Re: [Phys-L] playing for keeps

[John Denker <jsd@av8n.com>]

On Jun 29, 2013, at 1:26 PM, I wrote:

> > 3) Given an ellipse with semi-major axis "b" and semi-minor axis
> > "a", what's the formula for the area of the ellipse? I don't
> > remember this formula, but I can figure it out in less time than it
> > takes to ask the question.

On 06/29/2013 11:19 AM, Ludwik Kowalski wrote:
> I cannot do this; even for the area of a circle (A=PI*r^2, where PI
> is the perimeter/diameter, by definition).

My original challenge assumed you knew the formula for the
area of a circle.  If you don't know that, you can calculate
it.  Do the integral.  It's trivial in polar coordinates.

Given the formula for the area of a circle, you get the 
area of an ellipse for free, by using a scaling law.
The area of the ellipse has *got* to be proportional to "a", 
and also proportional to "b".  You know the formula for the 
special case where a=b, and then you scale it accordingly.

To see why this has *got* to be true, draw a large circle on
graph paper.  Each little cell on the graph paper is an element
of area, dx ∧ dy.  Now imagine stretching the paper by a factor
of 2 in the x direction.  The little cells are now rectangles,
and the circle is now an ellipse.  You know the formula for
the area of a rectangle, and the number of cells within the
figure did not change during the stretch.


> The only thing I can do "in less time than it takes to ask the
> question" is to justify the following answer: "the area of a circle
> is smaller, pehaps 20% or 30% smaller, than the area of a subscribed
> square, 4*r^2.

It's smaller by a factor of π over 4.

Also the ellipse is smaller than the circumscribed rectangle, 
by a factor of π over 4.

There is a picture that goes with this. 
  http://www.av8n.com/physics/scaling.htm#sec-ellipse

The picture is itself a mnemonic, because the picture -- and
the idea behind it -- are unforgettable.  The ellipse formula,
obviously, is forgettable, but the picture and the scaling 
law are not ... especially if you are in the habit of applying 
scaling laws to everything in sight, all day every day.

For example:  In a car, how does the stopping distance scale
as a function of speed?  This is a real-world life-and-death
issue that should grab students' attention.  IMHO this is what
physics is about and should be about.  The contrast between
this and the usual monkey-shooting makes my head spin.