## Problem on parameterizing the motion of a frisbee

A frisbee of radius $r$ translates rightward at speed $v$ meter/second. It rotates clockwise at $\omega$ radian/second. Initially the frisbee is centered at the origin.
1. Find the trajectory swept by the point initially at $(0,r)$.
2. Compute the speed as a function of time.
3. Describe when the speed is largest and smallest.
• ## Solution

#### Part a

Recall that
The importation types of motion in this problem are translation and rotation.
We write that $\bfx(t) = \bfx_\text{c}(t) + \bfx_\text{r}(t)$, where $\bfx(t)$ is the position of the designated point at time $t$, $\bfx_c(t)$ is the position of the center of the frisbee, and $\bfx_r$ is the position of the designated point relative to the center of the frisbee.
Now we find the position of the center of the frisbee at time $t$.
The velocity of the center of the frisbee has magnitude $v$ and is in the direction of $\bfi$. Hence, it's velocity is $v\ \bfi$.
Now we find the position of the designated point relative to the center at time $t$.
Relative to center, the designated point always has distance $r$. The point rotates around the center at the given and constant rate $\omega$.
At time 0, the angle of the designated point relative to the center of the disk is $\pi/2$.
After time $t$, the point has rotate $\omega t$ radians in the clockwise direction. Hence $\theta(t) = \pi/2 - \omega t$.
Thus \begin{align}\bfx(t) &= \bfx_\text{c}(t) + \bfx_\text{r} (t) \\ &= \langle vt + r \cos(\pi/2 - \omega t), r \sin (\pi/2 - \omega t) \rangle.\end{align}
This can be simplified to $$\bfx(t) = \langle vt + r \sin (\omega t), r \cos(\omega t)\rangle.$$

Recall that

#### Part c

We algebraically observe that the speed is largest when $\cos (\omega t) = 1$ and is smallest when $\cos(\omega t) = -1$. That is, when the angle swept is a multiple of $2\pi$, the velocity is maximal. When the angle swept is any other multiple of $\pi$, then the velocity is minimal.
That is, when the point is moving in the same direction as the frisbee translation, speed is maximal. When the point is moving in the opposite direction as the frisbee translation, the speed is minimal.