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Re: Kinematics

Just a terminal clarification. Dan had a handle on the calculation scheme,
if you follow the detail of what he's done. Ludwick stated that he had made
up the sample data. I prefer using "uncertainties" from day-one and the
term "errors" when the students are seasoned enough to be able to recognize
and account for them. Students come in already branded by "mathematics" and
will eventually notice that we are sometimes divided by our common language.

Tom Ford

At 02:42 PM 9/15/03 -0400, Ludwik wrote:
I agree with what Tom wrote but in asking the question
(see below) I was not addressing the effects of errors.
The questions I asked are valid even when errors are
negligible.

Unlike Dan, who imposed an assumption of a constant
acceleration before plotting the data I wanted to plot the
Vav and ACav versus time first and then decide if the
acceleration was constant or not. The issue was to plot
data properly (i.e. to assign Vav to proper t, and to use
the correct deltaT to calculate the acceleration. I think
my choices were "the best one can do."
Ludwik Kowalski


On Monday, September 15, 2003, at 01:34 PM, Tom Ford wrote:

> 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