Terminal Velocity Calculator

What is terminal velocity?

A body that is dropped speeds up at first, pulled down by its weight. As it gains speed, the air pushes back harder, because drag grows with the square of speed. Eventually the upward drag matches the downward weight, the net force becomes zero and the speed stops changing. That steady speed is the terminal velocity.

It depends on the object (its mass, its frontal area and how streamlined it is) and on the fluid it moves through (mainly its density). A skydiver, a raindrop, a feather and a hailstone all reach different terminal velocities for exactly these reasons.

How to use the calculator

• Mass of the falling object, from milligrams up to US tons.
• Cross-section area: the area of the object as seen from the direction of fall, also called the frontal or projected area.
• Drag coefficient: a number that captures the shape. It is preset to 0.294 and can be changed to anything above zero.
• Air density: preset to 1.225 kg/m³, the average near sea level. Change it for high altitude, other temperatures, or another fluid such as water (about 1,000 kg/m³).
• Gravity: preset to 1 g. You can switch the unit to m/s² or enter the pull of another planet.

The answer is given in metres per second and kilometres per hour, and in feet per second and miles per hour. All fields are required.

The terminal velocity formula

At terminal velocity the weight equals the drag force: m × g = ½ × ρ × v² × C_d × A. Solving for the speed gives:

v_t = √( 2 × m × g ÷ (ρ × A × C_d) )m = mass, g = gravity, ρ = fluid density, A = cross-section area, C_d = drag coefficient

Two consequences are worth remembering. The speed grows with the square root of the mass, so a body four times heavier falls only twice as fast. And it falls with the square root of the product of area and drag coefficient, so making an object bluffer or wider slows it down, which is how a parachute works.

Worked examples

A compact falling body. Take m = 80 kg, A = 0.5 m², C_d = 1.0 and sea-level air (ρ = 1.225 kg/m³, g = 9.80665 m/s²). Then v_t = √(2 × 80 × 9.80665 ÷ (1.225 × 0.5 × 1.0)) = √2,561.7 ≈ 50.6 m/s, which is about 182 km/h or 113 mph.

Four times the mass. With m = 320 kg and everything else unchanged, the result doubles to about 101.2 m/s.

Thin air. At roughly 3,000 m the air density is near 0.909 kg/m³. The first object then falls at √(2,561.7 × 1.225 ÷ 0.909) ≈ 58.8 m/s, about 16 percent faster than at sea level.

A raindrop. A drop 2 mm across has a mass of about 4.19 × 10⁻⁶ kg (4/3 × π × 0.001³ × 1,000) and a frontal area of π × 0.001² ≈ 3.14 × 10⁻⁶ m². Using C_d ≈ 0.47 for a sphere, v_t ≈ 6.7 m/s, close to the roughly 6.5 m/s measured for real drops of that size.

Under a canopy. A 90 kg load hanging from a 25 m² canopy with C_d = 1.5 settles at √(2 × 90 × 9.80665 ÷ (1.225 × 25 × 1.5)) ≈ 6.2 m/s, around 22 km/h.

How long does it take to get there?

The object approaches terminal velocity gradually rather than reaching it at a definite moment. With constant drag coefficient, the speed after t seconds of falling from rest is v(t) = v_t × tanh(g × t ÷ v_t) and the distance fallen is s(t) = (v_t² ÷ g) × ln cosh(g × t ÷ v_t).

For the 80 kg example above (v_t ≈ 50.6 m/s), 95 percent of the terminal speed is reached after about 9.5 s and a fall of roughly 300 m. A heavier or more streamlined object needs longer and falls farther before it levels off.

To follow how the speed builds up with time, use the acceleration calculator; for an average over a trip the average speed calculator and the speed calculator apply.

Assumptions and limits

• Constant air density. Real air thins with altitude, so a high-altitude fall starts faster and slows as it descends.
• Constant drag coefficient. In practice C_d varies with speed, tumbling and orientation.
• Quadratic drag only. At very low speeds, such as a fine dust particle, drag is proportional to speed instead and the formula does not apply.
• Buoyancy is ignored. That is negligible in air but matters for a body falling through water.

Results are estimates for learning and planning, not safety advice. Do not rely on them for skydiving, parachute selection or any activity where injury is possible. See the disclaimer.

If you also need the fluid's mass per volume, the density calculator helps. For the upward push in water see the buoyant force calculator, and for the collision at the end of a fall the impact force calculator.