How is it that always the same number of balls leave at the other end in Newton's cradle. I understand that the momentum needs to be conserved, but as momentum is defined as p=m*v couldn't you have a different number of balls move at a different speed instead of the same number of balls at the same speed?
The compression pulse that propagates through the metal spheres of Newton's cradle are not ordinary sound waves. They are approximate solitons (a nonlinear wave form that balances dispersion against nonlinearity). It is this property of soliton pulses that is responsible for the observed behavior.
More Details: Newton's cradle is a physical manifestation of the Fermi Pasta Ulam simulation conducted in the 1950s: https://en.wikipedia.org/wiki/Fermi%E2%80%93Pasta%E2%80%93Ulam_problem The expected thermalization (analogous to the expectation that more and more balls in Newton's cradle will start to move) fails to materialize because the dispersion is exactly balanced by the nonlinearity that results from the Hertz deformation law for elastic spheres. Instead, repeated occurrences of the initial conditions are observed.
In actuality the soliton solutions are only an approximate representation of the motions of the balls in Newton's cradle. As the energy get dissipated through friction, the balance between dispersion and nonlinearity breaks down and the expected thermalization finally occurs.
The simple answer is that not only momentum but also energy needs to be conserved.
This puts constraints on the number of balls that can be activated in the cradle.
Note that this does not always give a unique solution either. But it enforces that $ n $ balls to $ n $ balls is a "stable" solution.