Question

A satellite of mass $m$ revolves around the earth of radius $R$ at a height $x$ from its surface. If $g$ is the acceleration due to gravity on the surface of the earth, the orbital speed of the satellite is
$A)\text{ }\sqrt{gx}$
$B)\text{ }\sqrt{\dfrac{gR}{R-x}}$
$C)\text{ }\sqrt{\dfrac{g{{R}^{2}}}{R-x}}$
$D)\text{ }\sqrt{\dfrac{g{{R}^{2}}}{R+x}}$

Answer 1

This problem can be solved by using the direct formula for the orbital velocity of a satellite in terms of the radius of its orbit and the mass of the planet around which it revolves. By using this and the formula for the acceleration due to gravity in terms of the mass of the planet and its radius, we can combine both the equations to get the required answer.

Formula used:
$v=\sqrt{\dfrac{GM}{\left( R+h \right)}}$
$g=\dfrac{G{{M}_{E}}}{{{R}_{E}}^{2}}$

Complete step by step answer:
We will use the direct formula for the orbital velocity of a satellite orbiting around a planet.
The orbital velocity $v$ of a satellite revolving around a planet of mass $M$ and radius $R$ at a height $h$ above its surface is given by
$v=\sqrt{\dfrac{GM}{\left( R+h \right)}}$ --(1)
where $G=6.67\times {{10}^{-11}}N.{{m}^{2}}k{{g}^{-2}}$ is the universal gravitational constant.
Also, the acceleration due to gravity $g$ of the earth is given by
$g=\dfrac{G{{M}_{E}}}{{{R}_{E}}^{2}}$ --(2)
Where ${{M}_{E}},{{R}_{E}}$ are the mass and radius of the earth respectively and $G=6.67\times {{10}^{-11}}N.{{m}^{2}}k{{g}^{-2}}$ is the universal gravitational constant.
Now, let us analyze the question.
The mass of the satellite is $m$.
Let the mass of the earth be $M$.
The radius of the earth is given to be $R$.
The height of the satellite above the surface of the earth is given to be $x$.
Let the orbital velocity of the satellite be $v$.
The acceleration due to gravity on the surface of the earth is given to be $g$.
Therefore, using (1), we get
$v=\sqrt{\dfrac{GM}{\left( R+x \right)}}$ --(3)
Also, using (2), we get
$g=\dfrac{GM}{{{R}^{2}}}$
$\therefore GM=g{{R}^{2}}$ --(4)
Putting (4) in (3), we get,
$v=\sqrt{\dfrac{g{{R}^{2}}}{\left( R+x \right)}}$
Hence, we have got the required value of the orbital velocity as $\sqrt{\dfrac{g{{R}^{2}}}{R+x}}$.

So, the correct answer is “Option D”.

Note:
It is not required by the students to memorize the formula for the orbital speed of a satellite around a planet. In fact, they can derive it very easily by equating the gravitational force exerted by the planet on the satellite with the magnitude of the centripetal force required to keep the satellite in the circular orbit. This would actually be the better method, since the student will not have to memorize a formula unnecessarily which he or she might wrongly reproduce while answering the question. Deriving the formula is always better as there is a lesser chance of writing the wrong formula.