Solving 3 Body Elastic Collision Problem - Littlepig

[Littlepig]

Hi.
I'm having some problem solving this problem: Consider a 3 body elastic collision; 3 bodies on 1 axe; both have known initial velocity≠0 and mass≠0, Particle 2 is between of 1 and 3 so that v1>0 and (v2 >0 or v2<0) and v3<0.
Now, can I calculate the final velocity of all particles without assumptions??

Energy and momentum conservation only gives 2 equations for 3 variables.

However, if one make an assumption(like final velocity of 1 is equal final velocity of 2), one can solve it. This leads to my second question: assuming mass of 2 >> mass 1 and mass 2>> mass 3, can you suggest one assumption physically plausible in a way that final velocity of 2≠0 ? (particles have no special configuration, like rectangles or circles)

I was thinking of having some fraction of final velocity of 1 as final velocity of 2, but I'm not sure if it is plausible nor what fraction should i use use...xD

Thanks in advance,
Littlepig

 

[BobMonahon]

Think of the particles at point of collision: P1,P2,P3.
Now think of them as a system of [P1] and [P2,P3] - that is, a two-body problem.

The technique for solving the two-body problem is to convert to the Center-of-mass-at-rest frame of reference. In that COM/rest frame, total momentum is zero.

The only possible solution for the velocities after collision, if total momentum is zero, is that they reverse - that is [P1] has +v coming in, and -v going out. For two-body problem, you can prove this.

So you can solve 1/3rd of the problem: Convert to COM-at-rest frame, reverse the velocity of P1, and then convert back to the Lab frame.

Do you see that the same logic will hold for P3? And that then the same holds for P2?

Regards, BobM

 

[astrokreng]

Hello Littlepig,

Thank you for reaching out with your question about the 3 body elastic collision problem. I can tell you that it is not possible to solve this problem without making any assumptions. This is because, as you mentioned, there are only two equations (energy and momentum conservation) for three variables (final velocities of the particles).

In order to solve this problem, we must make some assumptions based on physical principles. One possible assumption could be that the final velocity of the middle particle (particle 2) is equal to the average of the final velocities of particles 1 and 3. This is based on the principle of conservation of kinetic energy, where the total kinetic energy before and after the collision must be equal.

As for your second question, assuming that the mass of particle 2 is much greater than the masses of particles 1 and 3, it is plausible that particle 2 would have a non-zero final velocity. This is because the larger mass of particle 2 would have a greater impact on the overall momentum of the system, resulting in a non-zero final velocity.

However, the specific fraction of the final velocity of particle 1 that should be used for particle 2 cannot be determined without additional information about the masses and initial velocities of the particles. I encourage you to continue exploring this problem and make reasonable assumptions based on physical principles. Good luck!