Determine max/min speeds of trochoid

Question

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

Answer 1

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 $\cos t=\pm 1$: $$ \max_{t} v(t) = |a+b|,$$ $$ \min_{t} v(t) = |a-b|.$$

Answer 2

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 $1$.

Given your parametrization, the velocity vector is

$$\mathbf{r}'(t) = \langle x'(t), y'(t) \rangle = \langle a - b \cos t, b \sin t \rangle$$,

and the square of its length, i.e., the square of its speed, is

$$|\mathbf{r}'(t)|^2 = (a - b \cos t)^2 + (b \sin t)^2 = a^2 + b^2 - 2 a b \cos t$$.

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.