The problem of excitation of vibrations during collisions

[reterty]

Homework Statement

Dear colleagues, my problem is following:
The ball of mass

with the initial velocity

collides with a system of two identical balls of mass

each, connected by a massless hookean spring with stiffness

and length

.
It is necessary to find the final velocities of the first ball and the center of mass of the second system, considering the impact to be absolutely elastic and occurring along the line through the centers of the balls and the spring.

Homework Equations

The Attempt at a Solution

The law of the momentum conservation has form: :

. I can always write the law of the energy conservation but there is the energy of the elastic deformation and I don't know how to derive the expression for the final value of deformation. Moreover, event in that case we will have only two equation with three unknown variables

 

[kuruman]

You get an additional equation using another conservation law that you did not mention.
(Edited to strike through irrelevant statement)
It is not clear what you mean by "final velocities". Is it perhaps "final" as in "immediately after the collision"?
Can you qualitatively describe the motion of the two-mass system after the collision?

 

[reterty]

Оk, I can solve this problem only in supposition
that immediately after the impact, two colliding balls get such momentums as if there is no spring at all.
Is it reasonable assumption?

 

[kuruman]

It is reasonable to assume that the collision is over before the spring is appreciably compressed. You can use energy and momentum conservation to find the velocities of the two balls participating in the collision immediately after the collision. Having that, you should be able to figure out the velocity of the system's CM.

 

[reterty]

Thank you