The problem of excitation of vibrations during collisions

In summary, the problem at hand is to find the final velocities of a ball with initial velocity v colliding with a system of two identical balls connected by a massless hookean spring. The collision is assumed to be absolutely elastic and occurring along the line through the centers of the balls and the spring. The law of momentum conservation and the law of energy conservation are both used to solve this problem, with an additional equation derived from another conservation law. The final velocities refer to the velocities immediately after the collision. The assumption is made that the collision is over before the spring is compressed, allowing for the use of energy and momentum conservation to find the velocities of the two balls and the system's center of mass.
  • #1
reterty
29
2

Homework Statement


Dear colleagues, my problem is following:
The ball of mass
0429e3dd940669f4c728ca27fe91530182.png
with the initial velocity
23e0281f80bad171b4adffa1649b3fef82.png
collides with a system of two identical balls of mass
78dab0d04dc1059f0ff1bd20a12558f682.png
each, connected by a massless hookean spring with stiffness
63bb9849783d01d91403bc9a5fea12a282.png
and length
2f2322dff5bde89c37bcae4116fe20a882.png
.
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: :
bed2e166f354302d759a83940b7bea7d82.png
. 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
 

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  • #2
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?
 
Last edited:
  • #3
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?
 
  • #4
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.
 
  • #5
Thank you
 

1. What is the problem of excitation of vibrations during collisions?

The problem of excitation of vibrations during collisions refers to the phenomenon of energy transfer and generation of vibrations in objects when they collide with each other. This is a common occurrence in everyday life, such as when a ball bounces off a wall or when two cars collide in a car accident.

2. Why does excitation of vibrations occur during collisions?

During a collision, kinetic energy is transferred from one object to another. This sudden transfer of energy can cause the molecules in the objects to vibrate, leading to the excitation of vibrations. The amount of energy transferred and the properties of the objects involved determine the magnitude and frequency of the vibrations.

3. What factors can affect the excitation of vibrations during collisions?

The properties of the objects involved, such as their material composition, shape, and surface roughness, can affect the excitation of vibrations during collisions. The speed and angle of collision, as well as the temperature and pressure of the environment, can also play a role in determining the intensity of the vibrations.

4. How does excitation of vibrations during collisions impact objects?

The excitation of vibrations during collisions can have various effects on objects depending on their material properties and the intensity of the vibrations. In some cases, it can cause objects to deform, crack, or even break apart. In other cases, it can lead to a change in the object's temperature, magnetic properties, or chemical reactions.

5. How is the problem of excitation of vibrations during collisions studied and addressed?

Scientists and engineers use various experimental and theoretical methods to study and address the problem of excitation of vibrations during collisions. These include computer simulations, analytical calculations, and physical experiments using specialized equipment. By understanding the underlying mechanisms and factors influencing the excitation of vibrations, scientists can develop strategies to minimize its negative effects and even harness it for beneficial purposes in certain applications.

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