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Instantaneous Velocity

A planet of m...

Question

A planet of mass $m$ moves along an ellipse around the sun so that its maximum and minimum distance from the sun are equal to $r_{1}$ and $r_{2}$ respectively. Find the angular momentum of this planet relative to the centre of the sun.

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Solution

The angular momentum
$L = m v r . . . . . (1)$
Conservation of angular momenta at point $1$ and $2$ ;
$m v_{1} r_{1} = m v_{2} r_{2}$
$\Rightarrow v_{1} r_{1} = v_{2} r_{2} . . . . . (2)$
Conservation of energy at $1$ and $2$ ;
$\frac{1}{2} m v_{1}^{2} - \frac{G M m}{r_{1}} = \frac{1}{2} m v_{2}^{2} - \frac{G M m}{r_{2}} . . . . (3)$
Using equation $(2)$ and $(3)$ we obtain
$v_{1} = \sqrt{2 G M (\frac{1}{r_{1}} - \frac{1}{r_{2}}) + v_{2}^{2}}$
$\Rightarrow v_{1} = \sqrt{2 G M (\frac{1}{r_{1}} - \frac{1}{r_{2}}) + {(\frac{v_{1} r_{1}}{r_{2}})}^{2}}$
$\Rightarrow v_{1}^{2} [1 - {(\frac{r_{1}}{r_{2}})}^{2}] = 2 G M (\frac{r_{2} - r_{1}}{r_{1} r_{2}})$
$\Rightarrow v_{1} = \sqrt{\frac{2 G M r_{2}}{r_{1} (r_{1} + r_{2})}}$
$∴ L = m v_{1} r_{1} = m \sqrt{\frac{2 G M r_{1} r_{2}}{r_{1} + r_{2}}}$ .

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