Chapter 5.3, Problem 14E

1. (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 $c=-gR+\frac{1}{2}{v}_0^2$.
2. (b) Use the solution for v in part (a) to show that the escape velocity of the rocket is given by $v_0=\sqrt{2gR}$. [Hint: Take y → ∞ and assume v > 0 for all time t.]
3. (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.
4. (d) Find the escape velocity from the Moon if the acceleration of gravity is 0.165g and R = 1080 mi.