(a) A mass $m$ is acted upon by a central force

$$\mathbf{F}=-\frac{k m}{r^{3}} \mathbf{r}$$

where $k$ is a positive constant and $\mathbf{r}$ is the displacement of the mass from the origin. Show that the angular momentum and energy of the mass are conserved.

(b) Working in plane polar coordinates $(r, \theta)$, or otherwise, show that the distance $r=|\mathbf{r}|$ between the mass and the origin obeys the following differential equation

$$\ddot{r}=-\frac{k}{r^{2}}+\frac{h^{2}}{r^{3}}$$

where $h$ is the angular momentum per unit mass.

(c) A satellite is initially in a circular orbit of radius $r_{1}$ and experiences the force described above. At $\theta=0$ and time $t_{1}$, the satellite emits a short rocket burst putting it on an elliptical orbit with its closest distance to the centre $r_{1}$ and farthest distance $r_{2}$. When $\theta=\pi$ and the time is $t_{2}$, the satellite reaches the farthest distance and a second short rocket burst puts the rocket on a circular orbit of radius $r_{2}$. (See figure.) [Assume that the duration of the rocket bursts is negligible.]

(i) Show that the satellite's angular momentum per unit mass while in the elliptical orbit is

$$h=\sqrt{\frac{C k r_{1} r_{2}}{r_{1}+r_{2}}}$$

where $C$ is a number you should determine.

(ii) What is the change in speed as a result of the rocket burst at time $t_{1}$ ? And what is the change in speed at $t_{2}$ ?

(iii) Given that the elliptical orbit can be described by

$$r=\frac{h^{2}}{k(1+e \cos \theta)}$$

where $e$ is the eccentricity of the orbit, find $t_{2}-t_{1}$ in terms of $r_{1}, r_{2}$, and $k$. [Hint: The area of an ellipse is equal to $\pi a b$, where $a$ and b are its semi-major and semi-minor axes; these are related to the eccentricity by $\left.e=\sqrt{1-\frac{b^{2}}{a^{2}}} .\right]$