Fun problem

I recently ran into a fun problem from an old Russian physics olympiad.
I found the solution a little surprising and intriguing. The problem
requires no more than high school classical mechanics and some
ingenuity. Here it is for your amusement.

Consider a point particle sliding on a flat table (ignore friction).
The table has a cylindrical hole of finite depth (vertical walls, flat
bottom). The particle can approach the hole with different velocities
and with different impact parameters (the particle's motion need not be
directed toward the center of the hole). As the particle falls into the
hole, it starts bouncing off the walls and the bottom (assume elastic
collisions). Sometimes it gets stuck in the hole forever, sometimes it
escapes (bounces out). Determine the relation between the depth of the
hole, its radius, the particle's initial velocity, and impact parameter
necessary for the particle to escape after it falls in.

Enjoy!

Igor

.

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    ... particle's trajectory forms with the hole's radius on HOR. ... bounces in the hole has the same length L. ... the point where it first reaches (and bounces on) the bottom. ... Subsequent bounces are at odd multiples of K. Crucially, the particle ...
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    ... Let's take as impact parameter the angle alpha that the ... particle's trajectory forms with the hole's radius on HOR. ... bounces in the hole has the same length L. ... Subsequent bounces are at odd multiples of K. Crucially, the particle ...
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