An Off-Center, Elastic Collision : Conservation of Kinetic Energy Without a Perfect Coefficient of Restitution (Journal Article)

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Abstract: We present an analysis of an elastic collision of a ball with a bar that is not necessarily on center. Although the analysis is accessible to students in most introductory high school or college physics courses, such a scenario is not typically treated in introductory textbooks, which are far more likely to treat totally inelastic collisions with a rotatable object, such as the case of a ballistic pendulum. In contrast to those textbook scenarios, which involve one or two conservation principles, this case study requires invocation of three conservation principles: linear momentum, angular momentum, and kinetic energy. Even though total kinetic energy is conserved, part of the translational kinetic energy is converted into rotational kinetic energy. Our approach uses (1) judicious selection of a reference frame and (2) dimensionless ratios of parameters, allowing a simple analysis that identifies key parameters upon which universal behaviors depend.