A body is projected vertically upwards from the surface of the earth with an initial velocity equal to half of the escape velocity. If R is the radius of the earth, the maximum height attained by the body from the surface of the earth is

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 . Initial kinetic energy of the body: K_i = 1 2 mv^2 = 1 2 m ( 2GM 4R ) = GMm 4R Initial gravitational potential energy at the surface: U_i = - GMm R At the maximum height h , the velocity becomes zero, so the final kinetic energy K_f = 0 . Final gravitational potential energy at height h : U_f = - GMm R+h According to the principle of