What I want: I have a rubber rope which is $5m$ in length when not stressed and is able to stretch about $100\%$ (to $10m$ long). I want to accelerate a constant mass horizontally, which has negligible friction. I'd like to have a function that tells me the velocity of the mass dependent on time, so for instance velocity $1 s$ after releasing it.
What I did: I've done some measurements of forces of the rope when pulling it to different lengths. Of course, when pulling $0cm$ (total length $5m$) I got a force of $0N$. Here is a graph of my results.
$x-axis$: displacement of one end of the rope
$y-axis$: measured force
I was also able to do a regression and found a function which describes how much force I get after I pull a given length. I name this function $F(s)$ for Force dependent on displacement. From this, it's easy to get the acceleration function, which is $a(s) = F(s)/m$ with $m = mass$ of the object I want to accelerate. But now I'm stuck. I somehow need to get $a(t)$ instead of $a(s)$, thus the acceleration by time, not by length, so I can then integrate that to get $v(t)$.
How do I convert the dependency of the function?