Hanging Spring

• A labrys (double axe) of length $L = 1$, is thrown horizontally with translational velocity $v = 1$, and angular velocity $\omega$ towards an up right wall. The labrys sticks to the wall if and only if the blade on top hits the wall before the handle does. The center of mass of the labris, is located a distance $x$ from its head.
  1. Neglecting gravity, and assuming that the wall is at a far distance, find the probabiity $P(\omega, x)$ of sticking to the wall as a function of the angular velocity, and the location of the center of mass.
  2. Show that if $x\leq\frac12$, the best throwing strategy is $\omega \approx 0$ with $P = 1/2$, while for $x > \frac12$, the best throwing strategy is to have $\omega \rightarrow \infty$ with $P = 1$.