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 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 $\tuple {x, y}$ 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:
- $\dfrac {\d y} {\d t} = \mathbf v_0 \sin \theta$.
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:
- $\dfrac {\d y} {\d t} = \mathbf v_0$
Hence the result.
$\blacksquare$
Sources
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.21$: The Cycloid: Problem $4 \ \text{(c)}$