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Kepler's Law

A planet orbi...

Question

A planet orbits the sun in a n 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 the following relations is correct.?

A

\frac{r_{P}}{r_{A}} = \frac{v_{A}}{v_{P}}

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B

\frac{r_{P}}{r_{A}} = \frac{v_{p}}{v_{A}}

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C

\frac{r_{P}}{r_{A}} = \sqrt{\frac{v_{p}}{v_{A}}}

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D

\frac{r_{P}}{r_{A}} = \sqrt{\frac{v_{A}}{v_{P}}}

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Solution

The correct option is A $\frac{r_{P}}{r_{A}} = \frac{v_{A}}{v_{P}}$
Angular momentum of planet at P, $L_{p} = m_{p} V_{p} r_{p}$ where $m_{p}$ is the mass of the planet.
Angular momentum of planet at A,
$L_{A} = m_{p} v_{A} r_{A}$
According to the law of conservation of angular momentum
$L_{P} = L_{A}, m_{P} v_{P} r_{P} = m_{P} v_{A} r_{A}$
$o r \frac{r_{P}}{r_{A}} = \frac{v_{A}}{v_{P}}$

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