For the longest time I've been puzzled by a problem of my own creation:
Suppose an ant crawls along a curve parametrized by the functions $x(t), y(t)$. Does there exist a function $s(t)$ in terms of $x$ and $y$ such that the ant, with position at time t $(x(s(t)), y(s(t)))$, crawls at constant unit speed? In other words, does there exist $s(t)$ such that the magnitude of the ant's instantaneous velocity $||v(s(t))||=1$ always?
The way I've gone about this so far is by deriving a differential equation from the condition $||v(s(t))||=1$, $$||v(s(t)||=\sqrt{x'(s(t))^2s'(t)^2+y'(s(t))^2s'(t)^2}=1$$ $$\Rightarrow s'(t)^2\bigl(x'(s(t))^2+y'(s(t))^2\bigr)=1,$$ but I'm not sure how to proceed from here. Perhaps I've overcomplicated it.
I'm interested to see what you all can remedy from this. Any help would be appreciated!