This topic reminded me of two things. The first is a reminiscence of my
youth, some of which was spent in bowling alleys (before they were
called "lanes"). I wasn't bowling; I was playing pool, and it turns out
that I was learning some classical mechanics from both games.
Back in those days the balls were returned to the bowler differently.
Instead of coming beneath the alley to a small, round trough at the end
of the alley as they do today, the balls rolled on a trough between the
lanes and up onto a ball rack, the top of which was a long horizontal
trough with slight indentations spaced approximately one ball diameter
apart. There were almost no sissy balls; all balls had the same weight,
16 pounds. The balls would line up in a neat row, just touching one
another, but occasionally a bowler would pull his ball out of the line,
leaving a gap. The next ball to arrive at the end of the line would
detach one ball from the far end of the line, just as in the so-called
"Newton's cradle". I did experiments with this apparatus and discovered
the counting balls phenomenon for myself. I do recall being fascinated
by it, but at that age I had no idea that it was physics that could
help me understand it better.
The second thing I was reminded of is an asymmetrical Newton's cradle
that Tony French described in his excellent textbook "Newtonian
Mechanics". It is to be found as Problem 9-17 on p. 361, if you have
access to a copy. The device is my second most favorite example of a
mechanical impedance matching device, my favorite being the bicycle.
"9-17 A collision apparatus is made of a set of n graded masses
suspended so they are in a horizontal line and not quite in contact
with one another (see the figure). The first mass is f(m0), the second
is f^2(m0), and so on, so that the last mass is f^n(m0). The first
mass is struck by a particle of mass m0 traveling at a speed v0...."
Unfortunately the limitations of this medium preclude my finishing this
problem legibly, so I'll leave it to your ingenuity to complete it.
I've told you enough to get you going. You will find it a rewarding
calculation, I'm sure.