Question

At a circular orbital radius R, a satellite has a tangential speed of 7500 m/s. A second satellite is in circular orbit around the same central body and has an orbital radius of 5R. The tangential speed of the second satellite is closest to :

Answer 1

Bear with me. So you have a satellite orbiting out at a radius R. According to Newton, the only significant force acting on the satellite at this point is gravity. But the satellite is moving around in a circle or else it would fall straight down and smash into the ground.

This means that the Force of gravity = the centripetal Force
\[F _{g}=\frac{ mv ^{2} }{ R }\]
And since\[F _{g} = \frac{ GmM _{earth} }{ R ^{2} }\]
...when you combine the two you end up with....
\[\frac{ v ^{2} }{ R } = \frac{ GM _{earth} }{ R ^{2} }\]
Then with a bit of algebra and rearranging you can see that
\[v =\sqrt{\frac{ GM _{earth} }{ R }}\]

Answer 2

So if
\[v =7500 m/s\] when the satellite is at radius R, how fast will it be moving if you move the satellite 5 times out?

Well if we plug in 5R for R in our equation we get
\[v =\sqrt{\frac{ GM }{ 5R }} =\sqrt{\frac{ 1 }{ 5 }}*\sqrt{\frac{ GM }{ R }} = \sqrt{\frac{ 1 }{ 5 }} * 7500\]