Question

A satellite is orbiting close to the surface of earth. In order to make it to move infinity Its velocity must be increased by
A. $20\% $
B. $30\% $
C. $41.4\% $
D. $60\% $

Answer 1

Hint: In order to solve this question, we must know about the orbital velocity and the escape velocity. The velocity with which a satellite revolves around the earth in its orbit is known as orbital velocity whereas the velocity the satellite needed to escape the gravitational field of earth and never return back into its orbit is known as escape velocity. So, in order to move the satellite into infinity from the surface of earth, we will need the escape velocity and we will find the percentage of change in velocity from escape velocity to orbital velocity.

Formula used:
If ‘R’ is the radius of earth and satellite is revolving near the surface of earth then orbital velocity will be ${v_{orbital}} = \sqrt {gR} $ and the escape velocity for earth’s gravitational field will be ${v_{escape}} = \sqrt {2gR} $ where ‘g’ is the acceleration due to gravity on earth.

Complete step by step answer:
Since, we have the orbital velocity and needed escape velocity to move the satellite to infinity as
${v_{escape}} = \sqrt {2gR} $
$\Rightarrow {v_{orbital}} = \sqrt {gR} $
Percentage of velocity needed to increase can be found as
$\Delta v = \dfrac{{{v_{escape}} - {v_{orbital}}}}{{{v_{orbital}}}}$
putting the values of velocities we get,
$\Delta v = \dfrac{{\sqrt {gR} (\sqrt 2 - 1)}}{{\sqrt {gR} }}$
$\Rightarrow \Delta v = \sqrt 2 - 1$
$\Rightarrow \Delta v = 0.414$
taking percentage of this increase velocity we get,
$\therefore \% \Delta v = 41.4$

Hence, the correct option is C.

Note: It must be remembered that, the radius of orbit of satellite will be the radius of earth as satellite is revolving near to the surface of earth and escape velocity will always be greater than orbital velocity and we need to give extra energy in order to make satellite escape from earth’s gravitational field.