= Vertical upward velocity of a rolling-point trochoid
{title2=$a>r$}
A point at distance $a$ from the axis of a radius-$r$ cylinder rolling downhill at constant axis speed $v$ has $\dot x=v[\cos\alpha+(a/r)\cos\theta]$, $\dot y=-v[\sin\alpha+(a/r)\sin\theta]$. For $0\le\alpha<\pi/2$, the existence of $\dot x=0$, $\dot y>0$ is equivalent to $a>r$. At $a=r$ the relevant zero-horizontal-velocity point is stationary, so a point within or on the rolling cylinder never has strictly vertical upward motion.
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