Escape Velocity.

The minimum velocity with which a body must be projected up so as to enable it to just overcome the gravitational pull, is known as escape velocity.

The work done to displace a body from the surface of earth (r = R) to infinity (r = ∞) is

Þ                    W = (GMm/R)

This work required to project the body so as to escape the gravitational pull is performed on the body by providing an equal      amount of kinetic energy to it at the surface of the earth.

If ν_e is the required escape velocity, then kinetic energy which should be given to the body is ½ mv_e²

there fore              ½ mv_e² = (GMm/R) Þ ν_e = (2GM/R)^1/2

                                                                         ν_e = (2gR)^1/2                          [As GM = gR²]

or                                [As g = 4/3 πρGR]

Note:

 (a) Escape velocity is independent of the mass and direction of projection of the body.

 (b) For the earth as g = 9.8 m/s² and R = 6400 km

                        ∴ ν = (2 x 9.8 x 6.4 x 10⁶)^1/2 = 11.2 km/sec