Why two balls pop out of Newton's cradle with unequal velocities

[adamg]

im working on a mathematical model of Newton's cradle. I am trying to explain mathematically why, when two balls are released, two balls pop up the other side etc.
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 don't know how to prove that n must equal 2. Thanks

 

[adamg]

basically, if you let v1,v2,...vn, be the velocities of the n balls that 'pop out' after the collision, therefore v1<v2<v3...<vn. From the conservation of momentum we get:

v1+v2+...vn=2v

and from the conservation of KE we get:

(v1)^2+(v2)^2+...(vn)^2= 2v^2

is there any way i can show that the only solution to this is vn=v(n-1)=2 and v1=v2=...v(n-2)=0?

 

[adamg]

alternatively can anyone prove that, if we have N balls, and we displace 2, 2 will 'pop out' the other side in another way? any ideas welcome! thanks