On the scaling of velocity

Question

If we rescale the time and space variables as $t/\epsilon^2$ and $x/\epsilon$, is the following scaling of the velocity $$v_\epsilon = \epsilon^{-1} v(t\, \epsilon^{-2}, x\, \epsilon^{-1})$$ correct? Does it produce small velocities for $\epsilon \to 0$?

Answer 1

It produces a small number for $v_\epsilon$, but that is hiding the fact that your unit of velocity has been multiplied by $\frac 1\epsilon$. If you start with meters and seconds, the unit of velocity is m/sec. If you take $\epsilon=\frac 1{1000}$ your unit of velocity is $\frac {mm}{\mu sec}$ or $1000$ times larger. A velocity of $1 \frac m{sec}$ becomes a velocity of $0.001 \frac {mm}{\mu sec}$