The fragmentation of a line of balls by an impact

Abstract

When a long line of stationary touching balls is hit on its end by another ball, the line fragments: some balls fly off at the far end, some in the middle hardly move, and the impacting ball rebounds backwards taking with it some nearby balls. Two laws for the contact force are studied, both elastic and cohesionless. First a simple law linear in the compression and then the nonlinear 3 over 2–power law of Hertz for touching spheres. For the linear force and for a line of N balls being impacted by a ball at velocity V, 1.5N1/3balls fly off from the far end, the furthest at a velocity 1.4VN–1/6, the others at similar but slower speeds, while the majority rebound, the impacting ball at –0.13V and the nth from the end at a velocity –0.16Vn–5/6at large n. For the nonlinear Hertz law, only two balls fly off from the far end with significant velocities, at 0.986V and 0.149V, the majority hardly move, and a few rebound, the impacting ball at –0.07V and the nth from the end at a velocity –0.084Ve–0.55n.