# Determine max/min speeds of trochoid

Find the minimum and maximum speeds of the point of a trochoid and the locations where each occurs.

I know a trochoid has equations $(x)t = at - b \sin{t}$ ; $y(t) = a- b \cos{t}$ for trochoid of radius a and b distance from center of circle (According to Wolfram)

The only speed equation I'm familiar with thus far in my study of parametric equations is the arc-length equation. Would I set that equal to zero and then try and solve for a & b (cause I did and that gets really messy- not sure I know how to do that)?

Any/all suggestions appreciated

• Speed is $v(t)=\sqrt{\dot{x}^2+\dot{y}^2}$. Set $\dot{v}=0$ to find optimal values. (The dots denote time derivatives) Sep 1, 2014 at 15:45

For a parametric curve, the speed is just the modulus of the tangent vector, hence if $$\gamma(t)=(at-b\sin t,a-b\cos t)$$ we have $$\dot{\gamma}(t) = (a-b\cos t,b\sin t)$$ so $$v^2(t) = a^2+b^2-2ab\cos t$$ and the stationary points for $v(t)$ are the ones for which $\cos t=\pm 1$: $$\max_{t} v(t) = |a+b|,$$ $$\min_{t} v(t) = |a-b|.$$

• Thanks very much. Great explanation
First of all, the minimum and maximum speeds depends on the choice of parametrization. An arc length parametrization is the one for which the speed is always $1$.
$$\mathbf{r}'(t) = \langle x'(t), y'(t) \rangle = \langle a - b \cos t, b \sin t \rangle$$,
$$|\mathbf{r}'(t)|^2 = (a - b \cos t)^2 + (b \sin t)^2 = a^2 + b^2 - 2 a b \cos t$$.
Since $\cos t$ takes values in $[-1, 1]$, the speed achieves minimum and maximum values when $\cos t$ is $-1$ and $1$, respectively; it achieves both of these infinitely many times.