A body is projected vertically upwards from the bottom of a deep mine of depth R 2 , where R is the radius of the Earth. If v_e is the escape speed from the surface of the Earth, the minimum initial speed required for the body to completely escape the Earth's gravitational pull is:

Step-by-step solution

The gravitational potential inside a solid uniform sphere at a distance r from its centre is given by: V(r) = - GM 2R^3 (3R^2 - r^2) For a mine of depth R 2 , the distance from the centre of the Earth is r = R - R 2 = R 2 . The potential at this point is: V ( R 2 ) = - GM 2R^3 (3R^2 - R^2 4 ) = - GM 2R^3 ( 11R^2 4 ) = - 11GM 8R For the body to just escape to infinity, its total mechanical energy must be zero. Applying conservation of mechanical energy: 1 2 mv^2 + m V ( R 2 ) = 0 1 2 mv^2 - 11GMm 8R = 0 v^2 = 11GM 4