Calculation
vend = √(2·m·g / (cw·A·ρ))
Once drag force reaches weight force, the body stops accelerating and falls at a constant terminal velocity.
Kräftegleichgewicht m·g = FW mit FW = cw·(ρv²/2)·Ap nach Dubbel Mechanik B 6.6.4, Gl. (35)
Terminal velocity of a body falling with air resistance
Once drag force reaches weight force, the body stops accelerating and falls at a constant terminal velocity.
MINTSI
02
Result
Select a target and calculate.
Understand the inputs
- Terminal velocity vend — Constant fall speed at force equilibrium between weight and drag.
- Mass m — Mass of the falling body.
- Gravitational acceleration g — Local gravitational acceleration, about 9.81 m/s² near sea level.
- Drag coefficient cw — Shape-dependent dimensionless coefficient; a spread-eagled skydiver (belly-to-earth) is about 1.0.
- Frontal area A — Largest cross-sectional area of the body normal to the fall direction.
- Air density ρ — Density of air; about 1.2 kg/m³ at sea level and 20 °C, decreasing with altitude.
Example
m=80 kg, cw=1.0, A=0.7 m² and ρ=1.2 kg/m³ give vend≈43.2 m/s (about 156 km/h).
Assumptions and limits
Steady-state fall conditions once force equilibrium is reached, constant cw and constant air density over the considered fall height; the acceleration phase up to terminal velocity and altitude dependence of ρ are excluded.
Understand Terminal velocity of a body falling with air resistance
This calculator determines the terminal velocity of a falling body, where drag and weight force balance, complementing the existing plain drag-force calculator with the steady-state falling case.
What does this quantity describe?
A falling body accelerates until growing drag force ½·cw·A·ρ·v² equals weight force m·g. Solved for v, this gives terminal velocity vend=√(2mg/(cwAρ)).
Formula and variables
vend = √(2·m·g / (cw·A·ρ))
vend = √(2mg/(cwAρ))
| Symbol / input | Meaning |
|---|---|
| Terminal velocity vend | Constant fall speed at force equilibrium between weight and drag. |
| Mass m | Mass of the falling body. |
| Gravitational acceleration g | Local gravitational acceleration, about 9.81 m/s² near sea level. |
| Drag coefficient cw | Shape-dependent dimensionless coefficient; a spread-eagled skydiver (belly-to-earth) is about 1.0. |
| Frontal area A | Largest cross-sectional area of the body normal to the fall direction. |
| Air density ρ | Density of air; about 1.2 kg/m³ at sea level and 20 °C, decreasing with altitude. |
Choose the inputs correctly
m is the mass of the falling body, cw its drag coefficient, A the frontal area, ρ air density and g gravitational acceleration.
How to use the calculator
Derive m, cw and A from the body's shape and mass (for a person, roughly 0.7 to 1.0 m² and cw≈1.0 belly-to-earth); use ρ=1.2 kg/m³ at sea level unless more precise values are available.
Worked example
m=80 kg, cw=1.0, A=0.7 m² and ρ=1.2 kg/m³ give vend≈43.2 m/s (about 156 km/h).
Understand the result and units
A smaller frontal area or lower cw (e.g. head-first orientation) markedly increase terminal velocity, as known from skydiving.
m is a mass, cw dimensionless, A an area, ρ a density, g an acceleration. vend is output as a speed.
Typical applications
Estimating skydiver terminal velocities, drop-test scenarios, and the terminal speed of falling objects in the atmosphere.
Assumptions, limits and common mistakes
Steady state once force equilibrium is reached, with constant cw and air density; the acceleration phase, altitude dependence of ρ and vortex-shedding effects at very high speeds are excluded.
Common mistake: Do not confuse initial acceleration with terminal velocity; the latter is only approximately reached after a certain fall distance.
Frequently asked questions
What is “Terminal velocity of a body falling with air resistance” used for?
Estimating skydiver terminal velocities, drop-test scenarios, and the terminal speed of falling objects in the atmosphere.
Where do the input values come from?
m is the mass of the falling body, cw its drag coefficient, A the frontal area, ρ air density and g gravitational acceleration.
What does the result not cover?
Steady state once force equilibrium is reached, with constant cw and air density; the acceleration phase, altitude dependence of ρ and vortex-shedding effects at very high speeds are excluded.