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. Find the distance from the satellite to the Earth's surface. The radius of the Earth and the free-fall acceleration on its surface are supposed to be known.

Explanation

Gravitational pull provides the required centripetal acceleration to the satellic. Thus if h be the sought distance, we have
so, `(mv^2)/((R+h))=(gammamM)/((R+h)^2)` or, `(R+h)v^2=gammaM`
or, `Rv^2+hv^2=gR^2`, as `g=(gammaM)/(R^2)`
Hence `h=(gR^2-Rv^2)/(v^2)=R[(gR)/(v^2)-1]`