Terminal velocity of raindrops

Why does a raindrop falling from a kilometre up arrive gently instead of at bullet speed?

Drop something from a kilometre and, with no air in the way, it arrives at the square root of 2gh — about 140 metres per second, faster than most pistol rounds. Rain falls from at least that height and lands on your face without hurting. The usual answer is "air resistance slows it down", which is true and explains almost nothing: why should resistance stop the drop at a particular speed rather than merely making it slower, and why does a heavier drop not simply overwhelm the air and keep accelerating?

Start with what makes the speed settle rather than merely reduce. Weight is fixed — it does not care how fast the drop is going. Drag is not: pushing through air means shoving aside a column of it, and the faster you go the more air per second you must shove, and the harder you must shove each parcel. Both factors scale with speed, so at everyday raindrop sizes drag grows roughly as the square of speed.

Now watch the drop from rest. Weight down, drag zero, so it accelerates. As it speeds up, drag climbs steeply, and the net force shrinks. It cannot overshoot: any excess speed produces a drag larger than the weight, which decelerates it back. That is a negative feedback loop with a fixed point, and the fixed point is where drag exactly equals weight. Ask yourself what happens after that — the answer is nothing at all. No net force, no acceleration, constant speed the rest of the way down.

How far down does this take? The natural timescale is the terminal speed divided by g. For a millimetre-ish drop falling at about 6.5 metres per second, that is under a second, and the distance is a few metres. So the kilometre in the question is a red herring: the drop reached its final speed almost immediately after leaving the cloud, and the remaining 999 metres change nothing.

Now the second half, and the more interesting one. Why is the settling speed low? Set weight against drag and see how each scales with the drop's diameter d. Weight goes as volume, so as d cubed. Drag goes as frontal area, so as d squared, times v squared. Balance them and v squared goes as d — so terminal velocity rises only as the square root of size. Quadruple the drop and you barely double its speed. Putting real numbers through mg = half rho C v squared A, with water at 1000 and air at 1.2 kilograms per cubic metre and a drag coefficient near 0.5, a 2 mm drop comes out around 6.6 metres per second, which agrees well with the standard measured value of about 6.5.

That square root is the whole reason rain is survivable. Falling speed is chained to a slowly growing function of size, and size itself is capped.

Why capped? Because a large drop is not a rigid ball. Airflow presses on its underside, and surface tension — the only thing holding it together — is weak in comparison once the drop grows past roughly five or six millimetres. The base flattens, then dishes inward, then the drop blows apart into fragments. So the atmosphere enforces its own ceiling: bigger drops do not fall faster, they cease to exist. Measured terminal velocities reflect this, rising from about 2 metres per second at half a millimetre to roughly 9 at five millimetres and then flattening — partly from breakup, partly because a big drop flattens into a shape with worse drag than a sphere, so it falls slower than the square-root law would predict.

Three refinements connect the steps.

First, the drag-goes-as-speed-squared rule is a regime, not a law. Very small droplets — the twenty-micron ones that make cloud and fog — are slow enough that viscosity dominates and drag goes as speed to the first power. There the balance gives velocity proportional to d squared, and the numbers are startling: about a centimetre per second, so a fog droplet effectively does not fall at all. That is why clouds float while rain falls, and it is a difference of regime, not of substance.

Second, drops are not tear-shaped. The classic raindrop outline is a folk image; a real falling drop is spherical when small and flattened underneath, like a bun, when large.

Third, the ceiling on drop size is a soft and contested edge rather than a hard number — drops of eight millimetres and more have been recorded in unusual conditions, and breakup depends on collisions with other drops as much as on aerodynamics alone.

A boundary worth marking: elsewhere in this collection, golf ball dimples are explained by a drag crisis — a sudden drop in the drag coefficient when the boundary layer goes turbulent — and a cyclist's descent by how mass and drag scale differently. Here the drag coefficient is treated as roughly constant, and the entire story is the equilibrium it produces.

• Why hailstones, which are ice and can grow far larger before anything breaks them up, do arrive hard enough to injure.
• How the same balance sets the descent rate of a parachute, and why area rather than mass is the lever there.

Every term the collection defines is gathered in the glossary.