Calculate terminal velocity (v_t = √[2mg / ρCdA]) and laminar Stokes' law velocity with the free in-browser Terminal Velocity Calculator. Computes free fall drag equilibrium, skydiver body orientations, raindrop velocity, time/distance to 95% terminal speed, and multi-fluid buoyancy.
195.20 km/h • 121.29 mph • 177.89 ft/s • Drop to 95%: 344.6 m
SI Speed (m/s)
Metric speed (km/h)
Imperial speed (mph)
Ballistics unit (ft/s)
1.83 · (v_t / g)
1.15 · (v_t² / g)
When an object falls freely through a fluid medium (such as air or water), it experiences two opposing vertical forces: the downward pull of gravity (\(F_g = mg\)) and the upward retarding force of aerodynamic drag (\(F_d\)). As the falling object accelerates, drag increases quadratically with speed until \(F_d = F_g\). At this point, the net acceleration drops to zero (\(a = 0\)), and the body descends at a steady, unchanging maximum velocity known as terminal velocity (\(v_t\)):
At terminal velocity, upward aerodynamic drag equals downward gravitational weight (\(F_d = mg\)), eliminating net force and acceleration.
In turbulent flow, drag force scales with the square of velocity (\(F_d = \frac{1}{2}\rho v^2 C_d A\)), causing rapid deceleration toward \(v_t\).
Terminal velocity is inversely proportional to the square root of fluid density (\(v_t \propto 1/\sqrt{\rho}\)), increasing dramatically at high altitudes.
Deriving the exact mathematical equations governing turbulent and laminar terminal velocities:
| Flow Regime | Governing Terminal Velocity Formula | Applicable Physical Regimes |
|---|---|---|
| Turbulent Quadratic Drag (\(Re > 1000\)) | $$v_t = \sqrt{\frac{2mg}{\rho C_d A}}$$ | Skydivers, baseballs, vehicles, raindrops > 1mm |
| Laminar Stokes' Law (\(Re < 1\)) | $$v_t = \frac{2r^2(\rho_p - \rho_f)g}{9\eta}$$ | Microscopic dust, aerosols, sediment settling, fog |
| Fluid Buoyancy Corrected | $$v_t = \sqrt{\frac{2(m - \rho_f V)g}{\rho_f C_d A}}$$ | Objects sinking in liquids (water, oil, honey) |
Choose Turbulent Drag for macro bodies (skydivers, parachutes, balls) or Stokes' Law for microscopic aerosols and sediment.
Input object mass (\(m\)) in \(\text{kg}\), \(\text{g}\), or \(\text{lbs}\), and projected frontal area (\(A\)) in \(\text{m}^2\), \(\text{cm}^2\), or \(\text{ft}^2\).
Pick shape presets (human belly, head-first dive, parachute, sphere) and fluid density (sea level air, stratosphere, water).
Inspect terminal velocity across \(\text{m/s}\), \(\text{km/h}\), \(\text{mph}\), \(\text{ft/s}\), plus time and distance required to reach \(95\%\) terminal speed.
How skydivers control terminal velocity by altering surface area (\(A\)) and drag coefficient (\(C_d\)):
Area: 0.7 m² • Cd: 1.0 • Standard stable arch position
Area: 0.18 m² • Cd: 0.7 • Streamlined vertical freefly dive
Area: 35 m² • Cd: 1.5 • Survivable soft touchdown landing
Integrating Newton's second law with quadratic drag reveals that free fall speed approaches terminal velocity asymptotically:
Because acceleration decreases exponentially as drag builds up, an object never reaches \(100\%\) of \(v_t\) in finite mathematical time. In practical physics, we calculate the time to reach \(95\%\) terminal velocity as \(t_{95\%} \approx 1.83 \frac{v_t}{g}\) and distance fallen as \(y_{95\%} \approx 1.15 \frac{v_t^2}{g}\).
Switches seamlessly between high Reynolds number quadratic drag and microscopic laminar Stokes' law.
Computes the exact duration and drop height required to reach near-terminal velocity under atmospheric drag.
Features presets for sea level air (\(1.225\)), stratosphere (\(0.004\)), fresh water (\(998\)), and custom fluids.
Pre-loaded drag coefficients for spheres (\(0.47\)), flat plates (\(1.28\)), skydivers (\(1.0\)), and parachutes (\(1.5\)).
Outputs terminal velocity simultaneously in \(\text{m/s}\), \(\text{km/h}\), \(\text{mph}\), and \(\text{ft/s}\).
Displays step-by-step LaTeX formula derivations with exact live variable substitutions.
Why dropping from the stratosphere breaks the sound barrier:
At an altitude of \(39\text{ km}\) (\(128,000\text{ feet}\)), atmospheric air density is less than \(1\%\) of sea level (\(\rho \approx 0.004\text{ kg/m}^3\)). Because \(v_t \propto 1/\sqrt{\rho}\), skydiver Felix Baumgartner accelerated to a peak terminal velocity of \(377.1\text{ m/s}\) (\(1,357.6\text{ km/h} \approx 843.6\text{ mph}\)), achieving Mach 1.25 and becoming the first human to break the sound barrier in free fall. As he descended into denser lower atmosphere, aerodynamic drag decelerated him back to \(54\text{ m/s}\).
Replaces inaccurate vacuum kinematics (\(v = \sqrt{2gh}\)) with realistic fluid drag equilibrium modeling.
Allows skydivers, drop-test engineers, and space agencies to size canopy surface areas for survivable impact speeds.
How droplet diameter dictates terminal fall speed through the atmosphere:
Stokes' law laminar settling; remains suspended in gentle cloud updrafts
Aerodynamic drag flattens raindrop base; surface tension prevents breakup
Mass 60g delivers 27 Joules of kinetic energy, shattering roof tiles and car glass
Settling dynamics in dense liquid media with Archimedes' buoyancy correction:
In liquids, buoyant force (\(F_b = \rho_f V g\)) significantly offsets gravitational weight, modifying terminal velocity:
Wastewater treatment clarifiers, gold mining sluice boxes, and sedimentologists rely on liquid settling velocity differentials to separate dense minerals from light sand.
Why dropping a coin from the Empire State Building is non-lethal:
A US one-cent coin has a mass of only \(2.5\text{ grams}\) and a broad diameter of \(19.05\text{ mm}\). Because it tumbles unpredictably through the air (\(C_d \approx 1.15\)), its terminal velocity caps at only \(11.0\text{ m/s}\) (\(40\text{ km/h} \approx 25\text{ mph}\)), delivering a minor sting with a kinetic energy of merely \(0.15\text{ Joules}\) upon ground impact.
How atmospheric drag decelerates returning orbital space capsules:
Spacecraft (e.g. NASA Orion, SpaceX Dragon) re-enter Earth's atmosphere at hypersonic velocities (\(v_0 \approx 7,800\text{ m/s} = 28,000\text{ km/h}\)). Blunt-body aerodynamic drag bleeds over \(99\%\) of kinetic energy into compressed air shockwaves, decelerating the capsule to a subsonic terminal velocity of approximately \(90\text{ m/s}\) before main parachutes deploy to slow final splashdown to \(7\text{ m/s}\).
Why small animals survive terminal velocity drops while large mammals break:
Survives a drop from any height unharmed
Walks away from multi-story falls
Feline righting reflex spreads body area
Fatal without parachute deceleration
How aerospace engineers experimentally determine drag coefficients:
In subscale and full-scale wind tunnels, force balance dynamometers measure net aerodynamic resistance (\(F_d\)) at calibrated airflow velocities (\(v\)), allowing precise calculation of the non-dimensional drag coefficient:
Automotive manufacturers utilize wind tunnels to reduce vehicle drag coefficients down to \(C_d \approx 0.20-0.24\), maximizing electric vehicle highway range.
Comprehensive answers to common questions about terminal velocity formulas, skydiver speeds, parachute sizing, and Stokes' law.
