Re: Kinematics

["John S. Denker" <jsd@AV8N.COM>]

On 09/16/2003 01:25 PM, Ludwik Kowalski wrote:
> 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.

I noticed that the data was in t-versus-x format.
Like Ludwik, I considered the hypothesis that this
was causing difficulties.

However, I concluded that the choice of independent
variable was not the main issue.  We have here a
wolf in sheep's clothing.  We should not conclude
conclude that sheep are carnivorous.
  sheep = t-versus-x format
  wolf = ill-posed problem

I stand by what I said previously (15:28 Monday, Sep
15, 2003): the problem is ill-posed.  Re-expressing
(or even re-taking) the data in x-versus-t format will
not make the problem any less ill-posed.

You cannot reliably infer the acceleration from samples
of the x and t coordinates, not without some nontrivial
regularization assumptions.

> 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.

We, Kemosabe?  I most certainly do hesitate.  If I
make such an assignment at all, I most certainly
flag it as an approximation.  Nontrivial additional
analysis will be required, to see whether this
approximation is adequately accurate for the task
at hand.

This issue comes up all the time in the numerical
methods business.  I get 1500 hits from
  http://www.google.com/search?q=forward-euler+backward-euler

> This is because we
> all "travel along the t axis at a constant rate."

No, that is neither necessary nor sufficient.  You
can re-parameterize the data in terms of a non-uniform
t_prime variable, and it doesn't make the problems any
worse.

The intermediate-value theorem says that there must
be *some* t in the interval [t1,t2] where the velocity
equals Vav.  Similarly there must be *some* x in the
interval [x1,x2] where the velocity equals Vav.  But
there is !!not!! any good reason to assume that the
special point is the middle of the t-interval or the
middle of the x-interval.

Yes, I'm assuming differentiability.  The problems get
even messier otherwise.