I have a question says that "If the orbital radius of a satellite increase four times the orbital velocity will be …" and the answer is that the velocity will be doubled !!
We studied that orbital velocity = $\sqrt{\frac{GM}{r}} = \frac{2\pi r}{t}$
I can't figure out how the answer is that the velocity will be doubled and how in the two laws the radius is directly proportional to velocity in one of them and inversely proportional in the other!!
Best Answer
The given answer is wrong. Multiplying $r$ by 4 divides $v$ by 2.
You have two equations for the orbital speed of a (small) body in a circular orbit. $$v=\sqrt{\frac{GM}r} \tag 1$$ and $$v=\frac{2\pi r}T \tag 2$$ where $r$ is the orbital radius, and $T$ is the orbital period.
Now from equation 1 it looks like $v$ is inversely proportional to $\sqrt r$, but it's also directly proportional to $r$ from equation 2. How can that work?
Simple! $T$ isn't independent of $r$. Let's equate the two equations. $$v^2 = \frac{GM}r = \frac{4\pi^2r^2}{T^2}$$ $$\left(\frac{GM}{4\pi^2}\right)T^2=r^3$$
You may recognise this as Kepler's 3rd law.