A satellite of mass m is initially revolving in a circular orbit of radius R around the Earth (mass M ). It is then transferred to a higher circular orbit such that its new orbital speed becomes exactly half of its initial orbital speed. The work done by external forces to achieve this transfer is:

Step-by-step solution

The orbital speed of a satellite is given by v = GM r . Let the initial speed be v_i = GM R . The final speed is given as v_f = v_i 2 . GM r_f = 1 2 GM R = GM 4R Thus, the final orbital radius is r_f = 4R . The total mechanical energy of a satellite in a circular orbit of radius r is E = - GMm 2r . Initial total energy, E_i = - GMm 2R Final total energy, E_f = - GMm 2(4R) = - GMm 8R The work done by external forces is equal to the change in total mechanical energy: W = E_f - E_i W = - GMm 8R - (- GMm 2R ) W = GMm 2