PX285 - C5 - a roller-coaster

the arc-length coordinate, $s$ , is used to describe the motion of a roller coaster, which represented the distance moved along the track from an arbitrary starting point
the height of the track at any point, $s$ , is given by $h (s)$
the lagrangian is:

L = \frac{1}{2} m {\dot{s}}^{2} - m g h (s)

using the euler-lagrange equation:

\begin{matrix} m g \frac{d h}{d s} - m \ddot{s} = 0 \\ \ddot{s} = - g \frac{d h}{d s} \end{matrix}

solutions

the above differential equation can always be solved by adopting some numerical approach, eg: finite difference approach - discretize $s$ and $t$
it can also be solved analytically
eg: $h (s) = α s$ :

\ddot{s} = - α g

- integrating both sides:
$$\dot s = - \alpha gt + v_{0}$$
- integrating again:
$$s = - \frac{1}{2} \alpha gt^{2}+ v_{0}t + s_{0}$$

also, for $h (s) = \frac{1}{2} c s^{2} :$

h^{'} (s) = c s

- the $h (s) - s$ graph is not a parabola - it is a cycloid
$$\ddot s = -gcs = -\omega^{2} s$$
where, the equation is in simple harmonic motion, and $ω = \sqrt{g s}$ is the angular frequency
- this is a 'perfect' SHM for all amplitudes, not necessarily small, neglecting friction