Terminal velocity calculator
Enter mass, shape and area, get the terminal velocity.
What is terminal velocity?
The steady speed a falling object reaches when air drag balances its weight: v = √(2mg ÷ (ρCA)), where m is mass, ρ the air density, C the drag coefficient and A the area facing the airflow (OpenStax University Physics).
Example: a skydiver
An 85 kg skydiver falling spread-eagle (C = 1.0, A = 0.70 m²) in sea-level air: √(2 × 85 × 9.807 ÷ (1.225 × 1.0 × 0.70)) = 44.1 m/s, about 99 mph. OpenStax, using 1.21 kg/m³ for air, gets 44 m/s.
What air density does it use?
1.225 kg/m³, the U.S. Standard Atmosphere value at sea level and 15 °C. Air is thinner higher up, so objects fall faster there; enter your own density if you know it.
Where do the drag coefficients come from?
The OpenStax table of typical values: a Toyota Camry 0.28, a sphere 0.45, a bicycle 0.90, a horizontal skydiver 1.0, a flat plate 1.12. Real values depend on shape and speed.
How is drag force calculated?
F = ½ C ρ A v². It grows with the square of speed — twice as fast means four times the drag.
How long does it take to reach terminal velocity?
The speed approaches the limit gradually; with this quadratic drag law v(t) = v_t · tanh(g·t ÷ v_t). For the skydiver in the example (v_t = 44.1 m/s) the speed reaches 95% of terminal after about 8.2 seconds. It never reaches 100% exactly, which is why a practical statement is "close to" terminal velocity.
Why do heavy and light objects fall at different speeds in air?
Without air, all objects fall at the same rate. With air, drag depends on the area and shape and grows with speed squared, while weight depends on mass. A heavier object of the same shape needs more speed before drag equals its weight, so its terminal velocity is higher: it grows with the square root of the mass.
Where the formula comes from
Two forces act on a falling object: weight mg downward and drag upward, F = ½ C ρ A v². The object speeds up until the two are equal. Setting mg = ½ C ρ A v² and solving for v gives v = √(2mg ÷ (ρ C A)). The formula therefore assumes that drag follows the quadratic law, which is a good model for objects of everyday size at high speed. It is not valid for very small particles, such as dust or fog droplets, where the drag is closer to proportional to speed, and it ignores buoyancy, which matters only for light objects. The mass is entered in kilograms or pounds and the area in square meters or square feet, and the calculator converts them.
Worked example: an 85 kg skydiver
With C = 1.0, A = 0.70 m² and sea-level air at 1.225 kg/m³, v = √(2 × 85 × 9.80665 ÷ (1.225 × 1.0 × 0.70)) = 44.1 m/s, or 98.6 mph. At that speed the drag is 834 N, equal to the weight of 834 N, which is the definition of terminal velocity. Head down, a diver presents a smaller area and a streamlined shape; with hypothetical values of C = 0.7 and A = 0.25 m² the result is 88.2 m/s (197 mph). Thin air also matters: at about 3,000 m the standard-atmosphere density is roughly 0.909 kg/m³, and the same belly-down skydiver falls at 51.2 m/s.
A raindrop as a rough check
Treat a raindrop as a water sphere 3 mm across (radius 1.5 mm), which has a mass of 14.1 milligrams. With C = 0.45 and a cross-section of π × 1.5² mm², the formula gives 8.4 m/s (19 mph). Real drops flatten as they fall and their drag coefficient changes with size, so this is an order of magnitude check on the model rather than a measurement.
Common mistakes
Entering the total surface area rather than the area facing the airflow (the cross-section) produces a speed that is far too low. Using grams or pounds in a field that expects the other unit is another trap; use the unit menus. A drag coefficient is not a constant for an object: a skydiver's changes with posture, and a ball's changes with speed and surface. And a parachute changes everything: a canopy is a large area with a high coefficient, which lowers the result to a few meters per second, so the skydiver example applies only to the free-fall phase.
Sources: OpenStax University Physics, Volume 1, section 6.4 (drag force and terminal speed; table of drag coefficients); standard gravity 9.80665 m/s² and the mile (NIST SP 811); air density 1.225 kg/m³ at sea level and about 0.909 kg/m³ at 3,000 m from the U.S. Standard Atmosphere 1976. Related: acceleration calculator, kinematics calculator and force calculator.