For large distances from the surface of Earth, the gravitational force is given by F(x) = GMm/(x + R)², where G = 6.7 × 10^-11 N-m²/kg² is the gravitational constant, M = 6 × 10²⁴ kg is the mass of Earth, m is the mass of the object in the gravitational field, R = 6.378 × 10⁶ m is the radius of Earth, and x ≥ 0 is the distance above the surface of Earth (in meters). 

a. How much work is required to launch a rocket with a mass of 500 kg in a vertical flight path to a height of 2500 km (from Earth’s surface)?

b. Find the work required to launch the rocket to a height of x kilometers, for x > 0.

c. How much work is required to reach outer space (x →∞)?

d. Equate the work in part (c) to the initial kinetic energy of the rocket, 1/2 mv², to compute the escape velocity of the rocket.