Faster Than a Ball Is Wide

A break hits the rack at 8 to 12 m/s. At 1/120 s a step, that is farther than a ball is wide, so a ball can start a step in front of another and end it behind.

Solve for the moment they touch

The simple test asks "do these two overlap now?" at the end of each step, so it only sees where the balls landed. A head-on pair stays overlapped for 4R, 11.4 cm, of travel, more than a step even at 12 m/s, so the test always sees it. A glancing pair overlaps for much less and can slip through between two looks. Either kind can also be seen too late, with one centre already past the other's, when the pair looks like it is parting.

Instead, solve for the moment the gap closes. Two balls Δp apart, moving at Δv relative to each other, touch when:

|Δp + Δv·t| = 2R

That is a quadratic in t, and its first root inside the step is the hit. Over a million hits of each kind at each speed:

                          8 m/s      12 m/s
one step                  6.67 cm    10.00 cm    a ball is 5.715 cm
head-on, missed           0%         0%
head-on, too late         14.3%      42.9%
glancing, missed          6.0%       14.9%
glancing, too late        26.7%      40.2%
time of impact, missed    0          0           of 2,000,000