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Satellites

Magnitude of ...

Question

Magnitude of binding energy of satellite is $E$ and kinetic energy is $K$ . The ratio $E / K$ is :

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Solution

The correct option is A $1$
Kinetic energy of satellite $(E)$
$K E (k) = \frac{1}{2} m v^{2}$
$= \frac{1}{2} m {(\sqrt{\frac{G M}{R}})}^{2} (∵ v = \sqrt{\frac{G M}{R}})$ [ $m =$ mass of satellite]
$= \frac{1}{2} m \frac{G M}{R}$
[Symbols have usual meanings]
Potential energy of the satellite,
$U = - m \frac{G M}{R}$
$∴$ Binding energy of the satellite
$E_{b} = K E + U$
$= \frac{1}{2} m \frac{G M}{R} + (- m \frac{G M}{R})$
$= \frac{1}{2} m \frac{G M}{R} - m \frac{G M}{R}$
$= - \frac{G M m}{2 R}$
$∴$ Magnitude of the binding energy $E = | E_{b} | = \frac{G M m}{2 R}$
The ratio of $E / K = \frac{G M m}{2 R} \times \frac{2 R}{G M m} = 1$

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