Question

A satellite is revolving in a circular orbit at a height $'h'$ above the surface of the earth of radius $'R'$. The speed of the satellite in its orbit is one-fourth the escape velocity from the surface of the earth. The relation between $'h'{\text{ and }}'R'$ is
A) $h = 2R$
B) $h = 3R$
C) $h = 5R$
D) $h = 7R$

Answer 1

We will use the relation of space velocity provided us to find the velocity of the satellite in terms of the escape velocity at the surface of the earth. We will then balance the centripetal force required for the orbiting of a satellite with the gravitational force it experiences.

Formula used: In this solution, we will use the following formula
$\Rightarrow v = \sqrt {\dfrac{{2GM}}{R}}$ where $v$ is the escape velocity of Earth for an object at its surface.

Complete step by step answer
We’ve been told that the speed of the satellite is such that in its orbit, its escape velocity is one-fourth the escape velocity from the surface of the Earth. Since the escape velocity at the surface of the Earth is
$\Rightarrow {v_E} = \sqrt {\dfrac{{2GM}}{R}}$
The velocity of the satellite will be
$\Rightarrow {v_S} = {v_E}/4$
Now for the satellite to orbit around the Earth, its centripetal acceleration must be equal to the gravitational force it experiences due to the earth. So the centripetal force and the gravitational force at a height $h$ above the surface of the Earth will be:
$\Rightarrow \dfrac{{m{v_s}^2}}{{R + h}} = \dfrac{{GMm}}{{{{\left( {R + h} \right)}^2}}}$
Since ${v_S} = {v_E}/4$, we can write
$\Rightarrow \dfrac{{m{v_E}^2}}{{16(R + h)}} = \dfrac{{GMm}}{{{{\left( {R + h} \right)}^2}}}$
And further ${v_E} = \sqrt {\dfrac{{2GM}}{R}}$, we can say
$\Rightarrow \dfrac{1}{{16}}\dfrac{{2GM}}{R} = \dfrac{{GM}}{{R + h}}$
Cancelling out terms on both sides we get
$\Rightarrow \dfrac{1}{{8R}} = \dfrac{1}{{R + h}}$
Cross multiplying the denominators, we get
$\Rightarrow R + h = 8R$

$\therefore h = 7R$ which corresponds to option (D).

Note
The relation of escape velocity given to us only serves the purpose to relate the escape velocity of the satellite to the escape velocity at the surface of the Earth whose value we know. We cannot use it any further and we would have to balance the centripetal force with gravitational acceleration since the satellite is in orbit and not escaping the gravity of Earth.