Free Fall, Explained: The Distance and Velocity Equations That Actually Work

Drop a stone off a bridge and two things happen at once: it falls farther every second, and it falls faster every second. Those two facts are not the same thing, and confusing them is the single most common mistake I see in introductory physics. This post walks through the equations that describe free fall under gravity, runs one clean numerical example end to end, and then explains the one assumption that makes all of it slightly wrong in the real world.

If you just want the numbers, the free fall calculator does the arithmetic for you. But the equations are short enough that it pays to understand them, because once you do, you can sanity-check any answer in your head.

What "free fall" actually means

In physics, free fall is the motion of an object pulled by gravity alone, with nothing else acting on it. No engine, no rope, and critically, no air pushing back. On Earth that pull produces a constant downward acceleration, written as g, with a value of about 9.81 m/s². Constant acceleration is the whole story here: every second that passes, the object's downward speed increases by another 9.81 m/s, no matter how fast it is already moving or how heavy it is.

That last part surprises people. A bowling ball and a marble released together hit the ground at the same instant, because mass drops out of the equations entirely. Galileo argued this four centuries ago, and Apollo 15 commander David Scott proved it on camera in 1971 by dropping a hammer and a feather on the airless Moon. They landed together. The only reason that experiment fails on Earth is air, which is exactly the caveat we will get to at the end.

The two equations you need

Free fall from rest, meaning the object starts with zero speed, reduces to two formulas.

Distance fallen:

distance = ½ · g · t²

Velocity reached:

velocity = g · t

Here t is the time in seconds since release, and g ≈ 9.81 m/s² on Earth. Notice the shapes are different. Distance has a t² in it, so it grows with the square of time, while velocity grows in a straight line. That difference is why a falling object covers so little ground in its first instant and then seems to suddenly plummet: at one second it has dropped under 5 metres, but by three seconds it has dropped more than 44.

The factor of ½ in the distance equation is not decoration. It comes from averaging the speed over the fall: the object starts at zero and ends at g·t, so its average speed is half of g·t, and distance is average speed times time. Drop the ½ and you double every distance you calculate. That mistake alone accounts for a remarkable share of wrong homework answers.

You can also run these backwards. If you know the height instead of the time, rearrange the first equation to get t = √(2h/g), and if you know the landing speed, the height it fell from is h = v²/(2g). Same physics, solved for whatever quantity you happen to be missing.

A worked example: three seconds of falling

Let's drop an object and watch it for exactly three seconds, ignoring air resistance, on Earth.

Distance. Plug t = 3 into the distance equation:

distance = ½ · 9.81 · 3²
         = ½ · 9.81 · 9
         = 44.145 metres

So after three seconds the object has fallen about 44.1 metres, roughly the height of a 14-storey building.

Velocity. Now the speed:

velocity = 9.81 · 3
         = 29.43 metres per second

That is about 29.4 m/s, or roughly 106 km/h. After only three seconds of falling, the object is moving at highway speed. This is the part that makes high falls so dangerous and short falls so deceptively gentle: the speed is climbing fast even while the distance is still modest.

Notice how the two answers grew at different rates. Between the second and third second, the speed went up by a steady 9.81 m/s, but the distance covered in that single second was far larger than in the first second, because of the t² term. The object spent the early part of the fall barely moving and the later part racing.

Changing the planet, changing the answer

Nothing in these equations is specific to Earth except the value of g. Swap in a different gravity and every result follows. The Moon's surface gravity is about 1.62 m/s², roughly a sixth of Earth's, so the same drop takes longer and lands much slower. Mars sits in between at about 3.71 m/s². This is why lunar astronauts could stumble and recover so gently: their three-second fall would cover only about 7.3 metres and arrive at under 5 m/s.

If you are comparing worlds or running a what-if, swapping g by hand for each one gets tedious fast. That is a good moment to reach for a scientific calculator to chain the square roots and squares, or to let the free fall tool hold the equations steady while you tap between gravity presets. The physics does not change; only the constant does.

The catch: air resistance and terminal velocity

Everything above assumes a vacuum. In real air, a falling object also feels drag, a backward push that grows stronger the faster the object moves. At first gravity wins easily and the object accelerates almost exactly as the equations predict. But as speed climbs, drag climbs faster, until the two forces balance. At that point acceleration stops, the object stops speeding up, and it falls at a steady terminal velocity.

For a dense, compact object over a short drop, terminal velocity is irrelevant and the vacuum equations match reality closely. A steel ball dropped a few metres behaves almost perfectly. But a sheet of paper, an open parachute, or a skydiver in spread-eagle position reaches terminal velocity quickly and from then on falls far slower than ½gt² would suggest. A human in freefall tops out around 53 m/s; the equations, left unchecked, would have them passing that speed after less than six seconds and accelerating forever.

So treat the textbook answer as an upper bound. It tells you the fastest and farthest an object could fall if air did not exist. The lighter and flatter the object, and the longer the fall, the more the real number lags behind. If you want to follow where that kinetic energy goes once the object lands, the kinetic energy calculator picks up from the impact speed these equations give you.

Putting it together

Free fall is one of the cleanest pieces of physics you will meet: two short equations, one constant, and a single assumption. Distance goes as ½gt², velocity goes as gt, and a three-second Earth drop lands you 44.1 metres down at 29.4 m/s. Keep the factor of ½, remember that distance and speed grow at different rates, and stay honest about air resistance for anything light or long, and you will never be badly wrong. For the arithmetic and the gravity presets, the free fall calculator is right there.

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Made by Toolora · Updated 2026-06-13