Two particles cling to each other like climbers on the same rope and fly toward a wall. When they hit, will they bounce off together, or will the rope snap? What does it depend on? Physicists considered the simplest case: everything happens on a single line, the particles attract only when they occupy the same point, and the wall is perfectly rigid.
If the pair moves slowly, the impact energy is small, and the particles cannot break their "rope." They bounce off the wall as a single unit, like a ball. The results for this case matched the calculations done earlier by other scientists — Dean Lee and Michael Pine. For fast pairs, things are different: the impact is so strong that the particles fly apart in opposite directions. It turns out that the probability of such a breakup is inversely proportional to the square of the pair's speed — that is, if the speed doubles, the chance of breaking up drops by a factor of four.
This problem is not just a mental exercise. Similar processes occur when molecules strike the surface of a solid. Understanding when a pair bounces off and when it falls apart helps in chemistry and surface physics. It is also a beautiful example of how simple rules give rise to complex behavior.
In a one-dimensional world, particles cannot go around each other — they can only collide and reflect or pass through each other. This makes the problem especially clean for calculations.