Impossibility of parametrising a non-singular curve by arclength

Question

Why can't you parametrise a nonsingular curve by its arclength?

Is it simply because the following arclength doesn't exist where $\alpha '(t) =0$?

$$s(t) = \int_{t_0}^{t} |\alpha '(t)| dt$$ Thanks.

Answer 1

If you take a look at this post, you can see that in order to get $\lvert \beta'(t) \rvert =1$ it is necessary to divide by $\alpha'(t)$ for each $t\in I$...

Alternatively, $\frac {dt}{ds}$ needs to exist for each $t$ in the domain of the curve.