Terminal Velocity Calculator

What Is the Terminal Velocity Calculator?

The Terminal Velocity Calculator computes the maximum speed a falling or descending object reaches when gravitational force equals aerodynamic drag. At terminal velocity, net force is zero and acceleration stops. The formula is vt = sqrt(2mg / (rho × Cd × A)), where m is mass, g is gravity, rho is fluid density, Cd is the drag coefficient, and A is the cross-sectional area. According to the HyperPhysics drag force reference, this quadratic drag model applies when the Reynolds number is above roughly 1,000, covering virtually all everyday objects at visible speeds.

The calculator includes drag coefficient presets for 10 common shapes (sphere, cube, cylinder, streamlined body, etc.) and atmospheric density presets for altitudes from sea level to 20 km. It displays terminal velocity in m/s, km/h, and mph simultaneously. In practice, skydivers use this to compare spread-eagle vs head-down body positions, and engineers use it to verify parachute and re-entry shield sizing.

When an object falls through air, drag force increases with speed until it exactly balances the gravitational force. At that point, acceleration drops to zero and the object falls at a constant speed: its terminal velocity. This Terminal Velocity Calculator computes that speed from mass, drag coefficient, cross-sectional area, and air density, returning results in m/s, km/h, and mph in real time. A drag coefficient reference table and planetary gravity presets are included for quick exploration.

The Terminal Velocity Formula

At terminal velocity, drag equals weight: ½ × rho × Cd × A × vt² = m × g. Solving for vt: vt = sqrt(2mg / (rho × Cd × A)). For a standard spread-eagle skydiver (m = 80 kg, Cd = 1.0, A = 0.70 m²) at sea level (rho = 1.225 kg/m³): vt = sqrt(2 × 80 × 9.81 / (1.225 × 1.0 × 0.70)) = sqrt(1569.6 / 0.858) = sqrt(1829) = 42.8 m/s = 154 km/h. This matches well-documented values for belly-to-earth skydiving, as referenced in the HyperPhysics drag force reference and the US Parachute Association Skydiver's Information Manual.

The Drag Coefficient (Cd)

The drag coefficient captures how aerodynamically clean a shape is. A smooth sphere has Cd = 0.47, a cube has Cd = 1.05, and a flat plate facing the flow has Cd = 1.28. A streamlined teardrop body can reach Cd = 0.04. The reference table in this calculator covers ten common shapes. For objects not in the table, published Cd values are available from the Engineering Toolbox drag coefficient database, which includes aerodynamic data for vehicles, aircraft components, and structural shapes. When the exact shape is unknown, Cd = 1.0 is a safe conservative estimate for non-streamlined objects.

Body Position and Skydiving

Altitude and Air Density

Air density falls with altitude. At sea level, rho = 1.225 kg/m³. At 2,000 m it drops to 1.007 kg/m³, at 4,000 m to 0.819 kg/m³, and at 10,000 m (cruising altitude) to just 0.413 kg/m³. Since terminal velocity scales as 1/sqrt(rho), an object that falls at 55 m/s at sea level would reach 55 × sqrt(1.225/0.413) = 94 m/s at 10,000 m under the same conditions. This is why high-altitude skydiving produces much higher freefall speeds before canopy deployment, and why Felix Baumgartner exceeded the speed of sound during his 2012 stratosphere jump from 39 km altitude where rho was less than 0.004 kg/m³.

Parachute Design and Safe Landing Speed

Terminal velocity physics drives every parachute design decision. A safe landing speed for a trained parachutist is approximately 5–7 m/s (18–25 km/h). For an 80 kg jumper with gear, the required Cd × A product is 2mg / (rho × vt²) = 2 × 80 × 9.81 / (1.225 × 36) = 35.7 m². A round canopy with Cd = 1.75 must have area A = 35.7/1.75 = 20.4 m², explaining why traditional round parachutes have diameters of 5–6 m. Modern ram-air rectangular canopies use Cd ≈ 1.4 and have areas of 20–30 m² depending on the jumper's weight. This calculator lets you verify any canopy specification instantly by entering the Cd, area, and your exit weight.

Terminal Velocity on Other Planets

Changing gravitational acceleration reveals dramatically different terminal velocities on other worlds. On Mars (g = 3.72 m/s², rho = 0.020 kg/m³), an 80 kg skydiver in spread-eagle position reaches vt = sqrt(2 × 80 × 3.72 / (0.020 × 1.0 × 0.70)) = sqrt(595.2 / 0.014) = sqrt(42,514) = 206 m/s = 742 km/h: far above survivable without a heat shield. This is why NASA's Perseverance rover required a supersonic parachute, retrorockets, and a sky crane for Mars entry, as documented in the agency's entry, descent, and landing engineering reports. Use the gravitational acceleration field to explore any body in the solar system.

Connecting Terminal Velocity to Other Kinematics Tools

Terminal velocity marks the end point of the acceleration phase in free fall. Before reaching it, the object accelerates under net force mg − Fd. The SUVAT calculator handles the constant-acceleration approximation for the early phase of a fall, before drag becomes significant. The projectile motion calculator models the full trajectory without drag, giving the idealised range and height: compare that with a drag-adjusted model to understand how much drag shortens real-world projectile paths. Together these tools give a complete picture of vertical motion from launch to equilibrium.

Reynolds Number and When the Formula Applies

The terminal velocity formula vt = sqrt(2mg / (rho × Cd × A)) uses a quadratic drag model, valid when the Reynolds number Re = rho × vt × L / mu is above roughly 1,000. At very low Reynolds numbers: tiny droplets, dust particles, pollen, micro-organisms: Stokes' law applies instead: the drag force is proportional to velocity (not velocity squared), and terminal velocity is vt = 2r²(rho_particle - rho_fluid) × g / (9 × mu), where r is the particle radius and mu is dynamic viscosity. For everyday objects moving at noticeable speeds (humans, balls, vehicles), Reynolds numbers are in the millions and the quadratic formula is the correct model. The calculator uses the quadratic drag formula throughout. If you are working with particle settling in a fluid, use Stokes' law for particles smaller than approximately 0.1 mm in water or 1 mm in air, and the quadratic formula for larger objects.

Metric and Imperial Unit Notes

Terminal velocity is displayed in all three common speed units simultaneously: m/s for SI calculations, km/h for general reference, and mph for US contexts. The mass input accepts both kg and lbs: the conversion factor is 1 lbs = 0.453592 kg. Cross-sectional area accepts m² and ft²: the conversion factor is 1 ft² = 0.092903 m². Air density is always entered in kg/m³ because the formula requires SI units throughout. Standard sea-level air density is 1.225 kg/m³ at 15°C and standard pressure. If you know the altitude in metres, the approximate density is rho = 1.225 × exp(-altitude / 10400), which gives 0.909 kg/m³ at 3,000 m and 0.736 kg/m³ at 5,000 m. Entering these values gives terminal velocities representative of high-altitude skydiving or mountaineering equipment drop scenarios.

Accuracy and Limitations of the Terminal Velocity Calculator

This calculator uses the terminal velocity formula v_t = sqrt(2mg / (ρCdA)), which is derived from equating gravitational force to quadratic aerodynamic drag. The formula is accurate for objects in the turbulent drag regime (Reynolds number above approximately 1,000), which applies to most objects larger than a marble falling faster than a few metres per second. For very small objects (fine dust, bacteria, fog droplets), Stokes drag (linear with velocity) applies and this formula will over-predict terminal velocity. Standard sea-level air density (1.225 kg/m³) is used unless overridden. The NASA terminal velocity educational resource derives the formula from first principles and gives benchmark values for skydivers, raindrops, and hailstones that can be used to validate calculator inputs.

The Most Common Terminal Velocity Calculation Mistake

The most common mistake is forgetting that terminal velocity depends on the square root of the drag area (Cd × A), not linearly on area. Doubling the cross-sectional area of a parachute reduces terminal velocity by a factor of sqrt(2) = 1.41, not by a factor of 2. This means a parachute needs to be four times the area to halve the terminal speed. The relationship is v_t ∝ 1/sqrt(CdA). Engineers who confuse the square-root relationship with a linear one undersize parachutes and airbag designs by a significant margin. The Physics Classroom free fall and air resistance guide explains how increasing drag area changes the terminal velocity formula and works through the parachute design example numerically.