil the accuracy. Sound takes time to return, about 0.13 seconds from 44 m, so the measured interval is slightly long. Air resistance also slows the stone a little. Both errors push the estimate the same way, so a timed depth of this sort is usually a small overestimate.

Free fall in orbit

An object in orbit is in free fall the whole time. It is accelerating toward the Earth continuously, but moving sideways fast enough that the surface curves away as quickly as it falls.

This is why astronauts float. They are not beyond gravity, which at the International Space Station's altitude is still about 89% of its value at the surface. They and the station are falling together, so there is nothing to press against and no sensation of weight. The correct name for the condition is free fall rather than zero gravity.

Where the 9.80665 comes from

The figure is a defined standard rather than a measurement. Real gravity varies across the Earth's surface, and the standard value was fixed so results are comparable.

Two things cause the variation. The Earth bulges at the equator, so the surface there sits further from the centre, and the rotation adds a small outward effect. Gravity measures about 9.780 m/s² at the equator against 9.832 m/s² at the poles, a spread of roughly half a percent. Altitude reduces it further, by about 0.3% at the top of Everest.

For everyday calculations the difference does not matter, and 9.81 or even 9.8 is close enough. It matters for precision weighing, where a balance calibrated in one place reads differently in another.

Common mistakes

Assuming heavier falls faster. Without air resistance, mass makes no difference at all.

Applying these figures to long falls. Beyond a few hundred metres air resistance dominates and terminal velocity takes over.

Forgetting the square root. Time depends on the square root of height, so four times the height gives twice the time.

Using g as 10 for precision work. It is convenient for mental arithmetic and about 2% high.