Velocity of an orbiting body

Question

The velocity of an orbiting body is given by:

$$v = \sqrt{\frac{Gm}{r}}$$

I was trying to derive this formula earlier but I was struggling with incorporating $G$ into my derivation.

I tried looking at centripetal force in order to derive the equation:

$$F = ma = \frac{mv^2}{r}$$

$$F = \frac{mv^2}{r}$$

So

$$v = \sqrt{\frac{Fr}{m}}$$

But this didn't seem to be the right way to go.

Answer 1

The force of gravity equals mass times acceleration so: $ F_g = ma $

We also know that the force of gravity equals the gravitational constant, G, multiplied by the mass of Earth and the mass of the satellite, all over the distance from the center of Earth to the satellite squared.

$F_g = \frac{GM_em }{r^2} $

And we also know that centripetal acceleration equals:

$a=\frac{v^2}{r}$

Therefore,

$\frac{GM_em }{r^2} = m(\frac{v^2}{r})$

Solving for $v$, we get:

$v=\sqrt{\frac{GM_e}{r}}$