Question

A satellite with kinetic energy E revolving round the earth in a circular orbit. The minimum additional kinetic energy required for it to escape into outer space is
A. $\sqrt 2 E$
B. $2E$
C. $\dfrac{E}{{\sqrt 2 }}$
D. $E$

Answer 1

The satellite orbiting the earth is held in its orbital constantly, due to the gravitational force. We can solve this problem when we find the total mechanical energy possessed by the satellite due to gravitational force so that an equal amount of energy can be supplied to the satellite to nullify the gravitational force and escape the orbit.

Complete step by step answer:
The satellite orbiting around the Earth is held by its gravitational force. Because of the gravitational force, there is a centripetal force acting on the satellite. This centripetal force constantly acting on the satellite, causes it to move in circular orbits at a constant velocity, v.
This velocity is given by –
$v = \sqrt {\dfrac{{GM}}{{R + h}}}$
where M is the mass of earth, R is the radius of earth and G is the gravitational constant and h = height at which the satellite is present.
Since, the value of h is very less compared to R, we can say, $R + h \approx R$
Thus,
$\Rightarrow v = \sqrt {\dfrac{{GM}}{R}}$
Due to this velocity, the kinetic energy possessed by the satellite in its orbit is –
$E = \dfrac{1}{2}m{v^2}$
where m = mass of the satellite.
Substituting,
$\Rightarrow E = \dfrac{1}{2}\dfrac{{GMm}}{R}$
Due to Earth’s gravitational field, the potential energy of the satellite is given by –
$P = - \dfrac{{GmM}}{{R + h}}$
Since, $R + h \approx R$
$\Rightarrow P = - \dfrac{{GmM}}{R}$
The total mechanical energy is the sum of potential energy and the kinetic energy.
Hence, we have
$T = P + E$
$\Rightarrow T = \dfrac{1}{2}\dfrac{{GMm}}{R} - \dfrac{{GMm}}{R} = - \dfrac{1}{2}\dfrac{{GMm}}{R}$
$\Rightarrow T + \dfrac{1}{2}\dfrac{{GMm}}{R} = 0$
From the equation of kinetic energy, we have –
$\Rightarrow T + E = 0$
$\therefore T = - E$

So, in order for the satellite to escape, energy equivalent to the total energy i.e. the kinetic energy must be supplied.

Hence, the correct option is Option D.

Note:
The minimum value of the velocity that the object should have, to escape the Earth’s gravitational influence, is termed as the escape velocity. The value of escape velocity for Earth is $11km{s^{ - 1}}$. This must be the minimum velocity that should be substituted in the above value for this equation to remain correct.