What is the Resulting Velocity of Ball 3 in a 3-Body Elastic Collision?

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Discussion Overview

The discussion revolves around the resulting velocity of a third ball in a three-body elastic collision scenario involving three balls of equal mass, each moving at specified velocities on a 2D Cartesian plane. Participants explore the implications of conservation of momentum and kinetic energy in this collision setup.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant describes a scenario where three balls collide at the origin, questioning the resulting velocity of ball 3 after the collision, given its initial velocity and the approach of balls 1 and 2.
  • Another participant emphasizes the necessity of conserving momentum in addition to kinetic energy during the collision.
  • A third participant expresses uncertainty about whether the velocity of ball 3 could increase post-collision, seeking clarification on the implications of the collision dynamics.
  • A later reply suggests that the momentum of balls 1 and 2 is exchanged upon collision with ball 3, asserting that ball 3's momentum remains unchanged due to equal and opposite impulses from balls 1 and 2.
  • It is noted that the scenario may differ if the balls are spinning, introducing additional complexity to the analysis.
  • One participant proposes that the described scenario could be experimentally tested using pucks on an air table under specific conditions.

Areas of Agreement / Disagreement

Participants express differing views on the outcome of the collision, particularly regarding whether ball 3's velocity can increase. There is no consensus on the final result of the collision, and the discussion remains unresolved.

Contextual Notes

The discussion does not resolve the assumptions regarding the effects of potential spinning of the balls or the specific conditions required for the proposed experimental setup.

austin.hornbac
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This may be intuitively obvious to you and I'm just missing it.

Say you had 3 balls each with a mass of 1g. Each moving at velocity of 10m/s on a 2D Cartesian plane. Ball 1 & 2 are moving toward the origin from opposite sides, so they are approaching each other at 20m/s. Ball 3 moving from negative Y values to positive values (so Up) at 10m/s. The 3 balls collide simultaneously at the origin so that Balls 1 & 2 contact Ball 3 at a 90° angle. Balls 1 & 2 don't collide with each other but rather each side of Ball 3.

Similar to hitting the cue ball in pool directly in the middle of 2 side by side balls, the cue ball would keep trajectory and the angle between the other 2 would be 90°. Only in reverse.

What would be the speed of ball 3 after the collision? I don't know if it would remain the same or not. I tried combining balls 1 & 2 to use it in the 2D collision with 2 objects equations but I don't know if that works or if I did it right.

My goal with this long winded explanation is to figure out if theoretically since the kinetic energy would be conserved in the system and the closing velocity between 1&2 is 20m/s could ball 3's velocity ever be above 10m/s.
 
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Remember that momentum also has to be conserved.
 
Thats just a fn or kinetic energy and mass. So I'm still not sure. Is the answer obvious that it couldn't possibly increase the velocity of ball 3 or ?
 
The momentum of balls 1 & 2 is exchanged - they reflect from ball 3.

The momentum of ball 3 is unchanged in your scenario: equal but opposite impulses occur simultaneously from balls 1 & 2. Thus ball 3 continues to roll along after the collision, as though nothing happened.

Of course this is all wrong if the balls are spinning, etc.

But you can reproduce this with pucks on an air table if your setup is just right.
 

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