Suppose a free rod, $l=2r$, is hit on a tip T and translates with $v= 1r/s$ and at the same time rotates with angular velocity $\omega= 1rad/s$. Is there an equation that can determine the position of point T at any time?
If there is none, can we describe this complex motion without using matrices?
I have drawn a sketch:
The curve looks a half ellipsis, but I can't be sure.
update
- what is the difference between an ellipsis and a cycloid? is a cycloid an ellipsis?
- if the rotational speed exceeds the translational v, the curve makes a loop. Can the equation still describe the motion?
Yes. You can write $T$ as $c + \frac{l}{2}(\sin \theta, \cos\theta)$ where $c$ is the position of the center of mass and $\theta$ is the orientation of the rod.
Assuming that $c(0) = (0,0),\ \theta(0) = 0$, and that the rod moves right and spins clockwise (your diagram doesn't look quite right, almost as if you had hit the tip opposite $T$), then we have
$T(t) = \left(t +\frac{l}{2} \sin t, \frac{l}{2} \cos t\right)$
which for $l=2$ looks like