Escape Velocity
√(2µ/r) for Earth, Moon, or Mars — from the surface or from altitude.
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The engineering
Escape velocity is an energy statement, not a speed limit: kinetic energy equal to the depth of the gravity well, direction irrelevant (ignoring the atmosphere and terrain in your way). Earth's 11.2 km/s versus the Moon's 2.4 is the one-number explanation of why lunar ascent stages are compact and Earth launch vehicles are skyscrapers.
The √2 relationship with circular speed is the working engineer's version: a satellite already in orbit needs only 41% more speed to leave entirely. Real missions speak in C3 (v² beyond escape), but C3 = 0 *is* this card.
Where this math comes from
The concept is Newton's cannonball fired one notch harder — fast enough and the miss never ends, faster still and the curve opens into escape. Pierre-Simon Laplace ran the logic to its extreme around 1796: a body whose escape velocity exceeded light speed would be invisible, the first sketch of a black hole, derived from this card's formula.
Tsiolkovsky's 1903 paper framed 11.2 km/s as the engineering target, and Luna 1 was the first machine to actually spend that much energy, leaving Earth's well in January 1959 (missing the Moon, discovering the solar wind as a consolation prize). Every interplanetary launch since buys its C3 against the same square root.
- 1687Isaac NewtonThe cannonball argument defines escape.
- 1796Pierre-Simon Laplace'Dark stars' — escape velocity meets light speed (circa).
- 1903Konstantin Tsiolkovsky11.2 km/s framed as spaceflight's price of exit.
- 1959Soviet Luna 1First spacecraft to reach Earth escape velocity.
See the full timeline of the math behind every calculator →
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