Maximum Rate of Change of Y Coordinate of Cycloid

Theorem
Let a circle $C$ of radius $a$ roll without slipping along the x-axis of a cartesian coordinate plane at a constant speed such that the center moves with a velocity $\mathbf v_0$ in the direction of increasing $x$.

Consider a point $P$ on the circumference of this circle.

Let $\left({x, y}\right)$ be the coordinates of $P$ as it travels over the plane.

The maximum rate of change of $y$ is $\mathbf v_0$, which happens when $\theta = \dfrac \pi 2 + 2 n \pi$ where $n \in \Z$.

Proof
From Rate of Change of Cartesian Coordinates of Cycloid, the rate of change of $y$ is given by:
 * $\dfrac {\mathrm d y} {\mathrm d t} = \mathbf v_0 \sin \theta$.

This is a maximum when $\sin \theta$ is a maximum.

That happens when $\sin \theta = 1$.

That happens when $\theta = \dfrac \pi 2 + 2 n \pi$ where $n \in \Z$.

When $\sin \theta = 1$ we have:


 * $\dfrac {\mathrm d y} {\mathrm d t} = \mathbf v_0$

Hence the result.