An ideal elastic collision between a sphere and a stationary plane reverses normal approach velocity. On the fully rough, tangentially elastic branch, the tangential contact-point velocity reverses too. If is the signed tangential impulse, translation and spin obey and , with positive spin anticlockwise. Restoring tangential translational plus rotational kinetic energy givesThe nonzero root is the displayed impulse; the other root is the smooth-contact branch. For , the map is and . It preserves kinetic energy and the angular momentum about the contact point, . An unspecified rough surface need not realize this idealization: finite friction or a dissipative tangential contact law requires additional constitutive data. Setting the outgoing contact slip to zero instead loses of kinetic energy for this uniform sphere.
A short transverse impulse at lever arm gives a pivoted rigid body angular velocity and contact-point velocity , where is its pivot moment of inertia. Thus plays the role of a translational mass in the impact direction. A point projectile elastically striking an initially stationary body stops immediately after impact precisely when its mass equals this effective value. The pivot can exchange linear momentum while doing no work.
A normal impulse on a free planar rigid body of mass and moment of inertia , at perpendicular lever arm , changes the contact point's normal velocity by . Its effective mass in that direction is therefore . This follows by adding the translational change to the rotational contribution .
For an incoming point mass , the inverse reduced contact mass is . A smooth elastic collision with initial relative approach speed has impulse , since the relative normal velocity reverses. This separates effective inertia at a contact from the body's total mass and explains the lever-arm dependence in an equal-speed elastic impact on a free rod.
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