Terminal Velocity Calculator

How to Use the Terminal Velocity Calculator

1. Enter the Mass (m) of the falling object and choose its unit (kg, g or lb).
2. Enter the Drag coefficient (Cd). Use 0.47 for a sphere, about 1.0 for a belly-down skydiver, 0.7 for a head-down skydiver, or 0.04 for a streamlined body.
3. Enter the Air density (ρ) in kg/m³, g/L or lb/ft³. Sea-level air at 15 °C is about 1.225 kg/m³; at 5 km altitude it drops to roughly 0.736 kg/m³.
4. Enter the Cross-sectional area (A) — the frontal area facing the flow — in m², cm², ft² or in².
5. Select the Gravity (g) for the location: Earth, Moon, Mars or Jupiter.
6. Read the terminal velocity in m/s, km/h, mph, ft/s and knots, along with the forces and the ballistic coefficient \(m/(C_d A)\).

What Is Terminal Velocity?

When an object is dropped, gravity accelerates it downward. But air resistance (drag) grows with the square of the speed: double the speed and the drag quadruples. As the object falls faster, the drag force keeps increasing until it exactly equals the object's weight. At that moment the forces cancel, the acceleration becomes zero, and the object continues falling at a constant speed — its terminal velocity.

Terminal velocity is not a fixed number for a given object; it depends on the medium and the object's orientation. A skydiver belly-down reaches about 50 m/s (180 km/h), but by going head-down and pulling the arms in (smaller area, lower drag coefficient) the same person can exceed 90 m/s. Open a parachute and the area explodes while \(C_d\) rises, cutting the terminal speed to a survivable 5–8 m/s. A small hailstone falls at 10–30 m/s, a raindrop only about 9 m/s.

Notice what the formula tells you physically. Terminal velocity grows with the square root of mass and gravity: heavier objects or stronger gravity mean faster fall. It shrinks with the square root of density, area and drag coefficient: denser air, a bigger frontal area or a blunter shape all slow the object down. Because mass and area act in opposite directions, a dense compact object (a cannonball) falls much faster than a light spread-out one (a sheet of paper) — this is why Galileo's feather and hammer only fall together in a vacuum.

An object never exactly reaches its terminal speed; it approaches it asymptotically. In practice, a falling body gets to about 95% of \(v_t\) after roughly \(3v_t/g\) seconds, which is a few seconds for most everyday objects in air.

Assumptions, Limitations and Practical Tips

• Quadratic drag only. The formula assumes drag is proportional to \(v^2\), which holds for typical falling objects at high Reynolds numbers. Very small, slow particles (dust, mist droplets) follow Stokes (linear) drag instead and fall slower than this formula predicts.
• Constant inputs. Real air density falls with altitude — a skydiver jumping from 10 km reaches higher speeds in the thin upper air than at sea level. For high-altitude falls, run the calculation with the density at different heights and see how \(v_t\) changes.
• Rigid, fixed-orientation object. \(C_d\) and \(A\) are treated as constant. A tumbling object or a flexible body (like a person) changes both, so treat the result as a good estimate, not an exact figure.
• No buoyancy. Buoyancy is negligible for dense objects in air but matters for light objects in water; in that case subtract the buoyant force from the weight first.
• Reference values. Useful drag coefficients: sphere 0.47, cube 1.05, streamlined body 0.04, cyclist 0.9–1.1, bald eagle in a dive as low as 0.16.

If you want to analyze the fall before terminal speed is reached — distances, times and intermediate speeds — combine this result with the Kinematic Equations Solver or the Final Velocity Calculator. The energy view of the fall is covered by the Kinetic Energy and Conservation of Energy calculators, while the Friction Force Calculator handles the analogous velocity-independent resistance on surfaces.