Maximum Rate of Change of X 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 $x$ is $2 \mathbf v_0$, which happens when $P$ is at the top of the circle $C$.
Proof
From Rate of Change of Cartesian Coordinates of Cycloid, the rate of change of $x$ is given by:
- $\dfrac {\d x} {\d t} = \mathbf v_0 \paren {1 - \cos \theta}$
This is a maximum when $1 - \cos \theta$ is a maximum.
That happens when $\cos \theta$ is at a minimum.
That happens when $\cos \theta = -1$.
That happens when $\theta = \pi, 3 \pi, \ldots$
That is, when $\theta = \paren {2 n + 1} \pi$ where $n \in \Z$.
That is, when $P$ is at the top of the circle $C$.
When $\cos \theta = -1$ we have:
\(\ds \frac {\d x} {\d t}\) | \(=\) | \(\ds \mathbf v_0 \paren {1 - \paren {-1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 \mathbf v_0\) |
Hence the result.
$\blacksquare$
Sources
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.21$: The Cycloid: Problem $4 \ \text{(b)}$