Randall D. Knight — Impulse and Momentum: Problem 67

Most collisions are partially elastic, which means that the objects bounce apart but with some of the mechanical energy dissipated as thermal energy. A partially elastic collision is characterized by a coefficient of restitution \( r \). The analysis of a partially elastic collision can be divided into four steps: First, the colliding objects compress, like springs. At the end of the compression phase, the two objects are moving with a common intermediate velocity \( v_{\text{int}} \). If the collision ended here, it would be a perfectly inelastic collision. You can calculate \( v_{\text{int}} \) and then the momentum change of each object. During the compression, each object experiences an impulse \( J_{\text{compress}} \) due to the other. You can use the momentum principle to calculate \( J_{\text{compress}} \) for each object. Second, the objects expand. During expansion, each experiences an impulse \( J_{\text{expand}} = r J_{\text{compress}} \), where \( r \) is the coefficient of restitution. It characterizes how well the objects are restored after deformation. A perfectly inelastic collision has \( r = 0 \), while a perfectly elastic collision has \( r = 1 \). Another application of the momentum principle gives the post-collision momentum, and thus the final velocity, for each object. A 100 g ball (ball A) traveling to the right at 30 m/s collides with a stationary 200 g ball (ball B). The materials are such that the coefficient of restitution is 0.25. What is each ball’s final velocity?