Fall far enough and you stop accelerating. Air resistance grows with speed until it exactly cancels gravity, and from that moment the fall continues at one constant rate: terminal velocity. This terminal velocity calculator computes that speed from the four quantities that set it — mass, air density, frontal area, and drag coefficient — the same formula behind skydiver speeds, raindrop sizes, and why dust floats but gravel plummets.
The Terminal Velocity Formula
Drag at speed v is ½ρv²ACd. Terminal velocity is where that drag grows to equal the weight mg, and solving for v gives v = √( 2mg ÷ (ρACd) ). Read the structure before the arithmetic: mass in the numerator (heavier falls faster), everything about the air and the object’s shape in the denominator (bluffer falls slower). The square root softens all of it — doubling mass raises terminal velocity only about 41 percent, not double.
Worked Example: The Skydiver
An 80 kg skydiver in a spread-eagle position presents about 0.7 m² of frontal area with a drag coefficient near 0.7, in sea-level air (ρ = 1.225 kg/m³). The formula gives √(2 × 80 × 9.81 ÷ (1.225 × 0.7 × 0.7)) = √2,616 ≈ 51.1 m/s — about 184 km/h, the famous belly-to-earth speed instructors quote. The same diver head-down (area near 0.18 m², Cd lower) pushes past 240 km/h, which is why body position, not courage, sets the speed of a skydive.
Why Small Things Fall Slowly
Scale an object down and its mass shrinks with the cube of size while its area shrinks only with the square — so the mass-to-area ratio collapses, and terminal velocity with it. A raindrop 2 mm across reaches only about 6 to 7 m/s, which is why rain stings rather than injures. Dust motes settle at centimetres per second and effectively float. An ant’s terminal velocity is so low that falls from any height are survivable — the square-cube law is a small creature’s built-in parachute, and the reason the “mouse survives any fall, horse splashes” ladder of examples works.
Air Density: Altitude and Weather
ρ at sea level on a standard day is 1.225 kg/m³. At 3,000 m it is near 0.91, at 10,000 m about 0.41 — thinner air drags less, so terminal velocity rises with altitude. Felix Baumgartner’s stratospheric jump exploited exactly this: in near-vacuum at 39 km he passed the speed of sound before the thickening air below slowed him to ordinary terminal speeds. Temperature and humidity shift density a few percent as well; the calculator’s density field exists so those cases are one input away.
Drag Coefficient and Frontal Area
Cd summarizes shape: about 0.04 to 0.1 for a streamlined teardrop, 0.47 for a smooth sphere, around 0.7 to 1.0 for a tumbling human, 1.05 for a flat plate face-on, up to 1.3 for an open parachute canopy — whose enormous area does most of the work anyway. Area must be the frontal area, the shadow the object casts in its direction of motion. Getting A and Cd right is the whole art of applying the formula; the arithmetic afterward is one square root.
Common Mistakes
Using diameter-based or total surface area instead of frontal area inflates the drag side. Mixing a Cd measured for one orientation with the area of another does the same at one remove. Assuming terminal velocity is reached instantly — objects approach it asymptotically, covering real distance while they accelerate (a skydiver needs roughly 450 m to reach 99 percent). And applying the sea-level density at altitude, the error the density input exists to prevent.
Terminal Velocity Beyond Earth
The formula travels well. On Mars the air is a hundred times thinner (ρ near 0.02 kg/m³ at the surface), so the same skydiver would “top out” several times faster — one reason Mars landers cannot rely on parachutes alone and add retro-rockets for the final descent. On Venus, wrapped in air denser than water-like haze at 65 kg/m³ near the surface, terminal velocities collapse to a gentle drift and probes descend almost like submarines sinking. Change one input — the density — and the calculator quietly re-derives falling on another world.
Related Tools
The Physics Calculators hub connects the falling-body set — contrast this with the drag-free SUVAT Calculator and the Newton’s Second Law Calculator that drag ultimately answers to.
Frequently Asked Questions
What is the terminal velocity of a human?
About 51 m/s (184 km/h, 114 mph) belly-to-earth in a spread-eagle position, and 240 km/h or more head-down. The worked example on this page derives the belly figure from mass, area, and drag coefficient.
How is terminal velocity calculated?
v = √(2mg ÷ (ρ A Cd)) — set drag equal to weight and solve for v. Mass and gravity push it up; air density, frontal area, and drag coefficient pull it down.
Why do heavier objects reach higher terminal velocities?
At the same shape and size, more mass means more weight for the same drag area, so drag needs a higher speed to balance it. The effect runs through a square root: four times the mass only doubles terminal velocity.
How long does it take to reach terminal velocity?
It is approached, never perfectly reached. A skydiver hits about 95 percent within roughly 10 to 15 seconds (a few hundred metres); small objects settle in well under a second.
What is the terminal velocity of a raindrop?
Roughly 6 to 9 m/s depending on drop size — drops larger than about 5 mm break apart, capping raindrop speeds near 9 m/s. Modest, because a drop’s mass-to-area ratio is tiny.
Does terminal velocity change with altitude?
Yes. Air thins with altitude, drag weakens, and the balance point moves to higher speeds — stratospheric jumpers exceed the speed of sound before descending air slows them.
Can terminal velocity be zero?
Only in the limit of negligible weight or infinite drag. Practically, very light high-area objects (a feather, a dandelion seed) get so close to zero that settling looks like floating.