Terminal Velocity Calculator

How to use this calculator

Enter the object’s mass in kilograms and its cross-sectional area in square metres. Select a drag coefficient from the presets (skydiver spread-eagle, skydiver head-down, sphere, cylinder) or enter a custom value. Choose air density from the altitude presets or enter a value manually. Press Calculate to see terminal velocity, drag force at terminal velocity, and the time constant for the approach.

What is terminal velocity?

Terminal velocity is the maximum speed a falling object reaches when the drag force from the air equals the gravitational force pulling it down. At this point, net force is zero and acceleration stops. The object continues falling at constant speed.

Every falling object accelerates initially. As speed increases, drag force increases. When drag exactly balances weight, acceleration reaches zero and the object moves at terminal velocity.

The terminal velocity formula comes from setting the drag force equal to the weight:

Solving for v:

Where:

• m = mass (kg)
• g = gravitational acceleration (9.81 m/s²)
• ρ = air density (kg/m³)
• Cd = drag coefficient (dimensionless)
• A = cross-sectional area (m²)

The drag force equation

Drag force is the resistance a fluid (air, water) exerts on a moving object. For turbulent flow (which applies to most objects falling at significant speed), the drag force is:

This quadratic dependence on velocity is important. Double the speed and drag force quadruples. This is why terminal velocity depends on the square root of weight: a four-times heavier object reaches only twice the terminal velocity (all else equal).

The drag coefficient Cd captures the shape’s aerodynamic efficiency. A streamlined teardrop shape has Cd near 0.04. A flat plate perpendicular to flow has Cd near 1.28. A sphere has Cd near 0.47. A cube tumbling in the air has Cd near 1.05.

Air density ρ varies with altitude, temperature, and humidity. At sea level (15°C), ρ = 1.225 kg/m³. At 10 km altitude (where commercial aircraft cruise), ρ drops to about 0.414 kg/m³. This is why terminal velocity is higher at altitude: less air resistance.

Terminal velocity in skydiving

Skydiving provides the most familiar human example of terminal velocity. A skydiver’s terminal velocity depends on body position, mass, and equipment.

Spread-eagle position: The skydiver faces down with arms and legs spread wide. Cross-sectional area is large (roughly 0.7 m²) and Cd is high (around 1.0). Terminal velocity for an average person (75 kg) is approximately 53 m/s (190 km/h). This is the standard freefall position used for stable skydiving.

Head-down position: The skydiver points head-down, arms against the body. Cross-sectional area drops to roughly 0.18 m² and Cd falls to about 0.7. Terminal velocity increases to approximately 90 m/s (320 km/h). Speed skydivers use this position.

Extreme positions: Wingsuit skydivers use purpose-built suits with fabric between the limbs, dramatically increasing area and Cd. Terminal velocity drops to around 40 m/s (144 km/h), allowing horizontal travel of 2-3 km for every 1 km of altitude lost.

World record: Felix Baumgartner’s 2012 stratospheric jump from 39 km altitude reached 377 m/s (Mach 1.25). The air is so thin at that altitude that terminal velocity briefly exceeds the speed of sound before increasing air density slows him.

The parachute effect

A parachute dramatically lowers terminal velocity by increasing both cross-sectional area and drag coefficient. A typical round parachute has A ≈ 50 m² and Cd ≈ 1.5. This reduces terminal velocity to approximately 5-6 m/s (18-22 km/h), a safe landing speed.

Ram-air parachutes (the rectangular canopies used in sport skydiving) are more aerodynamically sophisticated. They generate lift as well as drag, functioning as gliding wings with lift-to-drag ratios around 3:1. They reach terminal velocities of about 6-7 m/s vertically, but the pilot can control direction and even flare for a near-zero landing speed.

Terminal velocity for small objects

Small objects have much lower terminal velocities than large ones, because mass scales as the cube of size while cross-sectional area scales as the square. A small object has a higher area-to-mass ratio.

Object	Mass	Terminal velocity
Raindrop (large, 5 mm)	~0.065 g	~9 m/s
Raindrop (small, 1 mm)	~0.0005 g	~4 m/s
Hailstone (2 cm)	~4 g	~20 m/s
Baseball	145 g	~42 m/s
Golf ball	46 g	~32 m/s
Skydiver (spread-eagle)	~80 kg	~53 m/s
Skydiver (head-down)	~80 kg	~90 m/s

A grain of sand falls slowly (around 0.2 m/s), which is why dust particles stay suspended in air for extended periods. This principle is used in dust cyclone separators, where centrifugal force replaces gravity to separate particles by size.

Approaching terminal velocity

An object does not instantly reach terminal velocity: it accelerates toward it asymptotically. The equation of motion during the fall is:

This differential equation has the solution:

Where tanh is the hyperbolic tangent function. The time constant τ = v_t / g tells you how quickly the approach happens. After time τ, the object has reached about 76% of terminal velocity. After 2τ, about 96%.

Terminal velocity in water and other fluids

Terminal velocity exists in any fluid, not just air. The same formula applies with the appropriate fluid density.

Water is about 800 times denser than air. Objects falling in water reach much lower terminal velocities. A skydiver-sized object falling in water would have a terminal velocity of only about 1.5-2 m/s. This is why a belly-flop from a high diving board can cause significant injury: the water’s resistance is enormous compared to air.

For very small particles in viscous fluids (low Reynolds number), Stokes’ law replaces the turbulent drag formula:

Where μ is dynamic viscosity and r is the particle radius. The terminal velocity under Stokes’ law is:

This is the basis of sedimentation analysis in geology and material science. Fine clay particles have such low terminal velocities in water that they can remain suspended for weeks, creating the turbid appearance of mud puddles after rain.

Terminal velocity on other planets

The terminal velocity formula contains atmospheric density and gravitational acceleration, both of which vary between planets.

Mars: Gravity = 3.72 m/s², atmospheric density ≈ 0.02 kg/m³ (very thin CO₂ atmosphere). For the same skydiver, v_t = √(2 × 80 × 3.72 / (0.02 × 1.0 × 0.7)) = √(595.2 / 0.014) = 206 m/s. Despite lower gravity, the thin atmosphere means terminal velocity is much higher. This is why Mars entry vehicles need very large parachutes and supplemental retrorockets.

Venus: Gravity = 8.87 m/s², atmospheric density ≈ 65 kg/m³ (extremely thick CO₂ atmosphere, 90× Earth sea level). Terminal velocity for the same skydiver = √(2 × 80 × 8.87 / (65 × 1.0 × 0.7)) = √(1419 / 45.5) = 5.6 m/s. Falling through Venus’s atmosphere is almost like falling through water. The Soviet Venera probes used only parachutes at high altitude and then free-fell slowly through the dense lower atmosphere.

Jupiter: No solid surface, but the upper atmosphere has similar density to Earth. The extreme gravity (24.8 m/s²) would produce very high terminal velocities before reaching the dense lower layers.

Reynolds number and when the formula applies

The drag formula F = ½ρCd A v² applies in the turbulent flow regime. For very small, slow-moving objects, flow is laminar and Stokes’ law applies instead. The Reynolds number determines which regime applies:

Where L is a characteristic length (diameter for spheres), μ is dynamic viscosity.

• Re < 1: Stokes’ law (viscous, laminar flow)
• 1 < Re < 1000: transition region
• Re > 1000: turbulent drag law applies

For most objects of everyday size falling through air, Re is well above 1000 and the standard terminal velocity formula gives accurate results. For fine aerosol particles, pollen, and bacteria, Stokes’ law applies.

Engineering applications of terminal velocity

Particle separation: Industrial cyclones, centrifuges, and settling tanks exploit terminal velocity differences between particles of different sizes or densities. Heavier particles settle faster and can be separated from lighter ones.

Spray drying: Hot spray droplets must dry before reaching the bottom of the spray tower. The droplet’s terminal velocity determines the tower height needed. Smaller droplets have lower terminal velocity (longer residence time) and dry more completely.

Hail damage engineering: Hail impact energy is ½mv_t². Hailstones of 2 cm diameter reach about 20 m/s. The impact energy guides the design of roofing materials, vehicle panels, and greenhouse glass.

Parachute engineering: Military cargo drops require precise terminal velocity control to ensure safe landing of equipment. Heavy loads need larger parachutes; multiple smaller parachutes can be clustered, with their areas adding together.

Aerosol dispersion modeling: Atmospheric models tracking pollution, volcanic ash, or disease aerosols use terminal velocity to calculate how quickly particles deposit. Fine ash (< 1 μm) may stay aloft for months and circle the globe.