A particle of unit mass moves in a plane with polar coordinates $(r, \theta)$ and components of acceleration $\left(\ddot{r}-r \dot{\theta}^{2}, r \ddot{\theta}+2 \dot{r} \dot{\theta}\right)$. The particle experiences a force corresponding to a potential $-Q / r$. Show that

$$E=\frac{1}{2} \dot{r}^{2}+U(r) \quad \text { and } \quad h=r^{2} \dot{\theta}$$

are constants of the motion, where

$$U(r)=\frac{h^{2}}{2 r^{2}}-\frac{Q}{r}$$

Sketch the graph of $U(r)$ in the cases $Q>0$ and $Q<0$.

(a) Assuming $Q>0$ and $h>0$, for what range of values of $E$ do bounded orbits exist? Find the minimum and maximum distances from the origin, $r_{\min }$ and $r_{\max }$, on such an orbit and show that

$$r_{\min }+r_{\max }=\frac{Q}{|E|} .$$

Prove that the minimum and maximum values of the particle's speed, $v_{\min }$ and $v_{\max }$, obey

$$v_{\min }+v_{\max }=\frac{2 Q}{h}$$

(b) Now consider trajectories with $E>0$ and $Q$ of either sign. Find the distance of closest approach, $r_{\min }$, in terms of the impact parameter, $b$, and $v_{\infty}$, the limiting value of the speed as $r \rightarrow \infty$. Deduce that if $b \ll|Q| / v_{\infty}^{2}$ then, to leading order,

$$r_{\min } \approx \frac{2|Q|}{v_{\infty}^{2}} \text { for } Q<0, \quad r_{\min } \approx \frac{b^{2} v_{\infty}^{2}}{2 Q} \text { for } Q>0$$