Question

Two satellites of masses $100kg$ and $200kg$ are revolving around the earth at altitudes $6400km$ and $44800km$. The ratio of the orbital velocity of the satellites is:
(A) $1:2$
(B) $2:1$
(C) $1:4$
(D) $4:1$

Answer 1

Hint
Simply substitute the values given in the question above into the orbital velocity formula. Keep in mind that the mass of the smaller body is considered to be negligible compared to the heavier body around which the satellite is orbiting.
$\Rightarrow v = \sqrt {\dfrac{{GM}}{{R + h}}} $ where $G$ is the universal gravitational constant, $M$ is the mass of the central body which in this case is the earth, $R$is the radius of earth, $h$is the altitude of the satellite and $v$is the orbital velocity with which the satellite is revolving around the earth.

Complete step by step answer
Orbital velocity is the velocity of the satellite while it is orbiting a heavier mass.
For a satellite to orbit around earth, its centrifugal force must be equal to the gravitational force of attraction experienced by it due to earth’s gravity.
Let us take the radius of Earth to be $6400km$
Now the formula for the orbital velocity of satellites for earth is
$\Rightarrow v = \sqrt {\dfrac{{GM}}{{R + h}}} $ where $G$ is the universal gravitational constant, $M$ is the mass of the central body which in this case is the earth, $R$is the radius of earth and $h$is the altitude of the satellite.
In this formula we see that the orbital velocity is independent of the mass of the satellites and is only dependent on the mass of the earth.
So for the first satellite of mass $100kg$, the orbital velocity is,
$\Rightarrow {v_1} = \sqrt {\dfrac{{GM}}{{6400 + 6400}}} = \sqrt {\dfrac{{GM}}{{12800}}} $
The orbital velocity of the second satellite is
$\Rightarrow {v_2} = \sqrt {\dfrac{{GM}}{{6400 + 44800}}} = \sqrt {\dfrac{{GM}}{{51200}}} $
Therefore, their ratio is given by,
$\Rightarrow \dfrac{{{v_1}}}{{{v_2}}} = \dfrac{{\sqrt {\dfrac{{GM}}{{12800}}} }}{{\sqrt {\dfrac{{GM}}{{51200}}} }} = \sqrt {\dfrac{{51200}}{{12800}}} = \sqrt {\dfrac{4}{1}} $
$\Rightarrow \dfrac{{{v_2}}}{{{v_1}}} = \dfrac{2}{1} $
Therefore, the correct option is (B).

Note
At an altitude of $100 \,miles $ from the surface of the earth the density of air in the atmosphere is so small that at this height there is almost no friction between the satellite and air. That is why, the altitude of a satellite should be at least $100miles$if it is to orbit around the earth at a reasonable time.