A particle of unit mass moves under the influence of a central force. By considering the components of the acceleration in polar coordinates $(r, \theta)$ prove that the magnitude of the angular momentum is conserved. [You may use $\ddot{\mathbf{r}}=\left(\ddot{r}-r \dot{\theta}^{2}\right) \hat{\mathbf{r}}+(2 \dot{r} \dot{\theta}+r \ddot{\theta}) \hat{\boldsymbol{\theta}}$. ]

Now suppose that the central force is derived from the potential $k / r$, where $k$ is a constant.

(a) Show that the total energy of the particle can be written in the form

$$E=\frac{1}{2} \dot{r}^{2}+V_{\mathrm{eff}}(r)$$

Sketch $V_{\text {eff }}(r)$ in the cases $k>0$ and $k<0$.

(b) The particle is projected from a very large distance from the origin with speed $v$ and impact parameter $b$. [The impact parameter is the distance of closest approach to the origin in absence of any force.]

(i) In the case $k<0$, sketch the particle's trajectory and find the shortest distance $p$ between the particle and the origin, and the speed $u$ of the particle when $r=p$.

(ii) In the case $k>0$, sketch the particle's trajectory and find the corresponding shortest distance $\widetilde{p}$ between the particle and the origin, and the speed $\widetilde{u}$of the particle when $r=\widetilde{p}$.

(iii) Find $p \tilde{p}$ and $u \tilde{u}$ in terms of $b$ and $v$. [In answering part (iii) you should assume that $|k|$ is the same in parts (i) and (ii).]