Calculating orbital velocity, mysterious contradiction? (1 Viewer)

[The Savior]

So there are 2 formulas for calculating the orbital velocity of a satellite, v= √(GM/r) and v= 2πr/T. If r was increased in the first formula, then velocity would decrease, while in the second formula, if r was increased, then velocity would increase as well. Can someone help me explain why this is? Does it have something to do with the fact that in the second formula, if r is increased, then T has to be increased as well, due to Kepler's law which states that the ratio of the cube of r to the square of t is a constant?

 

[nightweaver066]

First formula is derived by equating centripetal force and gravitational force. So if you vary the "r", what you're doing is varying the distance between the satellite and the centre of the Earth.
This lets you calculate the required orbital velocity at some distance from the centre of the Earth.
(Just used Earth as e.g.)

Second formula is used when you consider some body already orbiting the Earth, and you know its distance to the centre of the Earth, and also it's period. Clearly, if it's orbiting a larger radius, then its orbital velocity must be greater in order to cover the larger distance in the same amount of time (the period).

 

[Fizzy_Cyst]

So there are 2 formulas for calculating the orbital velocity of a satellite, v= √(GM/r) and v= 2πr/T. If r was increased in the first formula, then velocity would decrease, while in the second formula, if r was increased, then velocity would increase as well. Can someone help me explain why this is? Does it have something to do with the fact that in the second formula, if r is increased, then T has to be increased as well, due to Kepler's law which states that the ratio of the cube of r to the square of t is a constant?

With the second equation, you can only say that v would increase with 'r' if T is constant, which it is not. We know from Keplers 3rd Law that r^3 is proportional to T^2. So, if r increases, then T would also increase by by r^3/2

Then this would end up showing that v is still inversely proportional to sqrt(r)