COM BYU 22

Analysis

The question asks if it is possible for one ball to stop after an elastic oblique collision between two identical balls.

Let the initial velocities be $\vec{u}_1$ and $\vec{u}_2$. Let the final velocities be $\vec{v}_1$ and $\vec{v}_2$. For one ball (say ball 1) to stop, we must have $\vec{v}_1 = 0$.

In an elastic collision between identical masses, the particles exchange the component of velocity along the line of impact (center-to-center line), while the perpendicular components remain unchanged.

Condition for Stopping

For ball 1 to stop completely ($\vec{v}_1 = 0$), two conditions must be met simultaneously:

1. Tangential Component: The component of $\vec{u}_1$ perpendicular to the line of impact must be zero. This means $\vec{u}_1$ must be directed entirely along the line of impact.
2. Normal Component Exchange: The component of $\vec{u}_1$ along the line of impact is exchanged with $\vec{u}_2$’s component. For ball 1 to have zero final normal velocity, it must receive a zero normal component from ball 2. This implies ball 2 must have no velocity along the line of impact, OR ball 2’s velocity along the line of impact must exactly cancel ball 1’s after exchange logic is applied (which simplifies to vector exchange).

Essentially, for $\vec{v}_1$ to be zero, the entire momentum of ball 1 must be transferred to ball 2. This happens in a head-on collision (impact parameter $b=0$) if ball 2 is at rest.

However, if both are moving, they must be moving in mutually perpendicular directions, and the velocity vector of the ball that stops must be directed exactly through the center of the other ball at the moment of impact.