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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 $C J s^{- 1}$ . The time taken for the satellite to reach the earth is: $\frac{G M m}{x C} [\frac{1}{R} - \frac{1}{r}]$ . Find the value of $x$ .

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Solution

The total energy at radius r is given by

$E = - \frac{G M m}{r} + \frac{G M m}{2 r} = - \frac{G M m}{2 r}$

Hence, let the time be t. Then energy lost will be

$C t = \frac{G M m}{2} (\frac{1}{R} - \frac{1}{r})$

$t = \frac{G M m}{2 C} (\frac{1}{R} - \frac{1}{r})$

Answer is $t = \frac{G M m}{2 C} (\frac{1}{R} - \frac{1}{r})$

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