For a dynamics class, we designed a roller coaster, and one of the requirements is to plot the velocity vs distance traveled, which I can do, but one section of my coaster will not plot.

The part with the helix of decreasing radius gives errors in the code. Upon starting the project, I did not know about the ArcLength function, and was calculating arc length using

Integrate[D[parametric[t],t].D[parametric[t],t],{t,a,b}]

and that returned the length for everything except the helix. For the helix, it returned the unevaluated integral.

ArcLength[h[t], {t,0,4pi}] works fine, but the error seems to come in when I do length[r_] = ArcLength[h[t], {t,0,r}]

What can I change so I can successfully plot the distance versus speed?