Escape velocity at altitude calculator

Enter the body's mass in kilograms, its reference radius in metres, and altitude in metres. The calculator adds radius and altitude to obtain distance from the centre, then evaluates the classical Newtonian expression. Results are returned in metres per second and kilometres per second. The calculation is deterministic, uses no network service, and rejects inputs whose combined centre distance is zero or negative.

Understand the distance used by the formula

Escape velocity depends on distance from the centre of mass, not altitude by itself. A published altitude usually measures height above a reference surface, while Newtonian gravity uses the radial coordinate measured from the body's centre. This calculator therefore forms r = R + h, where R is the reference radius and h is altitude, before applying the escape-speed equation. At Earth's surface, altitude is zero and the centre distance equals Earth's radius. Four hundred kilometres above that surface, the centre distance is the radius plus 400,000 metres. A negative altitude can describe a location below the chosen reference surface, provided the resulting centre distance remains positive. If radius plus altitude is zero or negative, the geometry has no valid radial distance and the capability returns an invalid-input error. Use consistent SI units: mass in kilograms and both lengths in metres. Converting kilometres to metres before submitting prevents a thousandfold distance error and a large, misleading change in the result.

How escape velocity is calculated

The calculator uses v = sqrt(2GM/(R+h)). Here G is the gravitational constant, fixed at 6.6743 × 10^-11 cubic metres per kilogram per second squared; M is the body's mass; R is its reference radius; and h is altitude. The expression comes from setting the object's total Newtonian mechanical energy to zero: its starting kinetic energy exactly balances the magnitude of its gravitational potential energy, leaving zero speed at infinite distance. The mass of the escaping object cancels, so it is not an input. Increasing body mass raises escape velocity, while increasing altitude increases centre distance and lowers escape velocity. The returned gravitational parameter is G multiplied by M, which makes the substitution easy to audit. Results include metres per second and kilometres per second, along with the actual centre distance used. This is an ideal two-body result. It assumes spherical symmetry, ignores atmosphere and thrust losses, and does not add or subtract speed supplied by the body's rotation.

Interpret and apply the result carefully

Treat the result as an instantaneous ideal speed threshold at the stated location, not as a complete launch delta-v budget. A real rocket climbs through an atmosphere, loses energy to drag and gravity while engines burn, may benefit from planetary rotation, and normally targets an orbit or transfer path rather than accelerating instantly to escape speed. The calculation is nevertheless valuable for comparing celestial bodies, checking textbook work, setting scales in simulations, and verifying mission-analysis code. Make sure mass and radius refer to the same body and that altitude uses the same reference surface as the radius. For a spacecraft already in orbit, escape speed at its altitude is not the additional burn required to escape, because the spacecraft already has orbital speed. Compare its current velocity vector and orbital energy with the threshold instead. API requests cost $0.002 per successful item, while invalid inputs such as missing fields, a non-positive mass, or a non-positive combined distance are rejected. Because the algorithm is pure arithmetic, identical inputs always produce identical JSON output.