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Assume that two balls $B_1,B_2$ of radius $r$ continuously move around inside of a square of size $d$. They bounce off the walls, i.e. the $x$-component of the velocity is multiplied with $-1$ when they hit the left/right wall, and similarly for the $y$-component. Apart from that, the velocity isn't changed. For which starting center positions $p_1,p_2$ and which starting velocity vectors $v_1,v_2$ of the balls will there be a collision after some time? I assume that it is "usually" the case, meaning that the space of all $(p_1,p_2,v_1,v_2)$ without collision has measure zero. I would not be surprised if this is a well-known question with a well-known answer, but I have literally zero knowledge about this area of math. I am also not sure about the tags.

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Martin Sleziak
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Martin Brandenburg
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  • Your question is improperly formulated, but assuming that you mean reflections after hitting the walls and not what you said, and assuming that collisions with a corner (which you didn't define) are the ones that produce multiplication with $-1$ of the velocity, then one can see the full trajectories in the following way: Reflect the square along its sides over and over again to tessellate the plane with its reflections. Then, the trajectory of a ball becomes a straight line on the plane. Now, multiply by the time dimension to get a graph of the trajectories as functions of time. –  Mar 08 '20 at 11:03
  • We get two lines (straight if the velocities are assumed constant but you didn't specify) in 3D space. The balls collide if and only if those two lines meet at some point. How often do two lines in space intersect? –  Mar 08 '20 at 11:04
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    I'm really curious under what circumstances one could expect a collision when the the second ball is following the first at the same velocity. There are obvious cases where the first ball bounces directly back at the second, but I wonder about nontrivial cases. – rschwieb May 27 '20 at 20:53

1 Answers1

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Suppose one ball is red, the other blue.
The centres of the balls travel within a square of side $A=d-2r$.
Reflect this smaller square along its side over and over until it tesselates the plane. There is always two balls in each square. The red balls fall into four groups with velocity $(\pm v_{1x},\pm v_{1y})$. Imagine these groups pass through each other on the boundary, but don't change direction. The pattern is periodic mod $2A$ in both the $x$ and $y$ directions, and balls no longer reflect at the boundary. Think of four red balls and four blue balls on a torus of side $2A$.
Take one red ball and one blue ball. Their position is $p_{i}+tv_{i}\pmod{2A}$ They will collide if $p_1-p_2+t(v_1-v_2)$ gets within $2r$ of a grid point $(2mA,2nA)$
There are some initial positions and velocities where that straight-line path stays away from grid points, but mostly they will collide.

Empy2
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