Question

If the kinetic energy of a satellite orbiting around the earth is doubled, then

• A. the satellite will escape into the space.
• B. the satellite will fall down on the earth.
• C. radius of its orbit will be doubled.
• D. radius of its orbit will become half.

Answer 1

The correct option is A the satellite will escape into the space.

Suppose the satellite is orbiting at an altitude of $r$ from the centre of earth.
Then its binding energy,
$E=\frac{-GMm}{r}+\frac{1}{2}m{v}^2$

Also, required centripetal force $=$ gravitational force
$\Rightarrow \frac{m{v}^2}{r}=\frac{GMm}{r^2}$
$\Rightarrow v^2=\frac{GM}{r}$

$\therefore E=\frac{-GMm}{r}+\frac{1}{2}m\times \frac{GM}{r}$

$\Rightarrow E=\frac{-GMm}{2r}<0$
$\Rightarrow$ System is bounded.

Also, its kinetic energy $=\frac{GMm}{2r}$

If kinetic energy is doubled, $i.e.$ it becomes $\frac{GMm}{r}$, then, new binding energy of the system,

$E^{{}^{\prime }}=\frac{-GMm}{r}+\frac{GMm}{r}$

$\Rightarrow E^{\prime }=0$
$\Rightarrow$ System is unbounded.

Therefore, satellite will escape into the space.