A particle is initially at rest at a height R (the radius of the earth) above the surface of the earth. Determine the minimum speed at which it must be projected to ensure it never returns.

Step-by-step solution

The initial distance of the particle from the centre of the earth is r = R + h = R + R = 2R . The gravitational potential energy of the particle at this height is U = - GMm 2R . For the particle to escape the earth's gravitational field and never return, its total mechanical energy must be greater than or equal to zero. 1 2 mv^2 - GMm 2R 0 1 2 mv^2 GMm 2R v^2 GM R v GM R The minimum speed of projection is GM R .