Question

With what speed $v_0$ should a body be projected as shown in figure, with respect to a planet of mass $M$ so that it would just be able to graze the planet and escape? The radius of the planet is $R$. (Assume that the planet is fixed).

Answer 1

By conservation of angular momentum we obtain
$m{v}_{0}d=mvr$
$\Rightarrow v=\frac{m{v}_{0}d}{mr}=v_0\left(d/r\right)$
By conservation of energy at $A$ and $B$, we have
$\frac{1}{2}m{v}_0^2-\frac{GMm}{\sqrt{D^2+d^2}}=\frac{1}{2}m{v}^2-\frac{GMm}{r}$
When $r\approx R$ we obtain,
$\frac{1}{2}m{v}_0^2-\frac{GMm}{\sqrt{D^2+d^2}}=\frac{1}{2}m.\frac{v_0^2{d}^2}{R^2}-\frac{GMm}{R}$
$\Rightarrow \frac{1}{2}m{v}_0^2(\frac{d^2}{R^2}-1)=GMm[\frac{1}{R}-\frac{1}{\sqrt{D^2+d^2}}]$
$\Rightarrow v_0=\sqrt{\frac{2GM(\frac{1}{R}-\frac{1}{\sqrt{D^2+d^2}})}{(\frac{d^2}{R^2}-1)}}$.