Answer:
[tex]v=\sqrt{\frac{gR_E}{2}}[/tex]
Explanation:
Satellites experiment a force given by Newton's Gravitation Law:
[tex]F=\frac{GMm}{r^2}[/tex]
where M is Earth's mass, m the satellite's mass, r the distance between their gravitational centers and G the gravitational constant.
We also know from Newton's 2nd Law that F=ma, so putting both together we will have:
[tex]ma=\frac{GMm}{r^2}[/tex]
[tex]a=\frac{GM}{r^2}[/tex]
If we are on the surface of the Earth, the acceleration would be g and [tex]r=R_E[/tex] (Earth's radius):
[tex]g=\frac{GM}{R_E^2}[/tex]
Which we will write as:
[tex]gR_E^2=GM[/tex]
If we are on orbit the acceleration is centripetal ([tex]a=\frac{v^2}{r}[/tex]), so we have:
[tex]\frac{v^2}{r}=a=\frac{GM}{r^2}=\frac{gR_E^2}{r^2}[/tex]
[tex]v^2=\frac{gR_E^2}{r}[/tex]
[tex]v=\sqrt{\frac{gR_E^2}{r}}[/tex]
And if this orbit has a radius [tex]r=2R_E[/tex] we have:
[tex]v=\sqrt{\frac{gR_E^2}{2R_E}}=\sqrt{\frac{gR_E}{2}}[/tex]
