An astronaut and a vengeful pole Imagine an astronaut floating in free-space with no significant nearby gravitational influences.  The astronaut takes an arbitrarily thin pole of uniform density with length $l$ and mass $m$, orients it vertically from his perspective, and then positions it some distance $h$ in front of him.  
Finally, once the pole is in place, he kicks it with force $F$ at some coordinate in space $p$, at distance $r$ from the pole's center of mass.  Under what conditions will another coordinate on the pole's contour pass through coordinate $p$ to collide with the astronaut's foot?
 A: The $h$ serves no purpose and could as well be 0. The 'kick' is best described with impulse $I$, not with 'force', albeit the actual impulse is irrelevant. It is important though that astronaut is not moving back recoiling from the kick.
The answer is: Where-ever he kicks, pole doesn't intersect that point again, meaning, it won't hit the astronaut's foot (but it may still hit astronaut on the head; perhaps a revised problem could be made about astronaut's head, or about a rod that has extra point mass attached in the middle).
To cut down on writing $r/l$ and the like everywhere, I normalize it by defining my own units so that $l=2 , m=1, I=1$ , my $r$ is your $2r/l$ , and my $r$ is in range 0..1.
then
$$v = I/m = 1$$
$$w = \frac{Ir}{\text{moment of inertia}} = \frac{Ir}{m(l^2)/12} = 3r$$
let coordinate system be centered on the centre of the pole before kick; $p_x=0, p_y=-r$ 
and equation of motion of other end of pole is:
$$x=t-\sin(t3r)$$
$$y=\cos(t3r)$$
Some time before the intersection, the $y$ has to be equal $p.y$ and $x$ has to be be <0
let's find $x$ when $y=p.y$:
$$t3r=\arccos(-r)$$
$$t=\arccos(-r)/(3r)$$
$$x=\arccos(-r)/(3r)-\sin(\arccos(-r))$$
and the rod would subsequently intersect $p$ if $x<0$ , but if we plot it, it never goes below 0 for $r$ in range 0..1 .
Plot of the graph on wolfram alpha
edit: sorry for excessive use of paragraphs, the form deletes newlines and the javascript is bugging out. edit2: ohh, that was noscript blocking some stuff, got that solved now. Forgot to define I, improved clarity a bit.
