| Chronology | Current Month | Current Thread | Current Date |
| [Year List] [Month List (current year)] | [Date Index] [Thread Index] | [Thread Prev] [Thread Next] | [Date Prev] [Date Next] |
In computer science, there is a set of problems that are
classified as NP-complete. Any such problem is in some
sense just as hard as any other NP-complete problem.
Meanwhile, there are lots of problems that are even
harder than NP.
There is a category of problems I like to call ESP-complete,
because solving the problem by gazing into a crystal ball
works about as well as anything else.
On 03/20/2014 06:31 AM, Rauber, Joel wrote:
PART I -- here is one kind of motion we care about.I'm not going to disagree with any of that. Everybody
1. When an object moves at a constant speed in a straight line, its position graph is a line.
2. In that case, the slope of that line stays constant. That slope is the speed.
3. Since the speed is not changing, the "velocity" graph is a horizontal line.
The slope is the velocity not the speed. If the object were moving at
a constant speed in the opposite direction the slope would be
negative, the sign indicating the direction of the velocity vector in
a 1D situation (which I'm assuming is what is happening here). Unless
you only allow objects to move in the positive direction of how you
oriented your coordinate axis.
on this lists understands what is being said there, and
knows it's correct.
My point, however, is that there is provably no logical
way for the student to understand that in the introductory
course.
The problem is, there are /at least/ four different notions
of "length" and the student has no way of knowing which one
we want him to use on any given day. Specifically, if we
move along a specified path from A to B, we have
a) The arc length, as measured by an odometer.
b) The chord length, i.e. the distance as the crow
flies from A to B.
c) The displacement pointing from A to B, with
direction as well as length.
d) The projection of the displacement vector onto the
x and/or y and/or z axes of some frame.
To say the same thing mathematically:
a) Arc length = Σ |Δr|
b) Chord = |Σ Δr|
c) Displacement = Σ Δr
d) Projection = x̂ • Σ Δr
I insist that all four of these are correct. Each makes
perfect sense in some situation or another. The problem
is, the student has no way of knowing which one is going
to be the "desired" answer on any given day. It's a
guessing game. It's ESP-complete.
I don't want to argue the point, but one could argue
that item (d) has the /least/ connection to the real
fundamental physics. That should be obvious from the
fact that it's frame-dependent, while the others are
not. This just cracks me up, because a huuuge part of
every introductory physics course I've seen emphasizes
viewpoint (d).
Even that wouldn't be so bad if it were consistent, but
in fact the student is expected to use (d) and (a) on
Tuesday and (c) on Wednesday and who-knows-what on Thursday.
This runs counter to every pedagogical principle known to
man. If it were wrong but consistent, the students could
learn it by rote, but if it's inconsistent, even that
doesn't work.
We all do it, all the time, and it drives students nuts,
and you can't blame them.
energy (physics definition versus DoE definition)
conservation (conservative flow, conserve wildlife, conservative force)
acceleration (scalar versus vector)
photon (standing-wave excitation number, running wave packet)
charge (charge on terminal versus gorge on capacitor)
spin (s^2 versus s_z)
adiabatic (isentropic versus isolated)
gravity (framative versus barogenic)
heat (four or five different meanings)
length (at least four different definitions)
time (at least three different definitions)
et cetera......................
The bad news is, it is tremendously hard to notice such
things. The good news is, these are all quite fixable,
once you notice them. You fix them using adjectives
and the like: Odometer distance, proper distance, arc
length, chord distance, crow-flies distance, path of
least action, direction and magnitude, components
and projections .........
Another thing that helps is to point out the problem.
Students are expecting physics to be logical. Alas
the logic can get derailed by various little things.
Even if the teacher managed to be 100% consistent in
the classroom, the students would still need to learn
how to deal with rampant nonsense in the real world.
===========
If I may pick on viewpoint (d) just a little bit more:
They say mathematics is the language of science. That
may be, but it speaks with an accent. Viewpoint (d)
makes the mathematics work great, but it obscures a
great deal of the physics. The grade-school notion
of a vector as having direction and magnitude is far
more elegant, far closer to the real physics ...
but the mathematics of direction and magnitude is a
nightmare.
_______________________________________________
Forum for Physics Educators
Phys-l@phys-l.org
http://www.phys-l.org/mailman/listinfo/phys-l