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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

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