Escape Velocity Calculator: v = √(2GM/R)
Escape velocity is the minimum speed a free-flying object needs at the surface of a gravitating body to escape its gravity without any further thrust. Derived by equating kinetic and gravitational potential energy, the formula is v_esc = sqrt(2 G M / R), where G is the gravitational constant (6.67430 x 10^-11 N m^2 kg^-2, NIST CODATA 2018), M is the body's total mass, and R is its radius. Escape velocity is independent of the object's mass: a proton and a spacecraft require the same speed to escape Earth. The concept underpins rocket mission planning, the physics of black holes (where escape velocity equals c), and whether a planet can retain a particular atmosphere over geological time. Lighter gas molecules that move faster than a planet's escape velocity gradually leak away into space. This calculator shows escape velocity in m/s, km/s, km/h, and mph for Earth, the Moon, Mars, Jupiter, the Sun, or any custom body you define. Select a preset or enter your own mass and radius to calculate instantly.
Escape velocity: --
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Escape velocity formula
v_esc = √(2 × G × M / R)
where G = 6.67430 x 10^-11 N m^2 kg^-2 (NIST CODATA 2018), M is mass in kg, and R is radius in metres. The result is in m/s.
Derivation
Setting kinetic energy equal to gravitational potential energy at the surface: (1/2) m v^2 = G M m / R. Solving for v: v = sqrt(2 G M / R). The object's mass m cancels, confirming escape velocity is independent of the escaping object's mass.
Escape velocities for common bodies
| Body | Mass (kg) | Radius (m) | Escape velocity (km/s) |
|---|---|---|---|
| Moon | 7.342 × 10^22 | 1,737,000 | 2.38 |
| Mars | 6.39 × 10^23 | 3,389,500 | 5.03 |
| Earth | 5.972 × 10^24 | 6,371,000 | 11.19 |
| Jupiter | 1.898 × 10^27 | 69,911,000 | 59.50 |
| Sun | 1.989 × 10^30 | 695,700,000 | 617.50 |
Sources: NASA Planetary Fact Sheet (2024); NIST CODATA 2018.
Escape velocity vs circular orbital velocity
For a circular orbit at the surface: v_orb = sqrt(G M / R). The ratio of escape to orbital velocity is always sqrt(2) approximately 1.414. So escape velocity is about 41.4% higher than the speed needed to orbit at the same altitude.
Unit conversions used
- 1 km/s = 1,000 m/s
- 1 km/h = 1/3.6 m/s
- 1 mph = 0.44704 m/s (exact, NIST)
Frequently asked questions
What is escape velocity and how is it calculated?
Escape velocity is the minimum speed an object must have at the surface of a body (ignoring air resistance) to escape its gravitational pull without any further propulsion. It is derived by setting kinetic energy equal to gravitational potential energy: v_esc = sqrt(2 G M / R), where G = 6.67430 x 10^-11 N m^2 kg^-2, M is the body's mass in kg, and R is its radius in metres. Earth's escape velocity at the surface is approximately 11.186 km/s.
How does escape velocity compare to orbital velocity?
For a circular orbit at the surface of a body, the orbital velocity is v_orb = sqrt(G M / R). Escape velocity is exactly sqrt(2) times the circular orbital velocity: v_esc = sqrt(2) x v_orb. At Earth's surface, circular orbital speed would be about 7.91 km/s and escape velocity is 11.19 km/s. In practice, orbital altitude raises the required orbital speed only slightly while escape velocity from low Earth orbit is about 11.0 km/s.
What is a black hole's escape velocity?
A black hole is defined as a body whose escape velocity equals or exceeds the speed of light c = 299,792,458 m/s. Setting v_esc = c in the formula gives the Schwarzschild radius: R_s = 2 G M / c^2. For Earth's mass, R_s is about 8.9 mm. For the Sun, R_s is about 2.95 km. Any mass compressed below its Schwarzschild radius becomes a black hole from which no light can escape.
Do rockets actually need to reach escape velocity to leave Earth?
No. Escape velocity is the speed needed for a ballistic (unpowered) trajectory to escape. Rockets continue burning fuel throughout their ascent, so they can escape at speeds below escape velocity as long as they maintain thrust long enough. In practice, reaching low Earth orbit takes about 9.4 km/s of delta-v including atmospheric drag losses, then additional burns raise the orbit further. Escape from Earth orbit to interplanetary space requires roughly 3.2 km/s more delta-v from LEO.
Why did Moon missions use multi-stage rockets?
Reaching the Moon requires escaping Earth's gravity well (about 11.2 km/s surface escape velocity), decelerating into lunar orbit, landing, lifting off from the Moon (escape velocity 2.38 km/s), and returning to Earth. The tyranny of the rocket equation means propellant mass grows exponentially with delta-v. Multi-stage rockets discard empty tanks to reduce mass, making it practical to achieve the roughly 15 km/s total delta-v budget for a lunar round trip with a useful payload.