Question

A star ship is circling planet of radius R. The astronauts find that the free fall acceleration at their altitude is half the value at the planets surface. How far above the surface are they orbiting?answer must be a multiple of R.

Answer 1

At the planet surface, the free fall acceleration is

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

where R - radius of the planet. Since they experience twice as less acceleration due to gravity at height r above the surface, we can write

$$\frac{g}2=\frac{GM}{(r+R)^2}\\ \space\\ \frac{GM}{2R^2}=\frac{GM}{(r+R)^2},\\ \space\\ 2R^2={(r+R)}^2,\\ r=\sqrt2R-R=R(\sqrt2-1).$$