Hailstone Terminal Velocity

A falling hailstone accelerates until drag balances weight, and after that it falls at a constant speed. Set the two equal — weight $\rho_i g \pi D^3/6$ against drag $\tfrac{1}{2}C_d \rho_a v^2 \pi D^2/4$ — cancel what cancels, and $v_t = \sqrt{4gD\rho_i/(3C_d\rho_a)}$ falls out. The diameter appears under a square root, which is the single most useful thing to know about hail: a stone four times as wide falls only twice as fast.

Some calibration, all at 900 kg/m³ ice, sea-level air and $C_d = 0.5$. A 20 mm marble arrives at 19.6 m/s, about 70 km/h. A 45 mm golf ball at 29.4 m/s. A 100 mm softball at 43.8 m/s, which is close to 160 km/h. Those figures match the published field measurements well enough that the simple sphere model has earned its place.

But the square root understates the danger, because damage follows energy. Kinetic energy goes as $mv^2$, mass goes as $D^3$, and $v^2$ goes as $D$ — so energy goes as $D^4$. Quadrupling the diameter doubles the speed and multiplies the impact energy by two hundred and fifty-six. That fourth power is why hail damage thresholds are so sharp: 20 mm stones bruise a roof and 50 mm stones destroy it, and the speed barely changed.

A thunderstorm is not still air, and this matters in both directions. Read the equation as a balance rather than as a fall and it tells you the updraught speed needed to suspend a stone of a given size — which is why hail size is used as a proxy for storm severity. A stone only grows large because an updraught held it in the growth zone long enough; a 30 m/s updraught can suspend a stone until it is about 47 mm across, and a supercell managing 50 m/s can hold one to well over 10 cm. On the way down, the stone falls through a downdraught that is itself moving downward, so ground impact speeds can exceed the still-air terminal velocity. And horizontal wind adds vectorially, which is what turns vertical hail into damage on one wall of a house and none on the other three.

The inputs are softer than they look, and the page should say so. $C_d = 0.5$ is the textbook smooth sphere. Measurements on real hailstones scatter from roughly 0.4 to 0.8 depending on surface roughness, lobes, spikes and whether the stone tumbles or spins — and since $v_t \propto C_d^{-1/2}$, that range alone is about ±20 %. Density runs from 800 kg/m³ for a stone with trapped air and rime up to 917 for clear solid ice. Real large hail is frequently not spherical at all, and a lobed stone's effective frontal area is not $\pi D^2/4$, so the diameter you measure with a ruler is not quite the diameter the equation wants.

Altitude is the one people forget. Air density falls roughly 10 % per kilometre near the ground, and $v_t \propto \rho_a^{-1/2}$, so identical hail lands about 8 % faster at 1,500 m on the Alberta prairie than at sea level — around 17 % more impact energy for the same stone. Hail-prone regions tend to be high ones, which is not a coincidence: the same elevation that puts the freezing level close to the ground also thins the air the stone falls through.