Escape Velocity Calculator

Calculate the speed needed to leave a body's gravity for good, v_esc = √(2GM/r), from Earth, the Moon, Mars, Jupiter, the Sun, a white dwarf, or a neutron star — in km/s, m/s, and as a fraction of c — with a log-scale comparison ladder from a small asteroid to a neutron star, and the exact link to this site's Schwarzschild radius calculator.

Escape velocity calculator

v_esc = √(2GM/r) is the speed needed to leave a body's gravity for good, with no further thrust. It's direction-independent — an energy condition on speed, not a statement about trajectory — and it says nothing about drag: a real rocket leaving Earth needs considerably more than 11.2 km/s of actual burn to fight its way through the atmosphere.

Mass (M)

Radius (r)

Escape velocity (v_esc)

11.186

v_esc ≈ 11.186 km/s

11186.2 m/s · 3.731303 × 10⁻⁵× the speed of light

Kilometers/second11.186 km/s

Meters/second11186.2 m/s

Fraction of light speed3.731 × 10⁻⁵ × c

Same equation, two directions

For this mass, the Schwarzschild radius is 8.87 × 10⁻⁶ km — the radius this mass would need to be compressed to for its escape velocity to reach exactly c. The current radius is 7.183 × 10⁸× the Schwarzschild radius. v_esc = √(2GM/r) and r_s = 2GM/c² are the same formula — one solved for speed, the other solved for the radius where that speed hits the speed of light. See this site's Schwarzschild Radius Calculator for the other direction.

Log scale of v_esc — from a small asteroid's stroll-pace escape velocity to a neutron star's, a span of almost nine orders of magnitude, not just four.

Fire a cannonball fast enough, straight up, sideways, or at any angle at all, and there’s a speed beyond which it never comes back — not because it out-races gravity, but because it has exactly enough energy to coast to infinity with nothing left over. That speed is the escape velocity, and it depends on nothing but the mass you’re leaving and how far from its center you start.

The formula

v_{\rm esc} = \sqrt{\frac{2GM}{r}}

This falls straight out of energy conservation: set kinetic energy equal to the magnitude of gravitational potential energy, ½mv² = GMm/r, and the test mass m cancels out entirely. That cancellation is the whole story — it’s why escape velocity applies equally to a pebble and a spacecraft launched from the same radius.

Direction doesn’t matter — and neither does the formula know about air

Two things about this formula surprise people the most:

It’s an energy condition, not a trajectory. Nothing in ½mv² = GMm/r says which way you’re pointed. Straight up, sideways, at 45°, even briefly downward before curving away — the same speed at the same r is enough, ignoring drag and other bodies. “Escape velocity” is really “escape speed.”
It says nothing about atmosphere. Earth’s escape velocity is about 11.2 km/s, but no real rocket burns straight to 11.2 km/s and coasts out — a real launch fights drag and gravity losses the entire way up through the atmosphere, so the actual delta-v budget is considerably higher than the bare formula suggests. This calculator gives you the idealized, vacuum, point-mass number; real spaceflight engineering adds a substantial margin on top of it.

The endpoint: where this formula becomes a black hole

Hold the mass M fixed and shrink the radius r, and v_esc climbs without bound. Shrink r far enough, and v_esc reaches c — the speed of light, the fastest anything can go. Set v_esc = c in the formula above and solve for r:

r_s = \frac{2GM}{c^2}

That’s the Schwarzschild radius, and it isn’t a coincidence that it uses the same G, M, and a factor of 2 — it’s the exact same equation, solved for the radius where escape speed hits the speed of light instead of solved for the speed itself. This calculator and this site’s Schwarzschild Radius Calculator are two views of one formula. Every result on this page includes the Schwarzschild radius for whatever mass you’ve entered, and how many Schwarzschild radii away the current radius sits — the smaller that ratio gets, the closer that mass is to being a black hole.

Reading the comparison ladder

The ladder plots escape velocity on a log scale from a small asteroid up through a neutron star — a span of almost nine orders of magnitude, not four: Bennu’s escape velocity is a gentle stroll (about 20 cm/s, a figure NASA’s OSIRIS-REx team liked to point out — you could jog fast enough to leave it), while a typical neutron star’s sits at roughly 60% of the speed of light. Between them: the Moon, Mars, Earth, Jupiter, the Sun, and a white dwarf, each a stepping stone in the same climb from “toy gravity” to “relativity is no longer optional.”