Escape Velocity Calculator

Escape velocity is derived by setting kinetic energy equal to the gravitational potential energy that must be overcome: ½mv² = GMm/r. The mass of the escaping object appears on both sides and cancels completely, leaving v = √(2GM/r), which depends only on the body being escaped.

So a pebble, a person and a fully fuelled rocket all need the same 11.18 km/s to leave Earth. What differs enormously is the energy required to reach that speed, since kinetic energy scales with mass — which is why launching a heavier payload costs more fuel even though the target speed is identical.

The figure describes an object given all its speed instantly at the surface and then coasting, like a cannonball. Real rockets do nothing of the sort: they accelerate continuously over minutes, gaining altitude as they go, and never travel at 11.18 km/s while still in the dense lower atmosphere — doing so would destroy them through drag heating.

Because gravity weakens with distance, a rocket that has already climbed needs less speed to escape from there. Continuous thrust means a rocket can leave Earth without ever reaching the surface escape velocity at any single moment. Escape velocity is a useful energy benchmark, not an operational requirement.

It sets which gases a world can hold onto. Molecules in an atmosphere move at a range of speeds determined by temperature, and the lightest ones — hydrogen and helium — move fastest. Where escape velocity is low, those fast molecules gradually leak away over geological time.

This explains a good deal of the solar system. The Moon at 2.38 km/s retains essentially no atmosphere. Mars at 5.03 km/s holds only a thin one and has lost most of what it once had. Earth at 11.18 km/s retains nitrogen and oxygen comfortably but still loses hydrogen steadily. Jupiter at 59.5 km/s retains even hydrogen and helium, which is why it remains overwhelmingly composed of them.

Rearranging for radius shows that any mass compressed small enough produces an escape velocity equal to the speed of light. That radius is the Schwarzschild radius, and a body compressed within it is a black hole — nothing, including light, can escape.

Newtonian gravity breaks down well before that point and general relativity is required, which is why this calculator refuses rather than returning a Newtonian answer for such inputs. The Newtonian derivation nonetheless gives the correct Schwarzschild radius, a coincidence noted as early as 1783 by John Michell, long before relativity existed.