Explanation of Radial Velocity Magnitude for Circular Motion

Both velocity and acceleration have direction as well as magnitude (size) and hence are vectors. Here vectors are denoted with bold face font.

Acceleration = dv/dt = (v₂ - v₁)/dt
Where dv = vector change in velocity. This is true for all velocities When a body is in a circular orbit the magnitude of the velocity is constant and |v₁| = |v₂| and the velocity vector is alway perpendicular to the radius vector from the center of the orbit to the object in orbit.

The velocity does not change magnitude but does change direction so there is acceleration. The change in the velocity is shown as the vector dv. The angle change in direction is dq and hence the magnitude of dv can be calculated using the 'skinny triangle' approximation (We assume dq is small).

|dv| = v dq

The acceleration of the orbiting object will be in the same direction as dv, and its magnitude given by:

a = |a| = dv/dt = v (dq/dt)

The size of the angular change is determined by the velocity and the radius of the circular orbit

dq = (v dt)/r

so dq/dt = v/r

As a result the radial acceleration of an object in orbit is given by

a = v . v/r = v²/r

And finally

a = v . v/r = v²/r