Re: [Phys-l] thoughts for future physics regents.

[John Denker <jsd@av8n.com>]

On 06/23/2011 12:37 AM, Bernard Cleyet wrote:
> A geometric:
>
> http://merganser.math.gvsu.edu/calculus/summation/odds.html
>
> which, I presume, is about what JD wrote.

That's part of what I was saying.  For another part, see
  http://www.av8n.com/physics/img48/arithmetic-odd-series.png

The area scales like the square of the length, for squares *and*
for triangles.  The square law is not just for squares!  In all
cases, the square law can be seen as the sum of odd numbers.

The square law for triangles is perhaps more interesting because
*any* polygon can be divided into triangles.  This tells us
immediately that the area of any polygon scales like the
square of the length.

Galileo noticed that for a particle undergoing uniform acceleration
from rest, the distance traveled in the Nth time period was proportional
to the Nth  _odd_  number.  That is subtly different from calling 
it proportional to N.  It shows how careful Galileo was with his 
measurements and his math.

  From the point of view of calculus, we think of the velocity as
  being proportional to time ... but if we are doing discrete
  math the difference between velocity ∝ (2N) and velocity ∝ (2N+1) 
  is significant.  One gives a perfect square law for the position 
  while the other doesn't.

  Galileo worked one or two generations prior to calculus, so he
  pretty much had to do discrete math ... and he did it right.

IMHO this is an amusing little bundle of results, including 
geometry, algebra, and physics.  Applications include summing
an arithmetic series, finding the area of a triangle or other
plane figure, scaling laws, and uniformly accelerated motion.

IMHO physics class should not fixate on the laws of motion to
the exclusion of other things like scaling laws.

Even more importantly, seeing the connections between all these
things makes each of them more useful.  The whole is much greater
than the sum of the parts.  Remember what William James said (in
the 1890s!) about the connections between ideas:
  http://www.av8n.com/physics/thinking.htm#quote-james