Question
Explain how the formula for critical velocity $(v_c)$ is derived?
✓
Answer
If a satellite of mass $'m \ '$ is revolving around the earth in an orbit of height $'h \ '$ with speed $v_c$ then as seen in the chapter on 'Gravitation', a centripetal force $\frac{ mV _{ c }^2}{ r }$ will act on it.
Here, $'r \ '$ is the orbital radius of the satellite from the centre of the earth.
This centripetal force is provided by the gravity of the earth.
Therefore, centripetal force $=$ gravitational force between the Earth and the satellite.
$ \frac{ mV _{ c }^2}{ R + h } =\frac{ GMm }{( R + h )^2}$
$V _{ c }^2 =\frac{ GM }{ R + h }$
$V _{ c } =\sqrt{\frac{ GM }{ R + h }}$
Here, $'r \ '$ is the orbital radius of the satellite from the centre of the earth.
This centripetal force is provided by the gravity of the earth.
Therefore, centripetal force $=$ gravitational force between the Earth and the satellite.
$ \frac{ mV _{ c }^2}{ R + h } =\frac{ GMm }{( R + h )^2}$
$V _{ c }^2 =\frac{ GM }{ R + h }$
$V _{ c } =\sqrt{\frac{ GM }{ R + h }}$
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