How To Calculate Free Fall

Free fall describes the motion of an object when gravity is the only force acting on it. There is no air resistance in the model, no thrust and no support, so the object accelerates downwards at a constant rate of about 9.81 metres per second squared.

The defining feature is that acceleration is constant and independent of mass. A hammer and a feather dropped in a vacuum fall at exactly the same rate, and the famous demonstration on the Moon confirmed it. In air they behave differently only because of drag.

Because the acceleration is constant, the distance fallen grows with the square of time. After one second an object has dropped about 4.9 metres, after two seconds about 19.6 metres, and after three seconds about 44.1 metres. Each second it falls much further than the last.

Speed, by contrast, grows linearly with time. Gravity adds 9.81 metres per second of speed every second, so after two seconds the object is moving at about 19.6 metres per second regardless of how heavy it is. That linearity is what makes the impact speed calculation a single multiplication.

Running the calculation from a known height means solving for time first. Rearranging the distance formula gives time equal to the square root of twice the height divided by gravity. From twenty metres that is the square root of about 4.08, which is a shade over two seconds.

Once the time is known, the impact speed is simply gravity multiplied by time. From twenty metres that gives about 19.81 metres per second, which converts to roughly 71 kilometres per hour. That is the speed at which a dropped object arrives, and it is why hard hats exist.

The formulas assume the object starts from rest. If it is thrown downwards with an initial speed, that speed has to be added to the gravity term, and the distance formula gains an extra term for the initial velocity. Dropping is the simple case; throwing is not much harder but needs the extra term.

Air resistance is the model's main limitation. For a dense, compact object falling a short distance the error is small, but for a light or large object it becomes dominant quickly. A feather, a parachute or a sheet of paper reaches terminal velocity and stops accelerating within seconds.

Terminal velocity is where drag exactly balances weight, and the object then falls at a constant speed rather than accelerating. A human in free fall reaches it at around fifty five metres per second, which is why skydivers stop getting faster after about twelve seconds.

Gravity varies slightly across the Earth's surface, from about 9.78 at the equator to about 9.83 at the poles, because the planet is not a perfect sphere and it rotates. For any calculation shorter than a few kilometres the difference is far smaller than the uncertainty in your measurement of the height.

The same equations describe any object in free fall, including one thrown upwards. On the way up the speed decreases at the same rate, the object momentarily stops at the highest point, and then it falls back down with the motion mirroring the ascent exactly.