kg
Please enter object mass.
Mass of the falling object
m²
Please enter cross-sectional area.
Area of the object facing the airflow
Dimensionless — use presets below
kg/m³
Used only when Custom is selected above
Terminal Velocity
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Sources & Methodology
Drag equation and aerodynamic constants verified against NASA Glenn Research Center, Engineering Toolbox, and Anderson's Introduction to Flight.
NASA Glenn Research Center — Terminal Velocity
NASA's aerodynamics education resource provides the terminal velocity formula, drag equation, and explanation of how drag coefficient, cross-sectional area, and air density determine the maximum fall speed of any object. The terminal velocity formula used in this calculator is sourced directly from NASA Glenn's treatment of the subject.
Engineering Toolbox — Drag Coefficients
The drag coefficient reference values for spheres (0.47), cylinders (0.82), flat plates (1.28), skydivers (0.7 to 1.0), cars (0.25 to 0.35), and other objects used in this calculator's preset list are sourced from Engineering Toolbox's comprehensive drag coefficient reference table.
Engineering Toolbox — Standard Atmosphere Air Density
Air density values at different altitudes (sea level 1.225 kg/m cubed, 1000 m 1.112, 2000 m 1.007, 4000 m 0.819, 8848 m 0.467) are sourced from Engineering Toolbox's standard atmosphere table, which conforms to the International Standard Atmosphere (ISA) model.
Methodology: Terminal velocity: Vt = sqrt(2mg / (Cd x rho x A)). Drag force at terminal velocity: Fd = mg (equals weight by definition). Drag force at any velocity v: Fd = 0.5 x Cd x rho x A x v squared. Ratio of drag to gravity at half terminal velocity = 0.25 (25%). At 90% of terminal velocity, drag = 81% of gravity. Mass conversions: 1 lb = 0.453592 kg, 1 g = 0.001 kg, 1 oz = 0.0283495 kg.
⏱ Last reviewed: April 2026
Air Resistance and Terminal Velocity — Physics Guide
In the real world, free fall is never truly "free" — air resistance acts on every falling object, opposing its motion and limiting how fast it can fall. Understanding how air resistance works, and how to calculate terminal velocity, is essential for aerospace engineering, parachute design, sports science, ballistics, and meteorology. This guide explains the drag equation, terminal velocity formula, and how factors like shape, size, mass, and altitude affect how objects fall through air.
The Drag Equation
The aerodynamic drag force on any object moving through a fluid (such as air) is given by the drag equation. This formula is the foundation for all terminal velocity calculations:
Fₑ = ½ × Cₑ × ρ × A × v²
Fₑ = Drag force (Newtons)
Cₑ = Drag coefficient (dimensionless) — shape factor
ρ = Air density (kg/m³) — 1.225 at sea level
A = Cross-sectional area facing airflow (m²)
v = Velocity (m/s)
Note: Drag increases with the square of velocity — doubling speed quadruples drag force.
Cₑ = Drag coefficient (dimensionless) — shape factor
ρ = Air density (kg/m³) — 1.225 at sea level
A = Cross-sectional area facing airflow (m²)
v = Velocity (m/s)
Note: Drag increases with the square of velocity — doubling speed quadruples drag force.
Terminal Velocity Formula
Terminal velocity occurs when drag force exactly equals gravitational force (weight). Setting Fₑ = mg and solving for v gives the terminal velocity formula:
Vₜ = sqrt( 2mg / (Cₑ × ρ × A) )
Example — Skydiver belly-down:
m = 80 kg, g = 9.80665 m/s², Cₑ = 1.0, ρ = 1.225 kg/m³, A = 0.7 m²
Vₜ = sqrt(2 × 80 × 9.80665 / (1.0 × 1.225 × 0.7))
Vₜ = sqrt(1569.06 / 0.8575) = sqrt(1829.8) = 42.8 m/s (95.7 mph)
m = 80 kg, g = 9.80665 m/s², Cₑ = 1.0, ρ = 1.225 kg/m³, A = 0.7 m²
Vₜ = sqrt(2 × 80 × 9.80665 / (1.0 × 1.225 × 0.7))
Vₜ = sqrt(1569.06 / 0.8575) = sqrt(1829.8) = 42.8 m/s (95.7 mph)
Drag Coefficients for Common Objects
| Object | Cₑ | Notes |
|---|---|---|
| Skydiver (belly-down) | 1.0 | Maximum drag position, stable |
| Skydiver (head-down) | 0.7 | Speed position, faster fall |
| Sphere | 0.47 | Baseball, golf ball, cannonball |
| Flat plate (perpendicular) | 1.28 | Maximum possible for flat shape |
| Cylinder (long axis horizontal) | 0.82 | Broadside to flow |
| Streamlined body (teardrop) | 0.04 | Optimal aerodynamic shape |
| Modern passenger car | 0.25–0.35 | Depends on design |
| Bicycle with rider (upright) | 0.9 | High frontal area |
| Open parachute | 1.3–1.5 | Designed for maximum drag |
| Badminton shuttlecock | 0.44 | Feather type |
How Air Density Affects Terminal Velocity
Since terminal velocity is proportional to 1/sqrt(ρ), lower air density means higher terminal velocity. This is why skydivers exit aircraft at altitude — they initially fall faster before air density increases near the ground. A skydiver at 4,000 m altitude (rho = 0.819) reaches a terminal velocity about 22% higher than at sea level. Felix Baumgartner's record-breaking 2012 jump from 39 km altitude, where air density was almost zero, allowed him to briefly exceed the speed of sound before deceleration as he descended into denser air.
💡 Design Insight: Parachutes work by dramatically increasing drag. A round parachute with diameter 8 m has area about 50 m² and Cₑ around 1.5, giving drag of 0.5 x 1.5 x 1.225 x 50 x v² = 45.9v² N. For a 90 kg person (weight 882 N), terminal velocity = sqrt(882/45.9) = 4.4 m/s (9.8 mph) — a safe landing speed. Without parachute (A = 0.7 m², Cₑ = 1.0), the same person falls at 42.8 m/s (95 mph) — fatal on impact.
Frequently Asked Questions
Terminal velocity formula: Vt = square root of (2mg divided by Cd x rho x A). Where m is mass in kg, g = 9.80665 m/s squared, rho is air density in kg/m cubed (1.225 at sea level), Cd is the drag coefficient, and A is the cross-sectional area in m squared. At terminal velocity, drag force equals gravity, giving zero net acceleration.
The drag force formula is Fd = 0.5 x Cd x rho x A x v squared. Drag increases with the square of velocity — doubling speed quadruples drag force. This is why terminal velocity exists: as an object speeds up, drag grows rapidly until it equals gravity, at which point acceleration stops and speed becomes constant.
A typical skydiver (80 kg, belly-to-earth, Cd = 1.0, A = 0.7 m squared) reaches terminal velocity of about 42 to 55 m/s (95 to 123 mph) depending on body position, mass, and altitude. In a head-down speed position (Cd = 0.7, A = 0.4 m squared), the same person reaches around 67 to 90 m/s (150 to 200 mph). Our calculator uses exact inputs so you can tune for any scenario.
The drag coefficient Cd is a dimensionless number describing how aerodynamically resistant an object is relative to its frontal area. Typical values: sphere 0.47, flat plate 1.28, streamlined teardrop 0.04, skydiver belly-down 1.0, skydiver head-down 0.7, modern car 0.25 to 0.35, open parachute 1.3 to 1.5, bicycle with rider 0.9.
Terminal velocity is proportional to 1 divided by the square root of air density. At higher altitudes where air is less dense, objects fall faster. At 4,000 m (rho = 0.819), terminal velocity is about 22% higher than at sea level (rho = 1.225). This is why Felix Baumgartner briefly exceeded the speed of sound during his 2012 stratospheric jump from 39 km altitude.
An object reaches approximately 63% of terminal velocity in one time constant (tau = m / (Cd x rho x A x Vt / 2)), and about 95% after three time constants. For a typical 80 kg skydiver, this is roughly 12 to 15 seconds after aircraft exit. Lighter objects with high drag (like a shuttlecock) reach terminal velocity much more quickly, sometimes in under a second.
Without air resistance, velocity increases forever as v = g x t with no upper limit. With air resistance, drag force grows with v squared until it equals gravity, producing zero net force and constant terminal velocity. In practice, real free fall always involves air resistance, making terminal velocity the relevant maximum speed for skydivers, raindrops, and all falling objects on Earth.
A typical 2 mm diameter raindrop (mass 0.004 g, area 3.14 mm squared, Cd = 0.47) reaches terminal velocity of about 6 to 9 m/s (13 to 20 mph). Larger drops fall faster. This low terminal velocity compared to what a pure free-fall calculation would suggest explains why rain does not hurt as much as expected from clouds at 1,500 m altitude. Large hailstones can reach 30 to 40 m/s.
Yes, for objects of the same shape and size. Terminal velocity is proportional to the square root of mass. Doubling mass increases terminal velocity by sqrt(2) = 1.414 times. A lead ball and a hollow plastic ball of the same diameter fall identically in a vacuum, but the heavier lead ball reaches a much higher terminal velocity in air because its weight overwhelms the drag force at a higher speed.
Net force = Gravity minus Drag = mg minus (0.5 x Cd x rho x A x v squared). At v = 0: net force = mg (maximum acceleration = g). As v increases, drag grows and acceleration decreases. At terminal velocity: drag = mg, net force = 0, acceleration = 0. The object continues falling at constant speed with no further acceleration.
Air density follows the International Standard Atmosphere model: rho = 1.225 x (1 minus 0.0000226 x h) to the power 4.256, where h is altitude in meters. Key values: sea level 1.225 kg/m cubed, 1000 m 1.112, 2000 m 1.007, 4000 m 0.819, 8848 m (Everest) 0.467. Lower density at altitude means higher terminal velocity for the same object.
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