Two satellites $M$ and $N$ go around the earth in circular orbits at heights of $R_{M}$ and $R_{N}$ respectively from the surface of the earth. Assuming the earth to be a uniform sphere of radius $R_{E}$ , the ratio of the velocities of the satellites $\frac{V_{M}}{V_{N}}$ is

{(\frac{R_{M}}{R_{N}})}^{2}

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\sqrt{\frac{R_{N} + R_{E}}{R_{M} + R_{E}}}

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\frac{R_{N} + R_{E}}{R_{M} + R_{E}}

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\sqrt{\frac{R_{N}}{R_{M}}}

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Solution

The correct option is A $\sqrt{\frac{R_{N} + R_{E}}{R_{M} + R_{E}}}$
Orbital speed $v = \sqrt{\frac{G M}{R_{E} + h}}$ ; as we can see from the equation, it is independent of the mass of the satellite.
$G$ and $M$ are constants, so
$v \propto \frac{1}{\sqrt{R_{E} + h}}$
$∴ \frac{V_{M}}{V_{N}} = \sqrt{\frac{R_{E} + R_{N}}{R_{E} + R_{M}}}$

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