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A planet is revolving around the Sun in an elliptical orbit. Its closest distance from the Sun is r and farthest distance is R. If the orbital velocity of the planet closest to the Sun is v, then what is the velocity at the farthest point?

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$\frac{vr}{R}$

$\frac{vR}{r}$

$v \sqrt{\frac{r}{R}}$

$v \sqrt{\frac{R}{r}}$

answer is A.

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Detailed Solution

As angular momentum is conserved, hence

$\begin{array}{l} I_{1} ω_{1} = I_{2} ω_{2} \\ or {MR}_{1}^{2} ω_{1} = {MR}_{1}^{2} ω_{2} or {MR}_{1} v_{1} = {MR}_{2} v_{2} \\ (∵ ω = \frac{v}{R}) \\ or R_{1} v_{1} = R_{2} v_{2} or VR = vr \\ ∴ V = \frac{vr}{R} \end{array}$

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