Aviation Calcs

Terminal velocity / re-entry

Terminal Velocity and Ballistic Re-Entry Calculator

Terminal velocity is the speed at which drag equals weight: vt = √(2βg/ρ), with ballistic coefficient β = m/(Cd A). Enter mass, drag coefficient, and reference area, and optionally an altitude (U.S. Standard Atmosphere 1976 density) or a density of your own. If you also enter an entry speed and a flight-path angle, the page reports the Allen–Eggers zero-lift peak deceleration, the altitude of that peak, and the speed there, using an exponential atmosphere with scale height 7,162 m. The model is a simple planning estimate, not a trajectory or heating analysis.

Density from

Leave at 0 for sea-level ISA density, 1.225 kg/m³, from the same U.S. Standard Atmosphere 1976 model as the standard-atmosphere calculator.

degrees below horizontal

Terminal velocity

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Ballistic coefficient and terminal speed

The ballistic coefficient β = m / (Cd A) is the mass per unit of drag area. Heavier, slimmer, or lower-drag bodies have a larger β and fall faster through a given density. When drag equals weight, the terminal speed is

vt = √( 2 β g / ρ )

with g = 9.80665 m/s². An 80 kg body with Cd = 1 and a 0.5 m² reference area has β = 160 kg/m² and a sea-level terminal speed of about 50.6 m/s (roughly 113 mph) in ISA density 1.225 kg/m³. At altitude the same body sees a lower density from the U.S. Standard Atmosphere 1976 model used on this site, and vt rises. Cd is not really constant from a belly-to-earth parachute through hypersonic entry; treat it as the value you intend for the regime you are asking about. For the definition of Cd, see the lift and drag coefficient calculator.

Allen–Eggers ballistic entry

For a zero-lift entry at speed VE and a constant flight-path angle γ below the horizon, in an exponential atmosphere ρ = ρ0 e−h/H, with drag much larger than weight during the pulse, Allen and Eggers (NACA Report 1381, 1958) give

V = VE exp( − ρ H / (2 β sin γ) )

The peak deceleration is independent of β,

amax = VE² sin γ / (2 e H)

and occurs at ρ = β sin γ / H, where the speed has fallen to VE / √e. This page uses H = 7,162 m (23,500 ft) and ρ0 = 1.225 kg/m³. A 7.8 km/s entry at 5° then has a peak of about 13.9 g; the altitude of that peak does depend on β (higher β, deeper peak). If β sin γ / H is greater than ρ0, the formal peak would lie below the surface and the model is not applicable — the page flags that. Gravity is omitted from the deceleration equation, lift is zero, and heating is not computed.

Sources

U.S. Standard Atmosphere 1976 (NASA-TM-X-74335) for density; Allen and Eggers, NACA Report 1381 (1958); Anderson, Hypersonic and High-Temperature Gas Dynamics; Regan, Dynamics of Atmospheric Re-Entry. For circular and entry-order-of-magnitude orbital speeds, see orbital velocity and period.

Frequently asked questions

What is the sea-level terminal velocity of an 80 kg body with Cd = 1 and A = 0.5 m²?

About 50.6 m/s, or roughly 113 mph. β = m/(Cd A) = 160 kg/m², and vt = √(2βg/ρ) with ρ = 1.225 kg/m³ and g = 9.80665 m/s².

Why does peak re-entry deceleration not depend on ballistic coefficient?

In the Allen–Eggers exponential-atmosphere, zero-lift, drag-dominated model, a_max = V_E² sin γ / (2 e H). β cancels at the maximum. β does set the altitude (and density) at which that peak occurs: ρ_peak = β sin γ / H.

Is this a heating or skip-entry calculator?

No. It reports terminal velocity and the simple Allen–Eggers deceleration pulse only. It does not compute heat load, stagnation temperature, lift modulation, skip entry, or a rotating-Earth trajectory.