The Paper-Stack Trampoline: Acoustic Restitution in Layered Media
A ball bouncing on a thicker cushion should bounce less. We show that this is not always so . Measuring the coefficient of restitution $e$ of a steel ball dropped on a stack of $N$ sheets of standard A4 printer paper, we find that $e$ first drops, then rises to a pronounced maximum, and only then decreases towards a minimum. The maximum occurs where the acoustic round trip through the stack matches the contact time of the impact. For a contact set by the stack itself, one recovers the classical bar-impact condition of a stack-to-ball mass ratio of order one. Bar impact, however, predicts only a weak recovery, insensitive to the surrounding gas. For paper stacks, the recovery is large and no maximum is resolved when the stack is evacuated: the air trapped between the sheets, not the paper alone, makes the stack a trampoline.