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.
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.