Re: [Phys-L] popcorn-maker quantification activity

[John Denker <jsd@av8n.com>]

Executive summary:  Expressing it as a "quantification activity"
is fine.  Expressing it in terms of "units" (invented or otherwise)
is not a good idea.

On 07/11/2018 03:51 PM, Brent Barker via Phys-l wrote:

> I'm writing an "intro to measurements" lab and want to find the
> activity where students are tasked with inventing units to describe
> the performance of a popcorn-popping machine. Does anyone have a
> reference for this, or who I might ask about this?

I interpreted this as a latter-day "Barometer Question"
i.e. a challenge to students to come up with as many
different measurables as possible.
  https://en.wikipedia.org/wiki/Barometer_question

Given a bucket of popcorn:
 -- You could measure the height of the bucket.
 -- You could measure the diameter of the bucket.
 -- If there is a contest for greatest puff-size, you
  could measure the diameter of your biggest puff.
 -- If you tie a string to the bucket and use it as
  a pendulum, you care about the length of the string.

All of the above have dimensions of length.  You can
measure them using all the same units.  Or all different
units (so long as the dimensions are OK).  In any case,
knowing the units doesn't tell you what's being quantified,
or why.

A "Barometer Question" is hard to grade.  Will you
reward solutions that are conventional?  Original?
Eccentric?  Perverse?

There are innumerably many ways of quantifying the merit
of a popcorn maker.

 -- For occasional household use, you might to buy the
  cheapest one that satisfies some minimum throughput
  requirements.
 -- Or (!) buy nothing and just use the microwave oven
  you already own.
 -- For backpacking, you might want one with minimal
  weight that works without electricity.
 -- For commercial use, you might want one that is well
  integrated with packaging equipment and works with
  near-zero operator intervention.

The general solution is to have an /objective function/
that depends on many variables.  Typically the objective
function has adjustable parameters, so you can program
it by specifying how much weight to give to each of the
desiderata.

In simplest cases, it suffices to use a first-order Taylor
series approximation, with feasibility constraints.  In
such cases the optimization reduces to /linear programming/.
There are fat books on the subject. It's interesting, because:
  a) Even though the objective function is linear (until
   you consider the constraints), the solution is a
   spectacularly nasty function of the parameters.  It's
   not just nonlinear;  it's discontinuous.
  b) If you're not careful, the complexity of the problem
   grows exponentially with the number of parameters.
    https://en.wikipedia.org/wiki/Karmarkar's_algorithm