If you do not understand the mathematics, you can step back.
Be logical, please. If I didn't understand mathematics, I would not have given you the basis of momentum equations.
Let the two balls hit at slightly different times. The Piston is moving in the –X direction.
1. Let the Ball B1 travelling in the +X hit the piston first. It will bounce back with a higher velocity (1200 compared with 1000). The piston will be slightly slower (99.9978 compared with 100).
2. The Ball B2 travelling in the –X direction then hit the piston. It is not hitting the piston travelling at 100 units but at 99.9978 units. It will bounce back with a lower velocity (800 compared with 1000 with the rounding).
As you can see, if you remove the need for the balls to simultaneously hit the piston, the result of the two separate collisions will send Ball B1 at 1200 in the –X direction. And the Ball B2 will travel at 800 in the +X direction.
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Making hidden assumptions often yield wrong results.
Where are your equations? Still no mathematics but words.
In the hypothesis that one ball hit first the piston, then we have twice a two bodies problem: the solution is still conventional and the equations are easy.
Let p1, p2, v1, v2 be the inital momentum and speed of ball1 and ball2, v'1 the ball1 speed after collision, v'2 the ball2 speed after collision, P piston momentum, v piston speed, v' piston speed after first collision, v'' piston speed after second collision.
First collision with ball1Conservation of momentum:
p1+P = p'1+P'
Conservation of kinetic energy:
p1
2/v1 + P/v = p'1
2/v'1 + P'
2/v'
Therefore, solving our two equations with two unknowns:
v'1= (m1-M)*v1/(m1+M) + 2*M*v/(m1+M)
v' = 2*m1*v1/(m1+M) + (M-m1)*v/(m1+M)
If we know the mass ratio r=M/m1=M/m2, it is simpler to write:
(1) v'1= v1*(1-r)/(1+r) + v*2*r/(1+r)
v' = v1*2/(1+r) + v*(r-1)/(1+r)
Second collision with ball2: we do the same, the initial piston speed being now v'.
(2) v'2= v2*(1-r)/(1+r) + v'*2*r/(1+r)
v'' = v2*2/(1+r) + v'*(r-1)/(1+r)
Replacing v' with its solution from (1), we get:
v'2= v2*(1-r)/(1+r) + [v1*2/(1+r) + v*(r-1)/(1+r)]*2*r/(1+r)
v'' = v2*2/(1+r) + [v1*2/(1+r) + v*(r-1)/(1+r)]*(r-1)/(1+r)
(3) v'2= v2*(1-r)/(1+r) + v1*4*r/(1+r)
2 + v*2*r*(r-1)/(1+r)
2 v'' = 2*v2/(1+r) + 2*v1*(r-1)/(1+r)
2 + v*(r-1)
2/(1+r)
2From equation (3) we obviously see that you neglected second order terms and it is the reason why your point 2 and conclusion is false.
You can't get right results by using only layman terms and approximate numerical values instead of equations, it's not enough rigor.
For correct resuls, do the math!