Re: Fun problem

Igor Khavkine wrote:
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.

Here is my try.

Let v be the particle's speed, h the depth of the hole , r its radius.
Let's consider the projection of the particles's trajectory on the flat
table HOR. Let's take as impact parameter the angle alpha that the
particle's trajectory forms with the hole's radius on HOR. . Let C be
the projection on HOR of the chord that the particle trajectory cuts
on the hole. The length L of C is L= 2r*cos(alpha) . Note that every
subsequent chord traced by the particle's trajectory on HOR while it
bounces in the hole has the same length L.

Denote by K the length of the (possibly jagged) path described by the
particle's projection on HOR between its entry point on the hole and
the point where it first reaches (and bounces on) the bottom.
Subsequent bounces are at odd multiples of K. Crucially, the particle
will be at HOR's level at even multiples of K. Since the particle's
speed is constant along the possibly jagged path t describes on HOR, we
have K=v*srqt(2g*h)/g (that's v times the time it takes to fall from
height h). Both K and L have now been expressed using the depth of the
hole, its radius, the particle's initial velocity, and impact
parameter.

The idea now is that the particle can leave the hole only grazing the
hole's boundary (it can't go higher). In order for it to graze, L/K
must be a rational number 2n/m, n and m being integers. The particle
will leave the hole when distance D it has walked along its path on
HOR is D=2nK and D=mL.

IV

.

Relevant Pages

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