Question:

One ball hitting the stack at speed v knocks out one ball at speed v. Why not two balls at 0.5v?

Answer:

The total momentum before the collision is 1mv.

(1) If one ball is knocked out at speed v, the total momentum after the collision is 1mv.

(2) If two balls are knocked out at speed 0.5v, the total momentum after the collision is also 1mv.

As for as momentum is concerned, both outcomes are possible because they both conserve the total momentum.

So why was the outcome (1) and not (2)?

Well, the outcome of any collision, besides obeying PCOM, also depends on the elasticity of the collision.

The total KE before the collision is 0.5mv².

Outcome (1) has a total KE of 0.5mv², meaning total KE is unchanged. Outcome (2) however has a total KE of 0.25mv², meaning half the total KE has been lost as heat or other forms of energy.

The reason why the outcome (2) did not occur was because the collision of metal balls are (quite) elastic. If the balls were made of plasticine, the outcome would have been 5 balls moving at 0.2v after the collision (representing a 80% loss of KE).

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