Question

Five identical balls each of mass $m$ and radius $r$ are strung like beads at random and at rest along a smooth, rigid horizontal thin rod of length $L$, mounted between immovable supports. Assume $10r<L$ and that the collision between balls or between balls and supports are elastic. If one ball is struck horizontally so as to acquire a speed $v$, the average force felt by the support is:

• A. $\frac{5m{v}^2}{L-5r}$
• B. $\frac{m{v}^2}{L-10r}$
• C. $\frac{5m{v}^2}{L-10r}$
• D. $\frac{m{v}^2}{L-5r}$

Answer 1

The correct option is B $\frac{m{v}^2}{L-10r}$

Since the collision is elastic and the balls are of equal mass, after the first collision, ball-2 will move with velocity $v$ and ball-1 will come to rest.
After the 4th collision, ball-4 will come to rest and ball-5 will move with velocity $v$ and hit the support at right.
Change in momentum of the ball-5 is $\Delta p=mv$

Time taken for this change is $\frac{L-10r}{v}$ where $L-10r$ is the effective length of the rod.

Average force: $F=\frac{\Delta p}{\Delta t}=\frac{mv}{\frac{L-10r}{v}}=\frac{m{v}^2}{L-10r}$