Why the Pucks Go at Right Angles
by Bob Day (bobday.nh@verizon.net)
Copyright (C) December, 2002 by Bob Day. All rights reserved.
"Consider two perfectly elastic pucks of equal mass, A and B, which lie on a
frictionless horizontal surface. B is standing still and A is moving toward B,
but not directly at its center. Prove that after A strikes B (in a frictionless
collision), the angle between the directions of the two pucks will be ninety
degrees." That was a problem in a physics course I took in college, and it
intrigued me because it was such a simple but nonintuitive (at least to my mind)
result. And it was so easy to check. I played around with a couple nickels on
my desk to see whether it was true, and sure enough, it was. Amazing!
Then I got down to the business
of proving it. I set up some equations. Conservation of momentum --
conservation of energy -- grind, grind, grind -- crank, crank, crank. And I
proved it. Ninety degrees. The equations said so. Then I said to myself,
"Well, yeah, but why do the pucks go off at ninety degrees?". I had proved it.
The equations said ninety degrees, but I still had no intuitive grasp of why it
was true.
Years later, I thought about it
again. Here's what I came up with:
Assume that puck A is traveling at a constant velocity of v units/sec
horizontally toward puck B, which is standing still. Before the collision, the
center of mass of the two pucks is moving horizontally to the right at 1/2 v,
and the pucks are each moving toward the center of mass at 1/2 v. After the
collision, the center of mass of the two pucks is still moving horizontally to
the right at 1/2 v, but the two pucks are now moving away from the center of
mass at 1/2 v, and the line between the two pucks (which goes through the center
of mass because of conservation of momentum) is no longer horizontal. One
second, say, after the collision, the center of mass is 1/2 v units from the
point of collision, X, and pucks A and B are each 1/2 v units from the center of
mass. Now, the collision point, X, and the positions of the pucks form three
corners of a quadrilateral which must be a rectangle, because the diagonals are
equal and bisect each other. Consequently, the paths of the pucks after the
collision are at ninety degrees to each other.
Much better. Now it made sense
to me -- that reasoning gave me a deeper understanding of why the angle had to
be ninety degrees -- more than just an equation saying so, "that's the way is,
take it or leave it".
Most of Physics today,
especially quantum theory, seems to be of the "grind grind grind, crank crank
crank" variety: The equations say "that's the way it is, so you just have to
accept it". It doesn't matter whether you "understand" it or not. Maybe we've
gotten so deep into things beyond our everyday experience that it can be no
other way. Or maybe it's just that we haven't yet found the concepts that would
provide us with a deeper understanding and we should keep looking.