Can someone please guide me to a resource that gives the time needed to move along a parametric curve, where the speed is specified by the position?
To be more specific, I have a 2D parametric curve $(x(u),y(u))$ with parameter values ranging from $u_{min}$ to $u_{max}$. I also have a function $s(u)$ (defined for the same $u$-values), that specifies the speed of the moving object along the curve. I would like to find the time it takes for a moving object to move along the curve.
Let $t$ denote time, $\ell$ denote arc length measured from $(x(u_{\min}), y(u_{\min}))$, and assume $u = \tau(t)$ expresses the parameter $u$ as a strictly increasing function of time. The chain rule gives $$ s(\tau(t)) = \ell'(t) = \sqrt{x'(\tau(t))^{2} + y'(\tau(t))^{2}}\, \tau'(t), \tag{1} $$ or in Leibniz notation, $$ s(u) = \sqrt{x'(u)^{2} + y'(u)^{2}}\, \frac{du}{dt}. \tag{2} $$ Assuming the speed $s$ is non-vanishing, rearranging (2) and integrating gives the time function in terms of the parameter $u$: $$ t - t_{\min} = \int_{u_{\min}}^{u} \frac{\sqrt{x'(\xi)^{2} + y'(\xi)^{2}}}{s(\xi)}\, d\xi. $$ In particular, the travel time for the entire curve is $$ t_{\max} - t_{\min} = \int_{u_{\min}}^{u_{\max}} \frac{\sqrt{x'(\xi)^{2} + y'(\xi)^{2}}}{s(\xi)}\, d\xi. $$