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

[John Denker <jsd@av8n.com>]

On 07/29/2013 05:38 PM, Philip Keller wrote:
>  suppose I were to ask this question of my students
> (what is the acceleration of a vertically launched ball at its peak?)
> having resolved to accept as correct the answer given by both groups -- the
> ones who are using the "slope of velocity" definition and the ones who are
> using rate of change of speed.  What answer should I mark correct for that
> second group?  Seems like the "right" answer for them is: undefined.  And
> if they took my advice and drew a graph, their speed graph would have a
> cusp.

Yes, the graph of speed versus time has a cusp.  The function 
is not differentiable at this point.

As for the scalar acceleration, there are several answers
that could be justified:

1) If you think of the acceleration as a derivative, then 
 the derivative does not exist, and the acceleration as a
 function of time is undefined at the critical point.  So
 we can easily defend an answer that says "undefined".

2) If you do your best to plot the acceleration versus 
 time, you can't make it a function, but you /can/ make
 it a connected curve, simply by coloring in the vertical
 segment from -g to +g.

 There are various unsophisticated ways to defend this,
 perhaps by saying it "looks nicer" ... and there are also
 some quite sophisticated and rigorous ways of defending
 this, in terms of regularizers.  So we can certainly
 defend an answer that says the solution set consists of
 "anything and everything on the interval from -g to +g".

 This conforms to the rule that says when given an ill-
 posed problem, if at all possible map out the entire
 solution-set.
   http://www.av8n.com/physics/ill-posed.htm#sec-how-to

3) You could also say the scalar acceleration is zero.
 This makes sense in terms of symmetry.  It can be formally
 justified in terms of limits and/or regularizers.  This
 is familiar as the signum function in mathematics and 
 computer science.  So we can defend an answer of "zero".

4) With a little more work, you could perhaps defend an
 answer of -g, or +g, or any other element or subset of
 the solution-set mentioned in item (2).  This is not
 my favorite answer, but it's not completely crazy.

5) Perhaps my favorite answer is "Who cares?"  You're
 asking about what happens on a set of measure zero.
 It could not possibly have any physical significance.

===========

The fact that there are four or five decent answers should
make you suspect that there is something deeply wrong with
the question.

The fact that "Who cares?" is a reasonable answer should
remove all doubt about the merits (or lack thereof) of
the question.