Chapter 13.2, Problem 13.114P

A space probe describes a circular orbit of radius nR with a velocity v₀ about a planet of radius R and center O. Show that (a) in order for the probe to leave its orbit and hit the planet at an angle θ with the vertical, its velocity must be reduced to αv₀, where

$\alpha =\sin \theta \sqrt{\frac{2(n-1)}{n^2-{\sin }^2\theta }}$

(b) the probe will not hit the planet if α is larger than $\sqrt{2/(1+n)}$.