What is terminal velocity?

Terminal velocity is the constant speed reached by a falling object when the upward aerodynamic drag force exactly equals the downward gravitational force (weight). At that point net acceleration is zero and the object stops accelerating, falling at a steady speed.

What formula does this calculator use?

The standard physics formula derived by setting drag equal to weight: v_t = sqrt(2·m·g / (rho·C_d·A)), where m is mass (kg), g = 9.80665 m/s², rho is air density (kg/m³), C_d is the dimensionless drag coefficient, and A is the projected cross-sectional area (m²). This follows directly from F_drag = (1/2)·rho·v²·C_d·A set equal to F_gravity = m·g.

What is the terminal velocity of a human skydiver?

A typical skydiver in the belly-to-earth position (mass ~80 kg, area ~0.7 m², C_d ~1.0) reaches roughly 54–58 m/s (195–210 km/h, 120–130 mph) at sea level. In the head-down "tracking" position the smaller area (~0.18 m²) pushes that toward 90–100 m/s (320–360 km/h). The calculator lets you tweak all three parameters to match any scenario.

Why does terminal velocity change with altitude?

Air density decreases with altitude according to the International Standard Atmosphere (ISA) model: rho = p·M/(R·T) where temperature and pressure both fall with height. Lower density means less drag force per unit of cross-section, so a higher speed is needed before drag equals weight. Enable the altitude input to apply the ISA tropospheric formula automatically.

What drag coefficient should I use?

Drag coefficient (C_d) depends on shape and, to a lesser extent, Reynolds number. Typical values: smooth sphere 0.47, long cylinder (crosswise) 0.82, flat plate 1.17, streamlined body 0.04–0.09, open hemisphere (cup) ~1.4, human standing ~1.0–1.3. For an open round parachute canopy the effective C_d is often quoted as 1.3–1.5 combined with the canopy area.

What is the Reynolds number shown in the results?

Reynolds number Re = v·L/nu quantifies whether flow around the object is laminar or turbulent (nu is kinematic viscosity of air, ~1.5×10⁻⁵ m²/s at 20°C; L is the square root of the cross-sectional area as a characteristic length). Re 100 000 is turbulent. This matters because C_d itself changes at the drag-crisis transition (~Re = 5×10⁵ for spheres), where the boundary layer trips turbulent and the wake narrows, sharply reducing drag.

Terminal Velocity Calculator

Terminal velocity is the maximum constant speed a freely falling object reaches when air resistance (drag) exactly balances gravity. Understanding it matters for skydivers designing deployment altitudes, engineers sizing parachutes, meteorologists modelling hailstone fall speeds, and anyone curious about why a cat falls differently from a bowling ball.

The physics formula

When an object falls through a fluid (here, air) it experiences two vertical forces:

Gravity (weight): F_g = m · g (downward)
Aerodynamic drag: F_d = ½ · ρ · v² · C_d · A (upward)

Terminal velocity is reached when F_d = F_g. Solving for v:

v_t = √(2·m·g / (ρ·C_d·A))

Symbol	Meaning	Units
v_t	Terminal velocity	m/s
m	Mass of the object	kg
g	Standard gravity (9.80665)	m/s²
ρ	Air density	kg/m³
C_d	Drag coefficient (shape + flow regime)	dimensionless
A	Projected cross-sectional area	m²

The constant g = 9.80665 m/s² is the ISO 80000-3 standard value used in all engineering calculations.

How it works

This calculator evaluates the formula in real time. Enter mass, drag coefficient, and cross-sectional area; it returns terminal velocity in m/s, km/h, and mph simultaneously. The solve-for-variable mode inverts the formula algebraically:

Solve for mass: m = v_t² · ρ · C_d · A / (2g)
Solve for area: A = 2·m·g / (v_t² · ρ · C_d)
Solve for C_d: C_d = 2·m·g / (v_t² · ρ · A)

These are useful when you know the desired landing speed and want to design the parachute area, or when you are working backwards from observed fall data to infer an unknown drag coefficient.

Air density and altitude

Air density ρ is not constant. Enable the altitude mode and the calculator uses the ISA (International Standard Atmosphere) tropospheric formula — valid from sea level to 11 000 m — to compute ρ automatically from altitude. The ISA assumes a temperature lapse rate of 6.5 K per 1 000 m and sea-level conditions of 288.15 K and 101 325 Pa. At 4 000 m (a typical opening altitude for tandem skydives) ρ ≈ 0.819 kg/m³ versus 1.225 kg/m³ at sea level, producing a terminal velocity about 22% higher.

Worked example — skydiver belly-to-earth

A 80 kg skydiver with arms and legs spread presents roughly 0.7 m² of frontal area and has C_d ≈ 1.0 at sea level (ρ = 1.225 kg/m³):

v_t = √(2 × 80 × 9.80665 / (1.225 × 1.0 × 0.7))
    = √(1569.064 / 0.8575)
    = √(1829.8)
    ≈ 42.8 m/s  →  154 km/h  →  95.7 mph

That is the classic “120 mph belly” figure quoted in skydiving training manuals. Switching to a head-down posture (area ≈ 0.18 m², C_d ≈ 0.7) gives ≈ 87 m/s (313 km/h, 195 mph) — showing how powerfully shape dominates the outcome.

Reynolds number note

The calculator also reports the Reynolds number Re = v·L / ν (where L = √A and ν ≈ 1.5 × 10⁻⁵ m²/s). This is informational: for Re below about 1 000 the flow is laminar and the Stokes drag law is more accurate; for Re above 10⁵ the boundary layer begins to trip, and near Re ≈ 5 × 10⁵ spheres experience the drag crisis where C_d drops sharply from ~0.47 to ~0.1. For macroscopic objects (people, baseballs, parachutes) Re is typically well above 10⁴, so the formula here is valid. For very small particles (raindrops under ~0.1 mm, fine dust) use Stokes’ law instead.