Question

The orbital velocity of an artificial satellite in a circular orbit just above the earth’s surface is v. For a satellite orbiting at an altitude of half the earth’s radius, the orbital velocity is?
A. $\dfrac{3}{2}v$
B. $\sqrt{\dfrac{3}{2}}v$
C. $\sqrt{\dfrac{2}{3}}v$
D. $\dfrac{2}{3}v$

Answer 1

Hint: Try to learn about the centripetal force and gravitational force. From them obtain the formula for orbiting velocity. Putting the first condition finds the value of v. Then putting the second condition finds the orbiting velocity in terms of v.

Formula used:
${{v}_{\text{orbital}}}=\sqrt{\dfrac{GM}{R}}$

Complete Step-by-Step solution:
Orbital velocity of a body can be defined as the velocity at which the body is revolving around another object in a given orbit. This velocity depends on the radius of orbit and the mass of the object it is orbiting.
The orbital velocity is given by the equation,
${{v}_{\text{orbital}}}=\sqrt{\dfrac{GM}{R}}$
Where, G is the universal gravitational constant
M is the mass of the object around which the body is revolving
R is the radius of orbit
Now, in the first case the body is orbiting the earth’s surface with orbital velocity v.
$v=\sqrt{\dfrac{GM}{R}}$
Where M is the mass of earth and R is the radius of earth
Now, in the second case the satellite is orbiting the earth at an altitude of half the earth’s radius.
So, the radius of orbit will be, $R+\dfrac{R}{2}$
Now the orbital velocity will be,
$\begin{align}
  & {v}'=\sqrt{\dfrac{GM}{R+\dfrac{R}{2}}} \\
 & {v}'=\sqrt{\dfrac{GM}{\dfrac{3R}{2}}} \\
 & {v}'=\sqrt{\dfrac{2}{3}}\sqrt{\dfrac{GM}{R}} \\
 & {v}'=\sqrt{\dfrac{2}{3}}v \\
\end{align}$
So, the new orbital velocity is $\sqrt{\dfrac{2}{3}}v$
The correct option is (C)

Note: The unit of orbital velocity will be $m{{s}^{-1}}.$If we put the values of G, R and M we can get the numerical value for the orbital velocity for different cases.
The orbital velocity depends on the mass of the object at the centre. For different objects the mass will be different and orbital velocity will vary for each of them.