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On 08/11/2007 05:00 PM, Folkerts, Timothy J wrote:
Along with my previous efforts to decide how to write symbols for
kinematic parameters, I have also been struggling with how to deal
with 1D motion.
That's another interesting and important question.
The book I am using (Serway) seems to really botch
things up. Basically, the whole chapter about 1D motion uses the
word "vector" when they usually actually mean "x-component of the
vector".
For example: " Average velocity can be either positive or negative."
" The average velocity is equal to the slope of the graph of position
vs time." " The graph of velocity vs time..."
The X-COMPONENT OF VELOCITY can be positive or negative. The
X-COMPONENT OF VELOCITY is the slope of x vs t. The X-COMPONENT OF
VELOCITY can be plotted on a graph.
Serway even admits that he botched the treatment of vectors in 1D
when he gets to the chapter on 2D: "This simple solution (using signs
to indicate direction) is no longer available in 2 or 3 dimensions.
Instead, we must make full use of the vector concept." In other
words, he didn't do things right the first time!
I expect that other books are similar.
How picky should we be? Is it so incorrect to say "plot v vs t" or
"velocity vs time" that it should be avoided? Even assuming that "v"
is the magnitude of velocity - rather than the true vector velocity -
is not right, because | v_vector | is always positive, but we are
perfectly comfortable plotting negative values on the graph.
Sticking to "v_x" or "x-component of velocity" the whole time would
seem to solve the problem without being too burdensome to either the
instructor or the students
Let's back up a moment. The fact is that in D=1, there is
an isomorphism between the vectors and the scalars. That
means that a lot of things that are not true in D>1 are
perfectly true in D=1. So, actually, I suspect what Serway
is doing is technically correct in D=1.
(In email I will use v to denote a vector. If I want the
magnitude I will write |v| explicitly.)
In particular, in D=1 projection of v onto the x-axis is
identical to v. I don't see anything incorrect or even
misleading about plotting v versus t. When the plot of v
lies below the v=0 contour, that can be understood in either
of two ways:
-- There is a downward-directed vector, with its tail at
v=0 and its tip at the ordinate being plotted.
-- Convert v to the corresponding scalar, observe that it
is a negative scalar, and plot it at the appropriate
negative position.
In D>1 it is nonsense to speak of vectors being positive or
negative ... but in D=1 it is perfectly acceptable.
On the other hand, it is quite possible to sow confusion even
while saying things that are technically true. When it comes
to D=1 kinematics, it is inevitable that students will over-
generalize what they learn about D=1 kinematics, and will have
to unlearn quite a bit when they get to D=3 kinematics ... and
unlearning is always hard, so it is worth diligently searching
for ways to minimize the amount of unlearning.
I am currently planning: 1) to do vectors first from Ch 3 and discuss
components of vectors 2) to cover 1D motion from Ch 2, but be more
careful about the terminology 3) to finish Ch 3 and 2D motion.
I've seen it done that way.
If I understand what TJF is saying, that's tantamount to skipping
one-dimensional kinematics entirely. Depending on the background
of the students, that's either a good idea or a bad idea.
For college students who have retained from HS even the vaguest
inkling of kinematics, it makes sense to dive right into D=3
kinematics and vectors at the same time. Kinematic ideas such
as force and velocity are the canonical illustrations of what
a vector is ... and the vectors quantify the kinematics, so it
all sashays along hand-in-hand.
For weaker students, this doesn't work so well. For HS students
who've never seen vectors *OR* kinematics before, you are AFAICT
pretty much obliged to start with D=1 kinematics.
My advice: Don't push D=1 kinematics too far. It starts out easy,
but it rapidly reaches diminishing returns. For example, at the
notorious "turn around" at the top of a D=1 trajectory, the
/direction of motion/ is undefined or at best discontinuous. It
is not worth studying this; just say that it is pathological in
D=1 but OK in D=3, and defer further discussion until the D=3
tools are available.
You cannot improve D=1 kinematics by talking about the x-component
of velocity. That wouldn't make sense because by hypothesis, we
are assuming the students don't yet know about components. (If
they did, we wouldn't be bothering with D=1 kinematics.)
Also it may be worth warning them from time to time that what you
are saying about D=1 kinematics, while perfectly true, is not
necessarily a reliable guide to real-world kinematics. In particular,
the idea of positive and negative velocities goes out the window
in D>1. (They will probably disregard the warnings, but at least
you can tell yourself you tried ......)
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