Ytdyklly

Two identical pucks of radius 10 cm are placed in a round arena of radius 1 m. They are positioned 50 cm away from the center of the arena on opposing sides. Assuming no energy losses during sliding and collisions, can you hit one of the pucks such that the two pucks touch exactly once and then never again? If so, how can you do it?

Notes:

• The collisions conserve energy and momentum.


• The walls of the arena don't move.


• As there is no friction the pucks can't exchange angular momentum.


• Please don't invent additional dimensions. Everybody knows that the universe is two-dimensional.

Without actually solving the problem formally I hypothesize that the answer is

No

My reasoning is that

The dynamics of perfectly elastic collisions is reversible in time. So if there existed a dynamical state where a collision occurs followed by no collisions forever, then that would be equivalent to being able to define a state where no collision occurs forever, but then a collision follows. Which doesn't make logical sense to my way of thinking.

I presume that there is probably a simple geometrical conjecture involving rational angles that could confirm the above. Proof that irrational angles don't produce a solution is as follows:

If both the balls bounces off the wall at an irrational angle (as a fraction of 360 degrees or 2pi radians), then the balls must collide eventually. Just consider stretching their future collision times along a single time axis. A ball a or b that does not collide with any other will hit the wall at constant time interval Ta or Tb and each collision with the wall will precess some angle Aa or Ab, where –pi<=Aa<=pi, around arena. So at the nth collision the ball a will collide with the wall at position nAa mod 2pi at time nTa. As angle Aa is an irrational fraction of 2pi, an n can be chosen such that the ball is arbitrarily close to the zenith (0 degree) position of the arena, and similarly an m can be chosen such that the ball will arbitrarily closely revisit that position after a further p (or 2p, or 3p…..kp) collisions – where m and k are functions of what you define as arbitrarily close. If both balls are bouncing irrationally then they each get arbitrarily close to the zenith after p and q collisions respectively at time intervals pTa and qTb. Now identify a fraction h/k which is arbitrarily close to pTa/qTb , and after kp and hq collisions each, the two balls will both be arbitrarily close to the zenith within an arbitrarily short time period and will collide. The same argument still holds if one ball is cycling a rational angle.

So the only remaining possibility is that both balls bounce at rational angle increments after the collision and manage to miss each other, which I still think is unlikely, although it is obviously possible to set two balls bouncing so they never collide...

If such a rational solution exists after one initial collision then:

Both the balls must precess rational angles, and the ratio of collision times Ta/Tb must also be rational.