Calculate escape velocity, the minimum initial velocity of an object to ensure that it will continue traveling into space and never fall back to Earth (assuming that no force is applied after takeoff). Take the limit as r → ∞in Exercise 41.

Data From Exercise 41

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 a separation 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.