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A planet orbits the sun in an elliptical path as shown in the figure. Let $v_P$ and $v_A$ be speed of the planet when at perihelion and aphelion respectively. Which of the following relations is correct ?
$\frac{{{r_P}}}{{{r_A}}} = \frac{{{v_A}}}{{{v_P}}}$
$\frac{{{r_P}}}{{{r_A}}} = \frac{{{v_P}}}{{{v_A}}}$
$\frac{{{r_P}}}{{{r_A}}} = \sqrt {\frac{{{v_P}}}{{{v_A}}}} $
$\frac{{{r_P}}}{{{r_A}}} = \sqrt {\frac{{{v_A}}}{{{v_P}}}} $
Solution
According to the law of conservation of angular momentum.
$\mathrm{L}_{\mathrm{P}}=\mathrm{L}_{\mathrm{A}} \Rightarrow \mathrm{m}_{\mathrm{P}} \mathrm{v}_{\mathrm{P}} \mathrm{r}_{\mathrm{P}}=\mathrm{m}_{\mathrm{A}} \mathrm{v}_{\mathrm{A}} \mathrm{r}_{\mathrm{A}} \Rightarrow\left[\frac{\mathrm{r}_{\mathrm{P}}}{\mathrm{r}_{\mathrm{A}}}=\frac{\mathrm{v}_{\mathrm{A}}}{\mathrm{v}_{\mathrm{P}}}\right]$
