Solving the Elastric 3-Body Collision Problem

  • Thread starter vibha_ganji
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In summary, the problem statement is trying to find the velocities of three balls after a collision. The textbook provides a hint to find the direction of the impulse delivered to each target ball, but does not provide a figure. The figure shows the three balls at the instant of the collision, with the first ball aiming directly at the contact point and all the balls being frictionless. The impulse delivered to each target ball is found to be 30 degrees to the initial direction of the first ball.
  • #1
vibha_ganji
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This is the problem statement from the textbook.

“A ball with an initial speed of 10.0 m/s collides elastically with two identical balls whose centers are on a line perpendicular to the initial velocity and that are initally in contact with each other (Fig. 6-33). The first ball is aimed directly at the contact point and all the balls are frictionless. Find the velocities of all three balls after the collision. (Hint: With friction absent, each impulse is directed along the line of centers of the balls, normal to the colliding surfaces.)”

The solution in the Instructor’s Manual states the following, “
All three balls are identical in mass and radii? Then balls 2 and 3 will move off at 30 ◦ to the initial direction of the first ball. By symmetry we expect balls 2 and 3 to have the same speed. The problem now is to define an elastic three body collision. It is no longer the case that the balls bounce off with the same speed in the center of mass. One can’t even treat the problem as two separate collisions, one right after the other. No amount of momentum conservation laws will help solve the problem; we need some additional physics, but at this point in the text we don’t have it.”
I am unable to understand how they concluded that the balls move off at 30 degrees to the initials direction of the first ball. I tried using the conservation of momentum but was unable to reach this conclusion.
 

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  • #2
Please post figure 6.33.
 
  • #3
vibha_ganji said:
I am unable to understand how they concluded that the balls move off at 30 degrees to the initials direction of the first ball. I tried using the conservation of momentum but was unable to reach this conclusion.
Thank you for the figure. Make a drawing of the three balls at the instant of the collision. Use the generous hint provided in the statement of the problem to find the direction of the impulse delivered to each target ball. What is the angle between the impulses and why?
 

1. What is the Elastic 3-Body Collision Problem?

The Elastic 3-Body Collision Problem is a mathematical problem that involves three objects colliding with each other and conserving both momentum and kinetic energy. It is often used to model real-world scenarios such as billiard ball collisions or the interactions of celestial bodies.

2. Why is solving the Elastic 3-Body Collision Problem important?

Solving the Elastic 3-Body Collision Problem is important because it allows us to accurately predict the outcomes of collisions between multiple objects and understand the underlying physics involved. This has applications in various fields such as engineering, physics, and astronomy.

3. What are some challenges in solving the Elastic 3-Body Collision Problem?

One of the main challenges in solving the Elastic 3-Body Collision Problem is the complexity of the equations involved. It requires advanced mathematical techniques such as differential equations and numerical methods to find a solution. Additionally, the problem becomes increasingly difficult as the number of bodies involved increases.

4. What are some real-world applications of the Elastic 3-Body Collision Problem?

The Elastic 3-Body Collision Problem has many real-world applications, such as predicting the paths of comets and asteroids, designing billiard tables, and understanding the interactions of particles in particle accelerators. It also has applications in computer simulations and video game physics.

5. Are there any limitations to the Elastic 3-Body Collision Problem?

While the Elastic 3-Body Collision Problem provides a good approximation for many real-world scenarios, it does have some limitations. It assumes that the objects involved are perfectly elastic and do not experience any external forces during the collision. In reality, there may be factors such as friction and air resistance that can affect the outcome of a collision.

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