A solid ball with a mass m is thrown on the ground such that...

A solid ball with a mass $m$ is thrown on the ground such that at the instant of contact it has an angular velocity $\omega_{1}$ and velocity components $\left(\mathbf{v}_{G}\right)_{x 1}$ and $\left(\mathbf{v}_{G}\right)_{y 1}$ as shown. If the ground is rough so no slipping occurs, determine the components of the velocity of its mass center just after impact. The coefficient of restitution is $e$.

Key Concepts

Angular Momentum and Rigid Body Kinematics

In collisions involving rotating bodies, angular momentum about the point of contact is affected by the impulses, and its conservation (or appropriate accounting for external torques during the impulsive contact) is used to link rotation with translation. Rigid body kinematics then ties together the motion of the mass center with the angular motion, especially under constraints like the no-slip condition.

Coefficient of Restitution

This is a measure of how elastic a collision is, defined as the ratio of the relative speed after impact to the relative speed before impact in the direction normal to the collision surface. It determines how much kinetic energy is conserved in the normal direction during the impact and is critical in setting the post-collision velocities for the involved bodies.

No?Slip (Rolling) Condition

This concept refers to the constraint whereby the point of contact between the rolling body and the surface has zero relative velocity during impact. It couples the translational and rotational motions, ensuring that the angular velocity and the center-of-mass velocity are related in a way that prevents sliding, which is essential for accurately determining the post-impact motion.

Impulse-Momentum Principle

This principle involves analyzing the effect of a short-time impulse on the momentum of a body. In collision problems, impulses (force applied over a very short period) change both the linear and angular momentum of a rigid body. Decomposing the impulse into components normal and tangential to the contact surface allows for the application of conservation laws and the restitution condition to determine the new velocities.

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