Question

Define critical speed of the satellite and obtain an expression for it. On what factor does it depend?

Answer 1

Hint: The velocity required for the satellite to revolve around the Earth is known as the critical velocity. The critical velocity is the horizontal velocity of the satellite.

Complete step-by-step answer:
When the satellite is released from the surface of the Earth. The horizontal velocity of the satellite required for it to revolve around the Earth. This velocity of the satellite is the critical velocity of the satellite.

Consider the satellite of mass m moving around Earth at a distance of h from the surface of the Earth. The mass and the radius of the Earth is M and R.

The centripetal force of the satellite is balanced by the gravitational force. So, it is given as:
$\begin{align}
  & \Rightarrow \dfrac{m{{v}^{2}}}{r}=\dfrac{GMm}{{{r}^{2}}} \\
 & \Rightarrow \dfrac{m{{v}^{2}}}{r}=\dfrac{GMm}{{{r}^{2}}} \\
 & \Rightarrow m{{v}^{2}}=\dfrac{GMm}{r} \\
 & \Rightarrow {{v}^{2}}=\dfrac{GM}{r} \\
 & \Rightarrow v=\sqrt{\dfrac{GM}{r}} \\
\end{align}$
Here, r is the distance between the satellite and the centre of the Earth.
Thus, the critical velocity of the satellite is given as:
$\Rightarrow v=\sqrt{\dfrac{GM}{R+h}}$
So, the critical velocity of the satellite is:
i. Directly proportional to the square root of mass of the satellite
ii. Inversely proportional to the square root of the distance

Note: The critical velocity of the satellite is independent of the mass of the satellite. So, the critical velocity of the satellite doesn’t change with the change of the mass of the satellite.
The critical speed of the satellite is constant for the planet. For Earth, the critical speed is 7.9 km/h.