Why Don't 8 Balls Move after Collision?

[sachin123]

Suppose there are 8 balls in line on a horizontal table.
Another 2 balls collide them from left side(both approach together) with velocity v.
The last 2 balls(from right) get knocked off with the same velocity as that of the balls that approached first.Why don't all the 8 balls move with a velocity v/4 or why doesn't only the last ball get knocked off with a velocity 2v?

The collision is elastic.

Something like this:
00 00000000
->
later,
00000000 00
->

 

[rcgldr]

This only happens in an ideal case, where there is no loss in kinetic energy. It occurs because of the way the elastic compression of each ball in the pack transfers the force from the colliding balls through the pack of balls to the ejected balls. This was covered in a earlier thread:

https://www.physicsforums.com/showthread.php?t=357358

 

[astrokreng]

There are a few key factors to consider in this scenario. First, we must understand the concept of conservation of momentum, which states that the total momentum of a closed system remains constant. In this case, the closed system is the 8 balls on the table.

When the two balls collide with the first 8 balls, momentum is transferred from the colliding balls to the first 8 balls, causing them to move with a certain velocity. However, since the collision is elastic, the total kinetic energy of the system is also conserved. This means that while momentum is transferred, the overall speed of the system remains the same.

Now, when the first 8 balls collide with the last 2 balls, the momentum is again transferred from the first 8 balls to the last 2 balls. However, since the last 2 balls are already moving with the same velocity as the first 8 balls, there is no change in their velocity. This is because the total momentum of the system must remain constant, and if the last 2 balls were to gain a higher velocity, the total momentum would increase, violating the law of conservation of momentum.

Additionally, the idea of the last ball being knocked off with a velocity of 2v is not entirely accurate. While the last ball may appear to be moving with a higher velocity, in reality, it is simply moving with the same velocity as the other 8 balls, as the system's overall speed remains constant.

In summary, the reason why the 8 balls do not all move with a velocity of v/4 or why the last ball does not get knocked off with a velocity of 2v is due to the conservation of momentum and kinetic energy in an elastic collision. The transfer of momentum from one set of balls to the other results in a redistribution of velocities, but the overall speed of the system remains constant.