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 = α M. 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 V n 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.”

1.

Let us denote by ui and vi the initial and final velocities of the ith object, respectively. Then conservation of energy and momentum require u1+u2=v1+v2 and u12+u22=v12+v22. These coupled equations have a unique set of solutions: v1 = u2 and v2 = u1. If they do not make a collision, then v1 = u1 and v2 = u2.

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Let us consider the last collision between C and B at Qk when α=αkmagic. At Qk the velocity of C changes from v to 0 and that of B changes from V to V. Energy-momentum conservation requires αv+V=V and αv2+V2=V2. The solutions to v and V are v=2V/(1+α) and V=V(1α)/(1+α). In a similar manner, we find that the velocities of A and C must be 4αV/(1+α)2 and 2V(1α)/(1+α)2, respectively, just before the collision at Pk. This guarantees that A stops at Pk and C moves with the velocity v=2V/(1+α) between Pk and Qk.

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25.

We can show that αV/(2+α)at0 by making use of Eq. (32) and the constraint π(ψ/2)NA<π+(ψ/2) that derives from Eq. (E6).

26.

Note that α = 0 represents the case that the ball is absent.

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