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Re: Kinematics



I would hope that the two comments, so far, do not represent an either/or
choice at the beginning of kinematics but rather a good sequence, if
presented to the students in the order described. It is always better to
throw the body in first and bring the mind in later, but this kind of first
experience should be trying to maintain a constant speed and direction.

Students asked to contemplate their motion graphs of position/time,
velocity time, and acceleration/time (as a way of continuing the
communication of understandings), should become disabused pretty well from
their inherited tendency to plot point to point by the time they get to
plotting the acceleration/time graphs of their intended constant velocity.

This is the proper time for them to notice that whether they or some motion
probe + computer combination are taking the data, position and time is all
there is until other understandings are later derived. This is the proper
time for them to notice that there are uncertainties is the measurement of
both position and time, so none of the plotted points are in exactly the
right place. This is the time for them to notice that as they strolled
along, they were not aware of the spectacular "jerks (q.v.)" which their
acceleration plot seems to portray. This is the place to say to the
persistent point-to-point plotter, "... and you mean to tell me that all of
nature was just waiting to make the change you have recorded in the slope
just in case YOU were WATCHING and wanted to take data? How does nature
keep track of all that stuff for all observers?"

Ludwik and his students were not yet ready to approach an instant when Dan
showed how this will be useful and pointed to some of the issues likely to
be addressed later, such as the uncertainty of the measurement, what that
might lead to, the whole notion of smoothing that will keep the message
from being forever buried in the noise, and the fact that five closely
space data points might belong to any curve

Tom Ford.

At 08:34 AM 9/15/03 -0500, Dan Crowe wrote:
If the acceleration is constant, then the average
velocity during a time interval (t1 < t < t2)
equals the instantaneous velocity at the midpoint
of the time interval [t =3D (t1 + t2)/2]. If the
midpoint of the time interval is used with the
data given by Ludwik, then the calculated
acceleration is inconsistent with the assumption
that it is constant.

I obtain the following values:

t (s) 0.75 2.0 2.9 3.55
Vav (m/s) 1.33 2.0 2.5 4.0
Aav (m/s^2) 0.536 0.556 2.31

The first two calculated values of acceleration
are consistent with the assumption that the
acceleration is constant, but the third value
is clearly inconsistent with that assumption.

If the acceleration is increasing (decreasing)
monotonically, then the average velocity equals
the instantaneous velocity at a time later
(earlier) than the midpoint of the interval.

If the final time reading was 4.0 s, then the
final average velocity would be 2.86 m/s and
would correspond to the instantaneous velocity
at t =3D 3.65 s, assuming that the acceleration
was constant. In that case, the acceleration
for the final interval would be 0.514 m/s^2,
which would be more consistent with the
assumption of constant acceleration.

Having students run provides a kinesthetic
experience that can aid learning, but it is
difficult for a runner to maintain a constant
acceleration for an extended time.

We perform a somewhat similar experiment
using a cart accelerated across a table by a
hanging mass. We use a spark timer to record
the displacement of cart every 0.1 s. This
experiment provides an acceleration that is
essentially constant, but does not provide
a kinesthetic experience.

Daniel Crowe
Oklahoma School of Science and Mathematics
Ardmore Regional Center
dcrowe@sotc.org


-----Original Message-----
=46rom: Ludwik Kowalski [mailto:kowalskil@MAIL.MONTCLAIR.EDU]
Sent: Saturday, September 13, 2003 7:44 PM
To: PHYS-L@lists.nau.edu
Subject: Kinematics


I have questions based on the first lab this year.
I took students outdoor. With chalk we marked
positions along a path (at x=3D0, 2, 4, 6, etc. meters
=66rom the origine. Two students at each mark
recorded time instances at which a runner was
passing by; all stopwatches were pressed at the
same moment, that is when x was zero. Here are
the data I am inventing to ask my questions:

1 x (m) 0 2 4 6 8
2 t (s) 0 1.5 2.5 3.3 3.8
3 dx 2 2 2 2
4 dt 1.5 1.0 0.8 0.5
5 Vav 1.33 2.0 2.5 4.0
6 ACav A B C

SPACES IN THIS TABLE SHOULD BE
PRESERVED IF YOUR FONT IS COURIER.

The goal is to calculate average accelerations,
A, B and C. Also to plot Vav and ACav versus time.

Lines 1 and 2 show row data (positions and times).
Lines 3 and 4 show steps in space and in time
Line 5 shows average velocities (dx/dt).

Question #1
In plotting Vav versus time what value of t should be
associated with each Vav? For example, should it
be (1.5+2.5)/2=3D2 s for Vav=3D2.0 and 2.6 for Vav=3D2.5?

Question #2
What value of dt should be used to calculate average
accelerations? For example, what should dt be in
B=3D(2.5-2.0)/dt ? Should it be 0.9?

Note: the concept of instantenous v and a has not
been introduced. dx and dt are deltas.
Ludwik Kowalski