How does momentum move in Newton's Cradle? In Newton's cradle, the ball on one side departs apparently as soon as the ball on the other side is struck--faster than the original ball would have gotten there moving at velocity $v$.  Imagine more balls, and it's more obvious. I used to line up long rows of marbles in the crack of a run, a watch the last one shoot away apparently as soon as the first one was struck, emphasizing that the motion transferred much faster than mv of the original moving marble. Which suggests that there's more to momentum than $mv$.  So how is the momentum transferred without moving mass m the full distance at velocity $v$?
 A: A mechanical wave travels through the balls so the transfer of energy/momentum is not infinite but depends on the speed of the mechanical wave through each of the balls.  It is the mechanical wave which is responsible for the transfer of energy/momentum at a speed which is much greater than the speed of the first ball which you called $v$.
An illustration of the mechanical (sound) pulse travelling through a material (and being reflected) has been used to measure the speed of sound in solids as described in this paper, first experiment - Figure 1 and this video, at 6:36.
A: As far as overall change in momentum, the initial $mv$ of the ball at the beginning is the same as the final $mv$ of the ball at the end.  So without making any assumptions about the middle steps, we know that initial and final momentum are the same, and momentum is conserved.
In the middle steps, momentum is transferred from one body to the next via a force, in accordance with Newton's second Law:
$$\rm{Force = [change\; in\; momentum]/[time]}$$
or
$$F=\frac {dp}{dt}$$
The force is transferred through the line of touching marbles at the speed of sound in marble (roughly 5000 m/s or 10,000 mph), which is far faster than the marble itself will roll.
