Question

Derive an expression for critical velocity.​

Answer 1

Answer:

Consider a satellite of mass m revolving round the Earth at a, height 'h' above the surface of the Earth. 

Let M be the mass and R be the radius of the Earth. 

The satellite is moving with velocity V and the radius of the circular orbit is r =R + hr=R+h. Centripetal force = Gravitational force

\therefore       \dfrac{Mv^2_c}{r}=\dfrac{GMm}{r^2}∴   rMvc2=r2GMm

\therefore       v^2_c=\dfrac{GM}{r}∴   vc2=rGM

\therefore       v_c=\sqrt{\dfrac{GM}{R+h}}∴   vc=R+hGM

This is the expression for critical velocity of a satellite moving in a circular orbit around the Earth, 

We know that,

g_h=\dfrac{GM}{(R+h)^2}gh=(R+h)2GM

GM=g_h(R+h)^2GM=gh(R+h)2

Substituting in equation (1), we get

\therefore     v_c=\sqrt{\dfrac{g_h(R+h)^2}{R+h}}∴  vc=R+hgh(R+h)2

\therefore     v_c=\sqrt{g_h(R+h)}∴  vc=gh(R+h)

where g_hgh is the acceleration due to gravity at a height above the surface of the Earth.