In a nutshell, the question is whether it can be faster to bounce a ball down an infinite flight of stairs than to bounce it down a ramp with the same slope.

To be more specific: this is a $2$ dimensional problem, so my ball is a disk and all the action takes place in $\mathbb{R}^2$. I imagine an infinite staircase with steps having depth $1$ and height $1$, the top corner of the top step is at the origin (and the staircase is oriented down and to the right). The ball has radius $1$, so when we toss the ball on the stairs and it will impact on an outside corner of a stair every time. Gravity acts downward with constant acceleration $g$. We compute the bounces by replacing the single corner point of the stair with an imaginary line tangent to the ball at the impact point.

Let $s(t)$ be the $x$-coordinate at time $t$ of the ball bouncing down the stairs, and let $r(t)$ be the $x$-coordinate at time $t$ of the ball rolling down the ramp.

QUESTIONS:

- Are there initial conditions for which $s(t) > r(t)$ for all sufficiently large $t$?
- Does the answer depend on the value of $g$?

IDEAS:

Can we determine the distribution of the incidence angles? Is it uniform? Does the distribution govern the long-term behavior of $s$?

We can replace the ball/stairs system with an equivalent one in which we track only the center of the ball. The center of the ball bounces down a "cobblestone ramp" made up of quarter circles (with radius $1$) heading down and to the right at slope $-1$.