Terminal Velocity

A falling body speeds up until drag, which grows as v², matches its weight; after that the net force is zero and the speed locks in at v_t = √(2mg ⁄ ρA C_d). For a belly-to-earth skydiver — roughly 80 kg, 0.7 m² of frontal area, C_d ≈ 1.0, air at 1.225 kg/m³ — that works out to about 43 m/s, near the familiar 190 km/h. Pull into a head-down dive and A collapses, pushing terminal velocity past 90 m/s; deploy a parachute and A jumps by two orders of magnitude, dropping it to a survivable 5 m/s.

Density matters as much as shape, which is why Felix Baumgartner exceeded the speed of sound in 2012 at 39 km altitude: with ρ perhaps 1% of sea-level air, v_t rises roughly tenfold. The formula also explains why small animals survive falls that kill large ones — mass grows with the cube of size while area grows with the square, so a mouse's terminal velocity is a fraction of a horse's. Note that v_t is an asymptote, not a speed reached at a definite moment; a skydiver is within a few percent of it after about 12 seconds.