Re: [Phys-l] Work and Energy: which first?

["Robert Cohen" <Robert.Cohen@po-box.esu.edu>]

John M. Clement wrote:

> How would a non calculus student take the average of a 
> decaying sine wave?
> The area under the curve can be approximated readily by a 
> series of triangles, but students would have much more 
> difficulty taking the average of each peak.

I'm not sure why the area under such a curve can be approximately more
readily (by a series of triangles perhaps) than the average.  If
students can approximate the *area* with triangles, why can't they
approximate the *average* with triangles (or rectangles, for that
matter)?  Maybe the difference is that I'm not trying to get an exact
answer.  If they could tell me that the average is not zero in such a
case, I'd be happy.**

This is the way I obtain an average.  Suppose I want to obtain the
average mass of five objects, two of which are 10 g and three of which
are 20 g.  Add them all up to get 80 g and divide by the total number 5
to get 16 g.

Equivalently, I could multiply (10g) by two, multiply (20g) by three,
and add them together before dividing by 5.

What happens if I have a continuous function?

If I have a continuous function, I multiply each value by the interval
over which the value is valid.  I then sum up everything and divide by
the total interval.  To me, if the intervals are infinitessimal, this is
calculus.  For my non-calculus students, of course, they can estimate
the integration in their head and I don't have to mention the word
"calculus".

The only difference, that I can tell, between what you are doing and
what I am doing is that you don't divide by the total interval (since
you don't need to) and you interpret the summation as an area on a
graph.  You can also interpret the "average" via a graph (i.e., the
horizontal line where the "area" is split in half) but you don't need a
graph to guess at an average.

Hey, actually, that is another advantage of the "average" method (i.e.,
you don't need to graph anything to get an approximate answer).

And, come to think of it, perhaps another advantage is that the
"average" method gets students to move away from the idea that
integration is defined as "area under a curve" and instead see "area
under a curve" as one application of integration (just as "averaging" is
one application of integration).

**I'm assuming the decaying sine wave represents a variation with
respect to time and, as such, one would be calculating the time-averaged
value.  Hmmmm, perhaps that is another advantage of the "average" method
(assuming students are aware of the difference between a time-average
and a position-average).  I'd bet if you plotted F vs. t (for a decaying
Hooke's law spring) and told them the total displacement, students would
incorrectly calculate the work by taking the area under the curve (the
smarter students might then divide by Delta t and multiply by the
displacement but they'd still be wrong).

----------------------------------------------------------
Robert A. Cohen, Department of Physics, East Stroudsburg University 
570.422.3428   rcohen@po-box.esu.edu    http://www.esu.edu/~bbq