(a) Use the substitution v = dy/dt to solve (13) for v in terms of y. Assuming that the velocity of the rocket at burnout is v = v₀ and y ≈ R at that instant, show that the approximate value of the constant c of integration is c = -gR + ½ v²₀.

(b) Use the solution for v in part (a) to show that the escape velocity of the rocket is given by v0 √2gR. Take and assume v → 0 for all time t.

(c) The result in part (b) holds for any body in the Solar System. Use the values g = 32 ft/s² and R = 4000 mi to show that the escape velocity from the Earth is (approximately) v₀ = 25,000 mi/h. 

(d) Find the escape velocity from the Moon if the acceleration of gravity is 0.165g and R = 1080 mi.