What continuous path $S$ minimises the time for a cycling route between two points $A,B\in \mathbb{R}^2$?
For simplicity, the points are assumed to both be on a horizontal line. In the diagram, the cyclist would cycle slowly for most of path $S_1$. On path $S_2$, they'd go quickly for most of the time with the assistance of gravity and you'd expect a shorter time.
The forces acting on the cyclist are: the constant cyclist input $F_{in}$, a constant rolling resistance $F_r$, drag from air resistance, which is proportional to their squared velocity $F_d=c_d v^2$, gravity $F_g$, a normal force $F_n$ to keep them on their path, and any excess force turns into acceleration $a$. Along their path their velocity is $v=\frac{\partial s}{\partial t}$ and acceleration is $a=\frac{\partial^2s}{\partial t^2}$, so this reads:
\begin{equation} F_{in} = F_r + c_d \big(\frac{\partial s}{\partial t}\big)^2 + mg\sin(\theta) + m\frac{\partial^2s}{\partial t^2}. \end{equation}
Where the slope angle $\theta$ is given by $\tan(\theta)=\partial y/ \partial x$. Find the path $S$ that minimises the time $T=\int_S \frac{ds}{v}$?
My Thoughts and Problems
- Firstly I'm not sure how to solve this equation for the velocity.
- Usually when asked to solve a path integral, the path is given. I have no idea how to optimise the path.
- When would a straight line be the fastest path?