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counting balls



I remember vividly when I first saw the "counting balls" demo. It was
when I was a little boy and my father took me with him to his bowling
league game. Bowling balls used to come up in a long rack with detents
and they exhibit this phenomenon better than those novelty toys.

1. What is the "counting ball demo"?

The counting balls are those lines of five ball bearings suspended from
two rails. Pull n balls off one side (n<5) and realease them, n balls
are ejected from the other side after the first n balls collide.

2. What does a "detent" do in a bowling ball rack?

The bowling ball racks used to be long pairs of rails with slight
depressions at intervals of about one diameter; those are the detents
of which I spoke.

I found the problem from French of which I wrote earlier today. It is
#9-17 for those of you fortunate enough to own a copy of this fine book.
Here, with some modifications to accommodate the ASCII medium, is his
problem (without permission):

A collision apparatus is made of a set of n graded masses suspended so
that they are in a horizontal line and not quite in contact with one
another... The first mass is mf, the second is mf^2, the third mf^3,
and so on, so that the last mass is mf^n. The first mass is struck by a
particle of mass m travelling at a speed vo. This produces a succession
of collisions along the line of the masses.

(a) Assuming all the collisions are perfectly elastic, show that the
last mass flies off with a speed vn given by

n
/ 2 \
vn = | ------- | vo
\ 1 + f /

(b) Hence show that, if f is close to unity, so that it can be written
as 1 +/- e ( with e << 1 ), this system can be used to transfer
virtually all of the kinetic energy of the incident mass to the last
one, even for large n.

(c) For f = 0.9, n = 20, calculate the mass, speed, and kinetic energy of
the last particle in terms of the mass , speed, and kinetic energy of the
incident particle. Compare this with the result of a direct collision
between the incident mass and the last particle in the line.

I will add that this device may be looked upon as an impedance matching
device for coupling energy by elastic collisions. I should also point out
that the balls may in practice be in contact initially. having them not
in cotact compells one to make the correct analysis as I mentioned before.

Leigh