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Orbital speed

A satellite is kept in its orbit by the gravitational force $$F_{\text{g}}$$ which acts as a centripetal force. For a circular orbit, $$F_{\text{g}}$$ must be exactly equal to the centripetal force $$F_{\text{c}}$$ for the specific orbit. This is the basis for deriving the equation for the orbital speed.

Orbiting speed requirement: $$F_{\text{g}}=F_{\text{c}}$$

Centripetal force (keeping satellite in orbit): $$F_{\text{c}}= ma=mv^2/r$$

Gravitational force (of primary on satellite): $$F_{\text{g}}=GMm/r_{\text{o}}^2$$

From Newton's second law,$$$\begin{align*}m\frac{v^2}{r}= G\frac{Mm}{r^2} \iff & v^2= G\frac{M}{r}\\\iff & v=\sqrt{ \frac{ G M}{r}}\end{align*}$$$

The formula applies to circular orbits and assumes that the primary is stationary.

At orbital speed, the gravitational force (red) is just strong enough to pull the satellite back onto its orbit and the speed along the orbit is just high enough to prevent the satellite from falling down to Earth.