A body is projected vertically upwards from the surface of the Earth with an initial velocity equal to half of the escape velocity. The maximum height attained by the body from the surface of the Earth will be : (where ' R ' is the radius of the Earth)

Step-by-step solution

Let the mass of the body be m and the mass of the Earth be M . The escape velocity from the surface of the Earth is v_e = 2GM R . Initial velocity of the body, v = v_e 2 = 1 2 2GM R = GM 2R . By the principle of conservation of mechanical energy: Initial Total Energy = Final Total Energy K_i + U_i = K_f + U_f 1 2 mv^2 - GMm R = 0 - GMm R+h 1 2 m ( GM 2R ) - GMm R = - GMm R+h GMm 4R - GMm R = - GMm R+h - 3GMm 4R = - GMm R+h 3 4R = 1 R+h 3R + 3h = 4R 3h = R h = R 3 Answer: R 3