Double the Drop and the Speed Rises 41 %, but the Energy Doubles

The air is missing from this calculation

Everything above assumes a vacuum, and how badly that assumption fails depends on the object and the distance. For something compact and dense over the first twenty or thirty metres the answer is close; for a sheet of paper or a feather it is wrong immediately, because drag grows with area while gravity grows with mass. Drag also grows roughly with the square of the speed, so a fall long enough to reach high speed stops accelerating altogether. In vacuum arithmetic a body reaches 50 metres per second after about 127 metres of fall; in air the acceleration has faded long before that, and the real answer at such distances needs the drag term rather than this formula.

Mass does not appear, and that is the whole point

Neither the time nor the speed contains the mass of the object, which is why in a vacuum a hammer and a feather land together — a result demonstrated on the Moon in 1971 precisely because there is no air there to spoil it. Change g instead and everything moves: the same 20-metre drop at the Moon's 1.62 m/s² takes 4.97 seconds and lands at 8.05 metres per second, 29 km/h. The mass reappears only when energy is asked for, since that is mgh, and reappears in air too, because a heavier object of the same shape reaches a higher terminal speed.

Frequently asked questions

Why is g 9.81 and not exactly 9.8?

Because it is a measurement rather than a definition, and it varies with where you stand. The Earth is not a perfect sphere and it spins, so gravity is stronger near the poles than at the equator, and it weakens with altitude. The conventional standard value is 9.80665 m/s², and 9.81 is that rounded. For a 20-metre drop the variation across the planet changes the answer by less than a hundredth of a second, so any of these values will do — the number matters when the calculation is a gravimetric survey, not when it is a falling object.

Does an initial downward push change much?

Less than intuition expects on short drops, because the speeds add in quadrature rather than directly. Throwing an object down at 5 m/s from 20 metres gives an impact speed of √(19.81² + 5²) = 20.43 m/s, a gain of about 3 %. The push matters much more for the time, which is where a thrown object gains its lead — and it matters not at all for a sideways throw, since horizontal motion has no effect on how long the fall lasts. That last fact is what makes a bullet fired horizontally and one dropped from the same height reach the ground together.