Re: [Phys-L] The Make-Believe World of Real World Physics

[John Denker <jsd@av8n.com>]

On 07/30/2013 10:09 AM, Anthony Lapinski wrote:
> Math cleverness? It's just d = vt.

Cleverness is relative.  We're not talking about rocket science, but
"some" level of thought is required.

In particular, not the contrast:
 a) If you travel at speed v1 for time t1 and speed v2 for
  time t2, the overall result depends on the average speed.
 b) If you travel at speed v1 for distance x1 and speed v2 for
  distance x2, the overall result depends on the average /inverse/
  speed.

Taking the average does not commute with taking the inverse.  In
other words, in case (b), if you average the speeds instead of 
averaging the inverse speeds, you get the wrong answer.  It's 
wrong by a huge amount if one of the speeds is much smaller than 
the other.

On the first day of class in high-school physics, I'll bet there
are a lot of students who have never heard the expression "Taking 
the average does not commute with taking the inverse" and could 
not explain what it means.  I'll bet there are lots of students
who blissfully average the speeds, and don't understand why it's
not always the right thing to do.

There are eleventeen ways of explaining this, which we can discuss
if anybody is interested ... graphs, abstract theory, numerical
examples, et cetera ... so by the end of the course the "inverse 
velocity" problems should be no big deal.

>  Are any other physics equations like this?

Only about a thousand that I know of.

Ohm's law is a fine example.  
  Resistors in series : add the resistances.
  Resistors in parallel : add the inverse resistances.

Ohm's law generalized to capacitors and inductors.

Second law of motion, especially when generalized to rotational
motion, where the moment of inertia (unlike the mass) can be
varying.

Equation for leverage, given any three points on a lever.

Newton's law of cooling.

The thermodynamics is the same, carried to the next level:
 -- change energy at constant pressure
 -- change energy at constant volume
 -- change energy at constant entropy
 -- et cetera