Scattering of a ball by a bat in the limit of small deformation

Abstract

The problem of the mechanical evolution of a shock between a cylindrically symmetric object and a spherical one is solved in the strict rigid (small deformations) approximation for arbitrary values of the initial conditions. The friction during the impact is assumed to satisfy the standard rules. Firstly, when it is assumed that the only source of energy dissipation is friction, the problem is fully solved by determining the conditions at the separation point between the two bodies. A relation determining whether the contact points of the two bodies slides between them or become at rest (to be $pure\,rotation$ state) at the end of the impact, is determined for this case of the purely frictional energy dissipation. In second place, the solution is generalized to include losses in addition to the frictional ones. It follows that, whatever the mechanism of the additional form of dissipation is, assumed that it did not affects the usual forms of the laws of friction, the complementary losses only can change the ending value of the impulse $I$ done by the normal force of the bat on the ball at separation. Then, the dynamical evolution of all the mechanical quantities with the value of $I$ during the shock process remains invariable. Thus, under the adopter assumptions of strict rigidity and validity of the standard rules for friction, the solution of the problem is also exactly found, whenever the total amount of dissipated energy is considered as known (by example, measuring the ending mechanical energy of the system). The analysis allows to determine the values of the tangential and normal coefficients of restitution for the class of shocks examined. Finally, the results are applied to the description of experimental measures of the slow motion scattering of a baseball by a bat. The evaluations satisfactorily reproduce the measured curves for the output center of mass and angular velocities of the ball as functions of the scattering angle and the impact parameter, respectively.