A particle $P$ with unit mass moves in a central potential $\Phi(r)=-k / r$ where $k>0$. Initially $P$ is a distance $R$ away from the origin moving with speed $u$ on a trajectory which, in the absence of any force, would be a straight line whose shortest distance from the origin is $b$. The shortest distance between $P$ 's actual trajectory and the origin is $p$, with $0<p<b$, at which point it is moving with speed $w$.

(i) Assuming $u^{2} \gg 2 k / R$, find $w^{2} / k$ in terms of $b$ and $p$.

(ii) Assuming $u^{2}<2 k / R$, find an expression for $P^{\prime}$ 's farthest distance from the origin $q$ in the form

$$A q^{2}+B q+C=0$$

where $A, B$, and $C$ depend only on $R, b, k$, and the angular momentum $L$.

[You do not need to prove that energy and angular momentum are conserved.]