terminal velocity

Terminal velocity is the speed at which a body falling through a fluid stops accelerating, because the drag force pushing up has grown to match the weight pulling down. From that point on the net force is zero, and by Newton's second law the velocity is constant — the body coasts down at the same speed indefinitely.

In the linear-drag regime (slow motion in a viscous fluid), setting m·g = b·v_t gives v_t = m·g / b. A ball-bearing dropped in a tall tube of glycerine reaches this speed within fractions of a second and then sinks at a steady rate. The approach is exponential, with time constant τ = m/b: after one τ the ball is at 63% of v_t, after three it is at 95%, after five it is within 1%.

In the quadratic-drag regime (fast motion in air), setting m·g = ½ρC_d A v_t² gives v_t = √(2m·g / ρC_d A). A 90 kg skydiver in a belly-down posture reaches about 55 m/s — the stable free-fall speed that has been verified thousands of times in sport parachuting. Pulling the ripcord increases A dramatically, drops v_t to something like 6 m/s, and allows a survivable landing.

Terminal velocity is why raindrops don't kill people, why small animals survive falls that would break larger ones, and why parachutes work at all. The dependence on size is the key: for geometrically similar objects, v_t scales as √(L), so bigger things fall faster. A cat dropped from a second-storey window is in trouble; a mouse dropped from the same height walks away.