orbital velocity vs velocity

Question

orbital velocity is sqrt GM/d, velocity is 2pir/T, what is the difference?

Answer 1

The period, T, is equal to sqrt(4π²r³/GM), so if we do 2π r/T, the 2π cancels out with the square root of 4π², r will cancel out with the square root of r², so you are left with just 1 r under the square root. At this point you have 1/sqrt(r/GM) this is just sqrt(GM/r). Now, we notice d and r mean the same thing, the distance between the centers of mass of the two objects, so we have sqrt(GM/d).

Essentially, with some algebraic manipulation, these two formulas are exactly the same, you can use either one depending on which is more convenient based on the information given.