### Abstract

We consider the one-dimensional scattering of two identical blocks of mass M that exchange energy and momentum via elastic collisions with an intermediary ball of mass m=αM. Initially, one block is incident upon the ball with the other block at rest. For α<1, the three objects will make multiple collisions with one another. In our analysis, we construct a Euclidean vector V_{n} whose components are proportional to the velocities of the objects. Energy-momentum conservation then requires a covariant recurrence relation for Vn that transforms like a pure rotation in three dimensions. The analytic solutions of the terminal velocities result in a remarkable prediction for values of α, in cases where the initial energy and momentum of the incident block are completely transferred to the scattered block. We call these values for α "magic mass ratios. "

Original language | English |
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Article number | 1.4897162 |

Journal | American Journal of Physics |

Volume | 83 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2014 Sep 22 |

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### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

**Magic mass ratios of complete energy-momentum transfer in one-dimensional elastic three-body collisions.** / Ee, June Haak; Lee, Jungil.

Research output: Contribution to journal › Article

*American Journal of Physics*, vol. 83, no. 2, 1.4897162. https://doi.org/10.1119/1.4897162

}

TY - JOUR

T1 - Magic mass ratios of complete energy-momentum transfer in one-dimensional elastic three-body collisions

AU - Ee, June Haak

AU - Lee, Jungil

PY - 2014/9/22

Y1 - 2014/9/22

N2 - We consider the one-dimensional scattering of two identical blocks of mass M that exchange energy and momentum via elastic collisions with an intermediary ball of mass m=αM. Initially, one block is incident upon the ball with the other block at rest. For α<1, the three objects will make multiple collisions with one another. In our analysis, we construct a Euclidean vector Vn whose components are proportional to the velocities of the objects. Energy-momentum conservation then requires a covariant recurrence relation for Vn that transforms like a pure rotation in three dimensions. The analytic solutions of the terminal velocities result in a remarkable prediction for values of α, in cases where the initial energy and momentum of the incident block are completely transferred to the scattered block. We call these values for α "magic mass ratios. "

AB - We consider the one-dimensional scattering of two identical blocks of mass M that exchange energy and momentum via elastic collisions with an intermediary ball of mass m=αM. Initially, one block is incident upon the ball with the other block at rest. For α<1, the three objects will make multiple collisions with one another. In our analysis, we construct a Euclidean vector Vn whose components are proportional to the velocities of the objects. Energy-momentum conservation then requires a covariant recurrence relation for Vn that transforms like a pure rotation in three dimensions. The analytic solutions of the terminal velocities result in a remarkable prediction for values of α, in cases where the initial energy and momentum of the incident block are completely transferred to the scattered block. We call these values for α "magic mass ratios. "

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U2 - 10.1119/1.4897162

DO - 10.1119/1.4897162

M3 - Article

VL - 83

JO - American Journal of Physics

JF - American Journal of Physics

SN - 0002-9505

IS - 2

M1 - 1.4897162

ER -