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etotheipi

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On a flat course, suppose a cyclist might be able to maintain 300W for around an hour. This gives a total allowed energy expenditure of ##1080t \text{ kJ}##, where ##t## is measured in hours. However, the height of a particular time trial course varies along its length, and let's suppose that the surface can be modelled by a function ##h(x)##, which we can choose later on. I want to try and figure out how the cyclist's power output should vary with ##x## according to the gradient of the slope, in order to minimise the time taken to complete the course (i.e. ##x=0## to ##x=d##). I will, initially, only assume a resistive term of ##kv^{2}##, for pushing the air out of the way.

I don't know what approach would be most appropriate for solving such an optimisation problem. The trajectory of the cyclist is parameterised by

##\vec{r} = \begin{bmatrix}x \\ y\end{bmatrix} = \begin{bmatrix}x \\ h(x)\end{bmatrix}##

such that

##\frac{d\vec{r}}{dt} = \begin{bmatrix}\frac{dx}{dt} \\ h'(x)\frac{dx}{dt}\end{bmatrix}##

We also know that ##P(x) = \vec{F} \cdot \vec{v} = \begin{bmatrix}F_x \\ F_y\end{bmatrix} \cdot \begin{bmatrix}\frac{dx}{dt} \\ h'(x)\frac{dx}{dt}\end{bmatrix} = F_x(x) \frac{dx}{dt} + F_y(x) h'(x)\frac{dx}{dt}##

and we could perhaps also put ##F_x = m\ddot{x}## and ##F_y = m\ddot{(h(x))}##. However, this all looks like it's going to get pretty messy quite quickly and I don't even know if it is possible to get anything out of this working.

If it turns out that it is a bit fiddly to do analytically, I wonder whether this problem would lend itself more to a computational solution? Thanks for your advice!