A payload of mass m is resting on the surface of a spherical airless planet of mass M and radius R . The minimum work required to lift this payload and establish it in a stable circular orbit at an altitude of 3R above the planet's surface is:

Step-by-step solution

Initial energy of the payload on the surface of the planet is: E_i = K_i + U_i = 0 - GMm R = - GMm R The payload is established in a circular orbit at an altitude of 3R . The orbital radius from the center of the planet is: r = R + 3R = 4R In a stable circular orbit, the necessary centripetal force is provided by the gravitational force: mv^2 r = GMm r^2 mv^2 = GMm r The final kinetic energy of the payload is: K_f = 1 2 mv^2 = GMm 2r The final potential energy is: U_f = - GMm r The final total energy of the payload