Question

An artificial satellite is launched into a circular orbit around the Earth with velocity v relative to the reference frame moving translationally and fixed to the Earth's rotation axis. The distance from the satellite to the Earth's surface. Find $x$. The radius of the Earth and the free-fall acceleration on its surface are supposed to be known.

Answer 1

Let the mass of satellite be '$m$' and the mass and radius of earth be $M,R$ respectively.

$V$ is the velocity of the satellite relative to earth

In an earth's orbit, centripetal force is equal to gravitational pull of the earth.

$\Rightarrow \frac{GMm}{{(R+h)}^2}=\frac{m{v}^2}{R+h}\Rightarrow (R+h)=\frac{GM}{v^2}\Rightarrow h=\frac{GM}{v^2}-R$

But, $g=\frac{GM}{R^2}$

Therefore, $\Rightarrow h=\frac{g{R}^2}{v^2}-R\therefore h=R(\frac{gR}{v^2}-1)$

Hence, $x=1$