Two satellites revolve around the sun, as shown in the figure. The first satellite revolves in a circular orbit of radius $R$ with speed $V_{1}$ . The second satellite revolves in an elliptical orbit, for which minimum and maximum distances from the sun are $R / 3$ and $5 R / 3$ respectively. Velocities of these positions are $V_{2}$ and $V_{3}$ respectively. The correct order of speeds is -

V_{2} > V_{1} > V_{3}

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V_{2} > V_{3} > V_{1}

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V_{1} > V_{2} > V_{3}

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V_{3} > V_{1} > V_{2}

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Solution

The correct option is A $V_{2} > V_{1} > V_{3}$
We know that, orbital velocity of a satellite moving in a circular orbit of radius $R$ around the sun of mass $M$ is given by

$V_{1} = \sqrt{\frac{G M}{R}}$

For an elliptical path taken by the satellite, using conservation of angular momentum,

$\frac{m V_{2} R}{3} = \frac{m V_{3} 5 R}{3}$

$\Rightarrow V_{2} = 5 V_{3} . . . (1)$

Using conservation of energy,

$(- \frac{G M m}{R / 3}) + \frac{1}{2} m V_{2}^{2} = \frac{1}{2} m V_{3}^{2} + (- \frac{G M m}{5 R / 3}) . . . (2)$

Solving $(1)$ and $(2)$ we get,

$V_{2} = \sqrt{\frac{5 G M}{R}}$ , and $V_{3} = \sqrt{\frac{G M}{5 R}}$

So, $V_{2} > V_{1} > V_{3}$

Hence, option $(A)$ is the correct answer.

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