A body is projected radially outwards from a height h = R above the surface of the Earth, where R is the radius of the Earth. If the body is projected with a speed equal to the escape velocity of the Earth's surface ( v_e ), what will be its speed at an infinite distance from the Earth?

Step-by-step solution

Let the mass of the Earth be M and the mass of the body be m . The escape velocity at the surface of the Earth is v_e = 2GM R , which gives GM R = v_e^2 2 . The body is projected from a height h = R , so its initial distance from the centre of the Earth is r = R + h = 2R . Initial potential energy, U_i = - GMm 2R = - 1 2 m ( GM R ) = - 1 4 mv_e^2 . Initial kinetic energy, K_i = 1 2 mv_p^2 . Since v_p = v_e , K_i = 1 2 mv_e^2 . Total initial energy, E_i = K_i + U_i = 1 2 mv_e^2 - 1 4 mv_e^2 = 1 4 mv_e^2 . At infinit