Escape Velocity Calculator

Escape velocity is the speed at which an unpowered projectile leaving a body has exactly enough kinetic energy to reach infinite distance with zero speed remaining. Setting one half m v squared equal to GMm divided by r and cancelling the projectile mass gives v equal to the square root of 2GM over r, which is why the value does not depend on what is escaping: a pebble and a crewed vehicle share the same figure. Surface values run from 2.38 km/s at the Moon and 5.03 km/s at Mars, through 11.19 km/s at Earth, to 59.5 km/s at Jupiter cloud tops and 617.5 km/s at the Sun. It is exactly the square root of 2 times the circular orbital speed at the same radius, so a 41.4 percent speed increase converts a circular orbit into an escape trajectory. Several caveats matter in practice. The figure assumes an instantaneous impulse in a vacuum, while real launch vehicles thrust over minutes and give up roughly 1.5 to 2 km/s to gravity losses and atmospheric drag. A continuously powered vehicle never needs to reach escape speed at all, since it can climb slowly under sustained thrust. The value is also a two-body idealization: escaping the gravity of Earth still leaves a spacecraft in orbit around the Sun, requiring roughly 42.1 km/s at that distance to leave the solar system. Pushing the formula until escape speed equals the speed of light returns the Schwarzschild radius, 2GM over c squared, about 8.87mm for the mass of Earth.

The physics is implicit in the Principia of Isaac Newton, published in 1687, where the thought experiment of a cannon fired horizontally from a high mountain shows that enough speed converts a fall into an orbit and, at greater speed, into a permanent departure. Universal gravitation combined with energy arguments yields the square-root expression directly, although the phrase escape velocity is much later. The idea found a striking early application in 1783, when the Reverend John Michell, rector at Thornhill in Yorkshire, presented a paper to the Royal Society reasoning that a sufficiently massive body would have an escape speed exceeding that of light and would therefore be invisible. Pierre-Simon Laplace advanced a similar argument in his Exposition du systeme du monde in 1796. Those dark stars were an anticipation rather than a derivation of modern black holes; the 1916 solution of Karl Schwarzschild to the Einstein field equations produced the same characteristic radius from entirely different physics. On the engineering side, the 1903 paper of Konstantin Tsiolkovsky on reaction devices established that such speeds were reachable only through high exhaust velocities and staged vehicles. The numbers were first met in practice in January 1959, when the Soviet Luna 1 probe exceeded the escape speed of Earth and entered solar orbit, followed by the American Pioneer 4 two months later. Since then escape velocity has served less as a launch target than as a bookkeeping quantity inside mission delta-v budgets.