Question

A satellite of mass 1000kg is supposed to orbit the earth at a height of 2000km above the earth’s surface. Find its speed in the orbit.

Answer 1

Hint: When the satellite is launched in the orbit of earth, the only force acting on the satellite is gravity. Due to its launch motion, it starts revolving around earth and due to revolution, it experiences a centripetal force which is balanced by the gravitational force. This is the reason why a satellite remains stable in earth’s orbit without the fuel. The speed of its revolution is called orbital speed.

Formula used:
$F_{centripetal}={\displaystyle \frac{m{v}^2}{r}}$, where ‘r’ is the radius of the orbit.

Complete step by step answer:
Let the radius of revolution of the orbit of the satellite be ‘r’, measured from the centre of earth. The centripetal force acting on the satellite is $F_{centripetal}={\displaystyle \frac{m{v}^2}{r}}$.
This force acts radially outwards and the force of attraction of earth balances the centripetal force.
i.e. ${\displaystyle \frac{m{v}^2}{r}}={\displaystyle \frac{GMm}{r^2}}$
Hence, $v=\sqrt{{\displaystyle \frac{GM}{r}}}$which is the orbital velocity of the satellite.

Here, ‘M’ is the mass of earth.
Since in the statement, it is given that the satellite is at a height of 2000km above the surface of earth, hence $r=6400+2000=8400km=8.4\times {10}^{6}m$
And $G=6.67\times {10}^{-11}N{m}^{2}k{g}^{-2},massofearth=6\times {10}^{24}$
Putting the values, we get:
$v=\sqrt{{\displaystyle \frac{6.67\times {10}^{-11}\times 6\times {10}^{24}}{8.4\times {10}^6}}}=6.9km{s}^{-1}$

Note:
It is important to note that the velocity of a satellite does not depend upon the mass of the satellite itself. It is just dependent upon the distance of the satellite from the center of earth. Hence two satellites at same orbit (means they have the same distance from the center of earth) will have the same velocity, irrespective of the mass. In fact even a lighter object like a meteor enters the orbit of a satellite, then both the object and the satellite will never collide. Hence it serves as an advantage and satellites remain safe.