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Two satellites $S_{1} {and S}_{2}$ are to be set into the orbits of heights $R/4 and R/6$ (above the surface of earth) respectively. They revolve around the earth in coplanar circular orbits in opposite sense (where, R = radius of earth). Which of the following statement(s) $is/are$ correct

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The ratio of their time periods is

{(\frac{15}{14})}^{3 /2}

The ratio of their time periods is

{(\frac{14}{15})}^{3 /2}

The ratio of their speeds of projection from the surface of earth is

\sqrt{\frac{23}{15}}

The ratio of their speeds of projection from the surface of earth is

\sqrt{\frac{21}{20}}

answer is D, A.

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Detailed Solution

h_{1} = \frac{R}{4} {,     h}_{2} = \frac{R}{6}

$T^{2} \propto R^{3}$

$\frac{T_{1}}{T_{2}} = {(\frac{R_{1}}{R_{2}})}^{3 / 2} = {(\frac{R + \frac{R}{4}}{R + \frac{R}{6}})}^{3 / 2}$

$= {(\frac{\frac{5}{4} R}{\frac{7}{6} R})}^{3 / 2} = {(\frac{15}{14})}^{3 / 2} \to (1)$

$V = \sqrt{\frac{2GM}{R} (\frac{h}{R + h})}$

$V_{1} = \sqrt{\frac{2GM}{R} (\frac{1}{5})} {, V}_{2} = \sqrt{\frac{2GM}{R} (\frac{1}{7})}$

$\frac{V_{1}}{V_{2}} = {(\frac{7}{5})}^{1 / 2} = {(\frac{21}{15})}^{1 / 2} \to (4)$

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