Terminal Velocity

Terminal velocity: v = √((2·m·g)/(ρ·C_d·A))

Terminal velocity is reached when aerodynamic drag equals weight, so net force (and acceleration) becomes zero. In the laminar regime (low Reynolds), Stokes drag applies: F_d = 6π·μ·R·v. In the turbulent regime, drag is quadratic: F_d = (1/2)·ρ·C_d·A·v², yielding v_term = √((2·m·g)/(ρ·C_d·A)). A 1 cm steel ball in water settles at ~2 m/s. A skydiver in belly-to-earth position reaches roughly 55 m/s (200 km/h); in a head-down dive, ~85 m/s (310 km/h). A raindrop tops out near 9 m/s — without air, even a pin would fall faster than any raindrop. Heavier or smaller-area objects have higher v_term; lighter or larger-area objects (parachutes, leaves) have lower v_term.

Applications

Skydiving and parachute design, ballistics of unguided bombs (gravity bombs), industrial spray atomization, sedimentation analysis (centrifugation, decantation in water treatment), meteorology of precipitation, and dust dispersion in occupational hygiene.

FAQ

Why does a parachute slow a fall so dramatically? It increases area A by ~50× and drag coefficient C_d, which reduces v_term by roughly √50 ≈ 7×, from ~55 m/s to ~5–7 m/s — safe for landing.

Does mass affect terminal velocity? Yes — v_term scales with √m for fixed area and shape. That's why a heavier skydiver in the same posture falls faster than a lighter one.

How long until terminal velocity is reached? Typically 5–10 seconds and 300–500 m for a human skydiver in belly position — depends on initial conditions and posture changes.