Answer:
[tex]v = \sqrt{\frac{GM}{r}}[/tex]
Explanation:
The universal law of gravitation is defined as:
[tex]F = G\frac{Mm}{r^{2}}[/tex] (1)
Where G is the gravitational constant, M and m are the masses of the two objects and r is the distance between them.
The centripetal force can be found by means of Newton's second law:
[tex]F = ma[/tex] (2)
Since it is a circular motion, the acceleration can be defined as:
[tex]a = \frac{v^{2}}{r}[/tex] (3)
Where v is the velocity and r is the orbital radius.
Replacing equation (3) in equation (2) it is gotten:
[tex]F = m\frac{v^{2}}{r}[/tex] (4)
Hence,
[tex]m\frac{v^{2}}{r} = G\frac{Mm}{r^{2}}[/tex]
Then, v can be isolated:
[tex]mv^{2} = G\frac{Mmr}{r^{2}}[/tex]
[tex]mv^{2} = G\frac{Mm}{r}[/tex]
[tex]v^{2} = G\frac{Mm}{mr}[/tex]
[tex]v^{2} = \frac{GM}{r}[/tex]
[tex]v = \sqrt{\frac{GM}{r}}[/tex]
So the relationship between speed and orbital radius is given by the expression [tex]v = \sqrt{\frac{GM}{r}}[/tex]
