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Potential Energy of an Object at a Height

A satellite o...

Question

A satellite of mass m is orbiting the earth in a circular orbit of radius $r$ . It starts losing energy due to small air resistance at the rate of $C J / s$ . Then the time taken for the satellite to reach the earth is

A

\frac{G M m}{2 C} (\frac{1}{R_{e}} - \frac{1}{r})

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B

\frac{G M m}{2 C} (\frac{1}{R_{e}} + \frac{1}{r})

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C

\frac{G M m}{C} (\frac{1}{R_{e}} - \frac{1}{r})

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D

will not each earth

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Solution

The correct option is A $\frac{G M m}{2 C} (\frac{1}{R_{e}} - \frac{1}{r})$
Total kinetic energy of satellite at $r = \frac{G M m}{2 r}$
Total kinetic energy of satellite on earth $= \frac{G M m}{2 R_{e}}$
Net loss in energy $E = f i n a l - i n i t i a l = \frac{G M m}{2 R_{e}} - \frac{G M m}{2 r}$
Given that it loses at the rate $r = C J s^{- 1}$ .
Therefore total time $= \frac{E}{r} = \frac{\frac{G M m}{2 R_{e}} - \frac{G M m}{2 r}}{C} = \frac{G M m}{2 C} (\frac{1}{R_{e}} - \frac{1}{r}) s$

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