Finding the function for a curve given its length and the length of its velocity? I am working on an assignment and am trying to find a function for $\vec{r}(t)$ with constant length $x$, whose velocity function also has a constant length $y$.
I'm not sure where to begin with this. I know that, say, $\langle \cos(t),\sin(t),0\rangle$ has a constant length of 1, but its velocity curve also has a constant length of 1. How can I scale the velocity without scaling the original function?
Thanks!
 A: So what you want to achieve is that the path of the curve stays the same, but the velocity changes.
To try to provide some intuition, suppose you are going to the grocery store. There is a fixed path for you to travel, how do you get there faster? You just speed up.
To put a finer point on this, if you plot the curve of your walk to the grocery store, you start at your house $H$ at time $0$, and end up at the grocery store $G$ at time $t_G$.
So $$\vec{r}(0)=H\text{ and }\vec{r}(t_G)=G$$. Now you construct a faster path $\vec{s}$ such that your velocity is twice as fast, so $$\vec{s}(0)=H\text{ and }\vec{s}(\frac{t_G}{2})=\vec{r}(t_G)=G$$.
To summarize for any arbitrary $t$ we have $\vec{s}(t) = \vec{r}(2t)$. And of course the $2$ is arbitrary as well.
A: The Speed is the magnitude of the velocity vector, in your example $\mathbf g(t)= (\cos(t),\sin(t),0)$ has as its velocity vector $\mathbf g'(t) = (-\sin(t),\cos(t),0)$, and thus the speed is going to be $||\mathbf g'(t)||=1$.
To increase the speed we can make use of the chain rule, in fact $\frac {d}{dt} (\mathbf g(2t))= 2\mathbf g'(2t).$ and thus the speed will be greater. Test this!
