Orbital velocity

A satellite in a circular orbit is falling the whole time, and the speed that keeps it there drops straight out of one substitution. Gravity supplies the centripetal acceleration and nothing else does, so $G M m / r^2 = m v^2 / r$, which rearranges to $v = \sqrt{GM/r}$.

The satellite's own mass cancels. A loose bolt and a space station in the same orbit travel at the same speed, which is why astronauts drift alongside their spacecraft instead of falling behind it.

Radius sets the speed, inversely: the higher the orbit, the slower the satellite. A low Earth orbit runs near 7.7 km/s, while a satellite at geostationary radius manages about 3.1 km/s.

Two comparisons worth keeping straight. Escape speed from the same radius is $v_{\text{esc}} = \sqrt{2GM/r}$, a CED derived equation and exactly $\sqrt{2}$ times the circular orbital speed. And orbital velocity is really a speed: the direction changes continuously, so only the magnitude holds still.

The CED sorts the conserved quantities by orbit shape. In a circular orbit the system's total mechanical energy, its gravitational potential energy, and the satellite's angular momentum and kinetic energy are all constant. In an elliptical orbit only the total mechanical energy and the angular momentum stay constant, while the gravitational potential energy and the kinetic energy each change.