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i said that if there are N balls in the cradle, each with mass m, and the two balls you displace have gained a velocity v by the time they hit the others, initially the momentum is 2mv, and KE is mv^2. Then i said that if n balls emerged from the other side with the same velocity, it would need to equal 2v/n for conservation of momentum. Then KE is equal to (2mv^2)/n and we see KE can only be conserved when n=2.

I was just wondering if anyone can help me prove this result for when the velocity of the n balls emerging from the left is not equal. i.e then, the average velocity of the n balls would need to be 2v/n, but i dont know how to prove that n must equal 2. Thanks