Challenge: simultaneous three-body collision

Problem

Challenge: simultaneous three-body collision. Three identical balls of mass $m$ move in a line: the first at $+v$, the second at rest, the third at $-v$. The first collides elastically with the second and, at the same instant, the third collides with the second. What is the final state?

Solution

For equal masses in a head-on elastic collision, velocities are exchanged. The setup is perfectly symmetric: the first ($+v$) and the third ($-v$) reach the second (at rest) simultaneously.

At the instant of collision, the second ball simultaneously receives the exchange from both: it gives/receives so that, by symmetry and by conservation of momentum and energy, it remains at rest. The first sets off with the velocity arriving from the right ($-v$) and the third with the velocity arriving from the left ($+v$): $\ev{v_1' = -v, \quad v_2' = 0, \quad v_3' = +v}$ This is the Newton’s cradle effect: the balls at the ends bounce back and the middle one doesn’t move. Check: momentum $-v + 0 + v = 0$ (as before) and kinetic energy $\tfrac{1}{2}mv^2 + \tfrac{1}{2}mv^2$ (as before) ✓.

Links

Topics: Quantità di moto e urti Concepts: Urto elastico · Conservazione della quantità di moto