Problem 3.8.11

Solution

Following the idea of problem $3.8.1$, we can initially consider the circulation of partial traveling waves of the rods' widths in each rod, reflecting back and forth like snakes, taking into account the method of problem $3.8.6$ (the sign of the deformation changes upon reflection). Each partial rectangular wave moves along the rod in the direction opposite to the other partial wave and has a particle velocity of $v/2$ in the corresponding direction.

At the moment of contact, these partial snake-waves at the point of contact already pass through unchanged.

Whoever understands this approach understands that the compression of the short rod ends after a time $l/c$, where $c$ is the speed of traveling waves in the rods.

After $2l/c$, the short rod will expand and acquire a velocity of $\boxed{v}$ to the left (in general terms: a partial compression wave from the long rod entered the short one with its opposite velocity and, after reflecting from the free boundary of the short one, managed to reach the contact boundary with deformation of the opposite sign).

Next, we write the law of conservation of momentum for the system:

$$Mv - mv = mv + MV,$$

where $V$ is the projection of the final velocity of the center of mass of the long rod onto the direction of the long rod's initial velocity.

Hence

$$V = v\left(1 - 2\frac{m}{M}\right)$$

and, taking into account $m/M = l/L$, for the magnitude:

$$\boxed{\left|V\right| = v \left|1 - 2l/L\right|.}$$