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Satellites

Two satellite...

Question

Two satellites A and B go around the earth in circular orbits at a height of $R_{A}$ and $R_{B}$ respectively from the surface of the earth. Assume the earth to be a uniform sphere of radius $R_{E}$ . The ratio of the magnitudes of the velocities of the satellites $\frac{V_{A}}{V_{B}}$ is:

\sqrt{\frac{R_{B}}{R_{A}}}

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\frac{R_{B} + R_{E}}{R_{A} + R_{E}}

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\sqrt{\frac{R_{B} + R_{E}}{R_{A} + R_{E}}}

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{(\frac{R_{A}}{R_{B}})}^{2}

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Solution

The correct option is B $\sqrt{\frac{R_{B} + R_{E}}{R_{A} + R_{E}}}$
Orbital speed $v = \sqrt{\frac{G M}{R_{E} + h}}$ ; $h$ distance from Earth's surface.
Here $G$ and mass of the earth $(M)$ are constants.
Orbital velocity for satellite $A$ is given by,
$v_{A} = \sqrt{\frac{G M}{R_{E} + R_{A}}}$
Orbital velocity for satellite $B$ is given by,
$v_{B} = \sqrt{\frac{G M}{R_{E} + R_{B}}}$
By taking ratio of $v_{A}$ and $v_{B}$
$∴ \frac{v_{A}}{v_{B}} = \sqrt{\frac{R_{E} + R_{B}}{R_{E} + R_{A}}}$

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