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

[austin.hornbac]

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.

 

[Nugatory]

Remember that momentum also has to be conserved.

 

[austin.hornbac]

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 ?

 

[UltrafastPED]

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.

 

[PJC01]

I can confirm that your understanding of the scenario is correct. In a 3-body elastic collision, the kinetic energy and momentum of the system are conserved. This means that the total kinetic energy of the system before the collision is equal to the total kinetic energy after the collision. Additionally, the total momentum of the system before the collision is equal to the total momentum after the collision.

In the scenario you described, Ball 3 would retain its original velocity of 10m/s after the collision. This is because the collision between Ball 1 and Ball 3 is perfectly elastic, meaning that there is no loss of kinetic energy in the collision. The collision between Ball 2 and Ball 3 would also be perfectly elastic, resulting in no change in Ball 3's velocity.

To calculate the velocities of the balls after the collision, you can use the equations for 2D collisions with two objects. In this case, you would use the velocities and masses of Ball 1 and Ball 2 combined to calculate the resulting velocity of Ball 3. However, since the collision between Ball 1 and Ball 3 is at a 90° angle, the equations would need to be modified to incorporate the angle of collision.

In summary, in a 3-body elastic collision, the velocities of the balls after the collision can theoretically be higher than their initial velocities, as long as the collision is perfectly elastic and the total kinetic energy and momentum of the system are conserved.