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Re: Kinematics



Most often the independent variable in our experiments
is time. In the situation I described the independent
variable was the distance along a straight line. That is
why my two questions were asked.

When we have a time interval between t1 and t2 we
do not hesitate to assign t=(t1+t2)/2 to Vav calculated
from the corresponding x1 and x2. This is because we
all "travel along the t axis at a constant rate." (Yes, I know
I am speaking like a poet; the "rate at which we travel
along the t axis" can not be defined.)

But our traveling along the x axis does not have to be
uniform. The best thing we can do is to assume that the
rate of covering the x axis is uniform between any two
consecutive x1 and x2. In reality there may be several
"back and forth" between x1 and x2 but we are not aware
of them, unless additional data are collected. Note that
I am assuming that the uncertainties (random errors in
measuring x and t) are negligible.
Ludwik Kowalski


On Monday, Sep 15, 2003, at 15:28 US/Pacific, John S. Denker wrote:

On 09/13/2003 09:43 PM, Ludwik Kowalski wrote:
I have questions based on the first lab this year.
I took students outdoor. With chalk we marked
positions along a path (at x=0, 2, 4, 6, etc. meters
from 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=2 s for Vav=2.0 and 2.6 for Vav=2.5?

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

This problem and all similiar problems are ill-posed.
There are some things we can do, and some things we
cannot do.

++ If you know some ordered pairs (time, position) you
can calculate the *average* velocity. For example:
Vav = (x2-x1)/(t2-t1)

++ By the intermediate value theorem, we know there is
*some* time in the interval [t1, t2] that has this
velocity.

-- Without nontrivial additional information, we have no
way of knowing the actual time corresponding to Vav.

-- Without knowing the actual time, we have no way of
calculating the acceleration (or even average acceleration).

Here is an example of what I'm talking about, that is,
a counterexample to the notion that Ludwik's problem
can be solved:

Suppose that very, very shortly before time t1, the
runner accelerates to 9000mph+Vbar. (Because this happens
so close to the end of the [t0,t1] interval, it has no
effect on the timing of the (t1,x1) event.) Then during
the entire [t1,t2] interval, the runner decelerates
uniformly to -9000mph+Vbar. The data shows that the
average velocity during the interval was Vbar, but there
was a treeemendous rearward acceleration that you could
not possibly have divined from the data. (The acceleration
is uniform, so average acceleration equals acceleration.)

================

The problem I'm pointing out cannot easily be prevented.
Without nontrivial additional information about the
situation, there is no physical basis for assuming things
like this can't happen. Indeed, it is known that the
world-line of an electron is fractal; the closer you
look, the more wild fluctuations you see.

================

If you want to convert this into a well-posed problem,
you will need to invoke some sort of regularization
principle. Since the data given above is made-up data
to begin with, we can fabricate a made-up regularization
principle to go along with it.

For example: Let the new assignment be to find the
*smoothest* set of accelerations that will explain the
data. Here "smoothness" is the regularization principle.
We get to specify what notion of smoothness we want,
perhaps L2-norm (i.e. RMS) or L-infinity-norm (i.e. max).

As an easier example: You could ask the students whether
the observed data is consistent with a constant acceleration.
(It is not. The first four data points are consistent, within
the implied uncertainty, but the last data point derails the
train.)