Question

Derive an expression for the critical velocity of a satellite.

Answer 1

The expression for critical velocity:

1. Consider a satellite of mass m revolving round the Earth at height h above its surface. Let M be the mass of the Earth and R be its radius.
2. If the satellite is moving in a circular orbit of the radius (R + h) = r, its speed must be equal to the magnitude of critical velocity v_c.
3. The centripetal force necessary for the circular motion of a satellite is provided by the gravitational force exerted by the satellite on the Earth.
   ∴ Centripetal force = Gravitational force
   ∴ ${\displaystyle \frac{{\text{mv}}_{\text{c}}^2}{\text{r}}=\frac{\text{GMm}}{{\text{r}}^2}}$
   ∴ ${\displaystyle {\text{v}}_{\text{c}}^2=\frac{\text{GM}}{\text{r}}}$
   ∴ ${\displaystyle {\text{v}}_{\text{c}}=\sqrt{\frac{\text{GM}}{\text{r}}}}$
   ∴ ${\displaystyle {\text{v}}_{\text{c}}=\sqrt{\frac{\text{GM}}{\text{R+h}}}=\sqrt{{\text{g}}_{\text{h}}\left(\text{R + h}\right)}}$
   This is the expression for critical speed at the orbit of radius (R + h).
4. The critical speed of a satellite is independent of the mass of the satellite. It depends upon the mass of the Earth and the height at which the satellite is the revolving or gravitational acceleration at that altitude.