A falling object speeds up until air drag grows to match its weight; from then on it falls at a constant terminal velocity, v = √(2mg / ρ·A·Cd). The formula comes from setting the drag force ½ρv²A·Cd equal to the weight mg and solving for v. Heavy, dense, streamlined objects reach a high terminal velocity; light, broad, draggy ones settle to a low one. It's why a hammer plummets but a feather drifts — and why a parachute, by hugely increasing the area, brings you down safely.
Reviewed: June 20, 2026 · Author: Naveen P N, Founder — AI Calculator · Verified against: the drag-equals-weight terminal velocity v = √(2mg/ρACd).
The terminal velocity equations
The terminal velocity is reached when the drag force, which grows with the square of speed, equals the constant weight. Solving that balance for v gives the square-root formula. Mass is in the numerator, so heavier falls faster; air density, area and drag coefficient are in the denominator, so any of them rising slows the fall. Because everything is under a square root, you need a four-fold change to double or halve the terminal velocity.
Worked example — a skydiver
Scenario: An 80 kg skydiver falls belly-down with a frontal area of 0.7 m² and a drag coefficient of about 1.0, in sea-level air (ρ = 1.225 kg/m³).
The skydiver levels out at about 42.8 m/s, roughly 154 km/h. Tuck into a streamlined dive (Cd ≈ 0.5) and the terminal velocity climbs to ~60.5 m/s; flare out for more drag (Cd ≈ 1.5) and it drops to ~34.9 m/s. Deploy a parachute — multiplying the area many times over — and the same balance gives a gentle few metres per second, which is exactly what makes landing survivable.
Frequently Asked Questions
v = √(2mg/(ρ·A·Cd)). 80 kg, 0.7 m², Cd 1.0 → ≈ 42.8 m/s (154 km/h).
The steady max speed where air drag balances weight, so the object stops accelerating.
Belly-down ~50–55 m/s (190 km/h); head-down streamlined can exceed 90 m/s (320+ km/h).
More weight, same drag for a given speed, so it must go faster before drag balances. v ∝ √m.
Both in the denominator — bigger area or Cd = more drag = lower terminal velocity. That's how a parachute works.