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A planet is revolving around the sun in 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 be v, then what is the velocity at the farther point?

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

$\frac{vr}{R}$

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

$\frac{vR}{r}$

answer is A.

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Detailed Solution

As angular momentum is conserved, hence,

$I_{1} ω_{1} = I_{2} ω_{2}$

${MR}_{1}^{2} ω_{1} = {MR}_{2}^{2} ω_{2} \Rightarrow {MR}_{1} v_{1} = {MR}_{2} v_{2} (∴ ω = \frac{v}{R})$
$\Rightarrow R_{1} v_{1} = R_{2} v_{2} \Rightarrow VR = vr$

$∴ V = \frac{vr}{R}$

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