Friction, Restitution, and Energy Loss in Planar Collisions

Abstract

Both particle and rigid body planar collisions are covered in this paper. For particles, the classical equations for oblique impacts are derived using Newton’s laws along with definitions of the coefficient of restitution and equivalent coefficient of friction. A general expression is obtained for the kinetic energy loss explicity containing the two coefficients. This expression for energy loss as a function of the friction coefficient possesses a maximum. The value of the friction coefficient at the maximum is a limiting value which can be used to determine whether or not sliding exists at separation. The maximum energy loss is independent of the physical mechanism of generation of tangential forces (friction) and serves as an upper bound for two-particle collisions. It is shown that to properly formulate and solve the rigid body problem, a moment must be considered at the common “point” of impact. A moment coefficient of restitution must be defined. This leads to six linear algebraic equations from which the six final velocity components can be calculated. An analytical solution is obtained for the general rigid body problem. In a reduced form, it is used to solve the problem of a single rigid body impacting a rigid barrier. This solution is then applied to a classical textbook problem. As shown for particle impacts, the concepts of limiting friction coefficient and maximum energy loss apply to rigid body impacts.