One dimensional Elastic collision of two identical particle

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Discussion Overview

The discussion revolves around the elastic collision of two identical particles in one dimension. Participants explore the implications of conservation laws for momentum and kinetic energy, questioning the validity of solutions where velocities are not swapped post-collision.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants propose that the solution where the velocities of the two identical particles are not swapped should be considered valid due to the symmetry of the system.
  • Others argue that traditional solutions, which involve swapping velocities, are more commonly accepted in textbooks and that the non-swapped solution may represent a different physical scenario.
  • A participant suggests that the non-swapped solution could represent a situation where one particle is moving and the other is at rest, raising questions about the physical interpretation of such a scenario.
  • Another participant emphasizes that if a collision occurs, momentum must be transferred, while if there is no collision, the particles do not interact.
  • There is a suggestion that in a one-dimensional context, the concept of a "miss" may not apply, leading to further discussion about the nature of collisions.

Areas of Agreement / Disagreement

Participants express differing views on the validity of the non-swapped velocity solution, with some accepting it as a valid interpretation while others maintain that traditional solutions are more appropriate. The discussion remains unresolved regarding the acceptance of the non-swapped solution as a physical scenario.

Contextual Notes

Participants reference the need for careful interpretation of mathematical solutions in the context of physical scenarios, indicating that assumptions about the nature of the collision may influence the validity of different solutions.

L0r3n20
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Hi everyone. I've a question that i wondered since the high school. Let's take two identical particles (same mass) that collide frontally. Assume it's an elastic collision. We have to conservate both the momentum and kinetic energy:

v_1 + v_2 = v'_1 + v'_1
v^2_1 + v^2_2 = v'^2_1 + v'^2_1

(where primes denotes the velocities after the collision). Now I do know the solution: the velocities are swapped among the two particles and here comes my question: since the one I wrote is a symmetric system, why should I not accept the solution where the two velocity are not swapped?
 
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L0r3n20 said:
Hi everyone. I've a question that i wondered since the high school. Let's take two identical particles (same mass) that collide frontally. Assume it's an elastic collision. We have to conservate both the momentum and kinetic energy:

v_1 + v_2 = v'_1 + v'_1
v^2_1 + v^2_2 = v'^2_1 + v'^2_1

(where primes denotes the velocities after the collision). Now I do know the solution: the velocities are swapped among the two particles and here comes my question: since the one I wrote is a symmetric system, why should I not accept the solution where the two velocity are not swapped?

You should accept that as a valid solution. What physical situation does it represent?
 
Every book I checked does not take into account such a solution. In fact, in order to obtain the desired solution they dived by (v_1 - v'_1) (and same for the v_2). As a physical solution I imagined a moving particle colliding with a static one. In such a situation, the solution where each particle keeps its velocity is represented by the first particle hitting the second one that remains still while the first one goes on. Am I right? Is this possibile?
 
L0r3n20 said:
Every book I checked does not take into account such a solution. In fact, in order to obtain the desired solution they dived by (v_1 - v'_1) (and same for the v_2). As a physical solution I imagined a moving particle colliding with a static one. In such a situation, the solution where each particle keeps its velocity is represented by the first particle hitting the second one that remains still while the first one goes on. Am I right? Is this possibile?

It's simpler than that. Imagine particle B is at rest and particle A is moving towards it at speed ##v##. Particle B closes its eyes and afterwards is still at rest, while particle is moving away at speed ##v##. What has happened? It's very simple!
 
If you haven't got it yet, here's another clue:

4ea4e79503e29-image.jpg
 
Ok thanks Perok that's very illuminating! :D
In other words IF there is a collision THEN we can assume momentum is transferred, ELSE we have a miss, right?
 
L0r3n20 said:
Ok thanks Perok that's very illuminating! :D
In other words IF there is a collision THEN we can assume momentum is transferred, ELSE we have a miss, right?

Yes. I'm suprised none of the books mention this, as it's a good example of having to interpret what is, after all, a perfectly valid mathematical solution. The equations are equally satisfied with no change to either particle.
 
I suppose that, being a 1D problem, you cannot admit a miss...
 
Both of you got a point. Now you convinced me. Many many thanks.
 

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