# Escape velocities

Escape velocities

Q: What are the escape velocities for the earth, solar system and galaxy?
[29 Nov 1999]

A: If we're a unit mass at distance r from an object of mass M and want to get to infinity then the potential energy required is the integral of MG / x2 between x = r and infinity, which is MG / r. To escape, our kinetic energy v2 / 2 must be equal to this.

If we're on the surface of the object, and it has surface gravity g, then g = MG / r2 so the potential energy required is gr and the escape velocity is sqrt(2gr).

If we're in a circular orbit around the object, then MG / r2 = rw2, where w is the angular velocity, so the potential energy required is r2w2 and the escape velocity is sqrt(2r2w2).

Thus we have

• Escape from earth: g = 9.81 m/s2, r = 6.4e6 m so v = 1.1e4 m/s = 2.5e4 mph.
• Escape from solar system: r = 1.5e11 m, w = 2 pi / year = 2e-7 radians/s so v = 4.2e4 m/s = 9.4e4 mph.
• Escape from galaxy: r = 30000 light years = 2.8e20m, w = 2 pi / (2e8 years) = 1e-15 radians/s so v = 4e5 m/s = 9e5 mph.

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