A State Transition Diagram for Simultaneous Collisions with
Application in Billiard Shooting
Yan-Bin Jia, Matthew T. Mason, and Michael A. Erdmann
Abstract
This paper models a multibody collision in the impulse space as a state
transition diagram, where each state represents a phase during which impacts are
``active'' at only a subset of the contact points. A state transition happens
whenever an active impact finishes restitution, or an inactive impact gets
reactivated, depending on whether the two involved bodies are instantaneously
penetrating into each other or not. The elastic energy due to an impact is not
only affected by the impulse at the corresponding contact point, but also by
other impulses exerted on the two involved bodies during the impact.
Consequently, Poisson's impulse-based law of restitution could result in
negative energy. A new law governing the loss of elastic energy during
restitution is introduced. Convergence of the impulse sequence generated by the
state transition diagram is established. The collision outcome depends on the
ratios of the contact stiffnesses rather than on their individual values. The
collision model is then applied in an analysis of billiard shooting in which the
cue stick impacts the cue ball, which in turn impacts the pool table.