A quincunx or Galton board consists of staggered pegs from which ping-pong balls bounce and eventually display a binomial / normal distribution in catch-bins. I am wondering if the downward progress of each ball has been studied.
Assume just one ball falls, initially with some nonzero horizontal velocity and zero vertical velocity. Gravity increases its vertical velocity over time; assume no friction. Perhaps the ball radius and the peg radii are identical. The peg is immobile. Ball-peg collision is elastic. The ball's velocity vector is reflected from the common ball-peg tangent at a collision (like a lightray from a mirror).
(First bounce in animation below.)
Over time, the ball makes (sometimes slow) downward progress. E.g., in the example below, traversing the $10$ rows of pegs would take about $4.5$ time units in free-fall in the absence of collisions, but the $25$ collisions retard its progress considerably: it takes about $19$ time units to escape below.
(Repeats once; reload to replay animation. Browser dependent.)
Q1. Is there a simple argument to show that the ball actually does make downward progress, in that its expected vertical position at time $t_2 > t_1$ is below its expected position at $t_1$? Gravity is pulling the free paths downward, but collisions can shoot the ball upward.
Q2. Any insight into quantitative bounds, say, on the expected number of collisions per unit time, as a function of the ball/peg radii? Have analogous lattice reflections been studied?