With what initial velocity v₀ must we fire a rocket so it attains a maximum height r above the earth? Use the results of Exercises 35 and 39. As the rocket reaches its maximum height, its KE decreases from 1/2 mv²₀ to zero.

Data From Exercise 35

The gravitational force between two objects of mass m and M, separated by a distance r, has magnitude GMm/r², where G = 6.67 × 10⁻¹¹ m³kg−1s⁻¹.

Show that if two objects of mass M and m are separated by a distance r₁, then the work required to increase the aseparation to a distance r₂ is equal to W = GMm(r₁⁻¹ − r₂⁻¹).

Data From Exercise 39

An object of mass m moves from x₁ to x₂ during the time interval [t₁, t₂] due to a force F(x) acting in the direction of motion. Let x(t), v(t), and a(t) be the position, velocity, and acceleration at time t. The object’s kinetic energy is KE = 1/2 mv².