[Phys-L] Re: Intro. Physics and Real-World Problems (thrust and drag)

[John Denker <jsd@AV8N.COM>]

On 07/03/05 13:50, Jeff Weitz wrote:
> ... intro teachers need to balance the need for rigor
> and careful, complete definitions on the one hand with the need to
> include problem settings that use the language of the world outside our
> classroom.  The real-world connection is what draws a great many of our
> students into the discussion in the first place.  It does also bring in
> messiness and approximations. ....

I heartily agree with all that.

There will always be tradeoffs between simplicity and
accuracy.  In an intro-level course, simplicity will
be favored.  There will never be an exact right way
to make the tradeoff ... by which I mean the optimum
will be partly determined by taste, and will vary from
class to class and even from day to day.

Not only are approximations necessary in order to keep
the students from getting brain-blisters ... let's
face it, approximations are necessary to keep the
cognitive load on the _teachers_ within reasonable
bounds.  It is not practical to require teachers to
have domain expertise in fluid dynamics and every other
field to which physics ideas can be applied.

=======

One might be tempted to post a big disclaimer on the
wall of the classroom:  "Everything we do here is an
idealization and an approximation" ... but even that
isn't quite right.  Some elements of introductory
physics (e.g. conservation of momentum) really are
exact, so far as we know, and other parts close enough
for all practical purposes.

This creates a genuine dilemma when formulating story
problems.  We apply an exact law to a real-world scenario
and come up with an inexact conclusion.  That's just
plain tricky.
  -- Exact law does not imply exact application.
  -- Inexact application does not imply inexact law.

This raises some interesting metaphysical questions
about why we think the law is exact, even when all
real-world applications are inexact.

This is particularly hard for students who are taking
(or have just taken) high-school geometry, where it
is emphasized that there is *no* notion of proximity
or continuity when stating theorems.  If you mis-state
the premises of a theorem even slightly, the conclusion
does not hold at all.  If you apply a theorem to a case
where the premises don't exactly hold, the conclusion
does not hold at all.

If you remember half of your password, you won't be
able to half-way log in.

So why is it that we can _approximately_ apply the
laws of physics?  We derive laws that apply to massless
strings and frictionless pulleys ... but then we apply
them to real-world strings and real-world pulleys, and
get approximately-correct answers.  This is genuinely
tricky.

Part of the answer is that often there _is_ a notion
of continuity in physical situations.  For example, it
the thrust is misaligned from the horizontal by some
small angle theta, the horizontal component of thrust
will be approximately equal to the total thrust.  Its
magnitude will be off by an amount that is second order
in theta.

This leads directly to the notion of _controlled approximation_.
Everybody makes approximations ... but what sets the pros
apart from the bush leagues is the ability to make a
controlled approximation.  Consider the contrast:
  a) Assume thrust is horizontal.
  b) Assume thrust is nearly horizontal.
  c) Ascertain that being slightly non-horizontal has
   effects that are second order in small quantities, and
   therefore unimportant for present purposes.

Another example:
  a) Assume massless strings and frictionless pulleys.
  b) Assume nearly-massless strings and nearly-frictionless pulleys.
  c) Ascertain that if your mass is off by 1% (due to strings)
   and your force is off by 1% (due to pulleys), your predicted
   acceleration will be off by about 2%, maybe less, because of
   the nice continuity properties of the F=ma law.

I really want to emphasize that this sort of continuity cannot
be taken for granted.  If you randomly mis-type 1% of the
characters when writing a long C++ program, your program will
not work right, not even approximately.

On the other hand, computers -- spreadsheets in particular --
are useful for teaching about continuity of physics laws.  You
can change the input numbers by 1% in each direction, and see
how that affects the output numbers.

Bottom line:  Recommendations for teachers:
  1) When writing story problems, don't demand exact answers.
   1a) Bad: asking about "equal" versus "greater than".
   1b) Better: asking about "approximately equal" versus
    "substantially greater than".
  2) Similarly, don't define things that don't need to be
   defined.
   2a) Bad: Define thrust to be horizontal.
   2b) Better: Say "in this idealized situation, the thrust is
    horizontal."
  3) More generally: be clear, even at the most introductory
   levels, about
   3a) exact statements, versus
   3b) inexact statements.
  4) Be clear, even at the most introductory levels, about
   4a) "brittle" laws, i.e. laws that hold only when their
    premises hold exactly, versus
   4b) "robust" laws, i.e. laws have nice continuity properties,
    so that they can be applied to slightly messy situations.

Remember: *You* know which laws are brittle and which are robust,
but the students weren't born knowing that ... and this issue
isn't well covered in typical textbooks.
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