A particle of unit mass moves with angular momentum $h$ in an attractive central force field of magnitude $\frac{k}{r^{2}}$, where $r$ is the distance from the particle to the centre and $k$ is a constant. You may assume that the equation of its orbit can be written in plane polar coordinates in the form

$$r=\frac{\ell}{1+e \cos \theta}$$

where $\ell=\frac{h^{2}}{k}$ and $e$ is the eccentricity. Show that the energy of the particle is

$$\frac{h^{2}\left(e^{2}-1\right)}{2 \ell^{2}}$$

A comet moves in a parabolic orbit about the Sun. When it is at its perihelion, a distance $d$ from the Sun, and moving with speed $V$, it receives an impulse which imparts an additional velocity of magnitude $\alpha V$ directly away from the Sun. Show that the eccentricity of its new orbit is $\sqrt{1+4 \alpha^{2}}$, and sketch the two orbits on the same axes.