The Paper-Stack Trampoline: Acoustic Restitution in Layered Media
Abstract
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.
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BibTeX
@misc{liesker2026paperstack,
title = {{The Paper-Stack Trampoline: Acoustic Restitution in Layered Media}},
author = {Manou Liesker and Colton Kawamura and Maria Kieft and Joshua A. Dijksman and Antoine Deblais},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31213},
eprint = {2609.31213},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31213},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Paper-Stack Trampoline: Acoustic Restitution in Layered Media
AU - Manou Liesker
AU - Colton Kawamura
AU - Maria Kieft
AU - Joshua A. Dijksman
AU - Antoine Deblais
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.31213
UR - https://arxiv.org/abs/2609.31213
AB - 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.
ER -