1. General Relationships

Let initial velocity be $u$ and final velocity be $v_f$.

The average velocity for uniform acceleration is:

$$ v_{av} = \frac{u + v_f}{2} $$

The velocity $v_{mid}$ at the midpoint of the distance $S$ is given by the kinematic relation:

$$ v_{mid} = \sqrt{\frac{u^2 + v_f^2}{2}} $$

This is the Root Mean Square (RMS) of the initial and final velocities.

2. Establishing the Range

We know from the inequality of means that for any real numbers, $RMS \ge AM$ (Root Mean Square $\ge$ Arithmetic Mean). Thus:

$$ \sqrt{\frac{u^2 + v_f^2}{2}} \ge \frac{u + v_f}{2} \implies v_{mid} \ge v_{av} $$

The minimum value $v_{mid} = v_{av}$ occurs when $u = v_f$ (constant velocity motion).

To find the upper bound, we consider the extreme case of unidirectional motion where one of the velocities is zero (either starting from rest or coming to a stop). Let $u = 0$.

Then, average velocity $v_{av} = v_f / 2$, which implies $v_f = 2v_{av}$.

Substituting into the $v_{mid}$ formula:

$$ v_{mid} = \sqrt{\frac{0^2 + (2v_{av})^2}{2}} = \sqrt{\frac{4v_{av}^2}{2}} = \sqrt{2}v_{av} $$