Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

Re: more acceleration



On 11/21/2003 06:32 PM, cliff parker wrote:
> A particularly bright student responded as follows.
>
> "This is a tough one. Because the forces are balanced, there should
> be constant velocity (no acceleration). However velocity is never
> constant in this situation, so the best way to describe acceleration
> is that it is transferring from positive acceleration to negative
> acceleration."
>
> I had not thought about the problem as thoroughly as he did. I like
> it when my students stretch my thinking! I know we have had
> discussion of situations similar to this over the past few days but I
> was wondering what others thought the best answer to this question
> would be. Is acceleration simply undefined at this point?

There are deep conceptual issues in play here, possibly
hiding behind superficial terminological issues.

What do we mean by "constant at a point"? At a single
point, whatever it is can only have one single value. A
value is a value. If it can't have more than one value,
we can't even define a notion of constancy or change.

Our notions of constancy and change are defined w.r.t
a _neighborhood_ of the point. We can say that something
has a velocity at a point, but we can't measure or even
define the velocity unless we have examined the object's
position at neighboring points.

Similar remarks apply to the acceleration. We can't
measure or even define it unless we have examined the
object's velocity at neighboring points.

So I would say to the student something like this: you
are correct to say "the velocity is never constant" in
the sense that it cannot remain constant over any
_macroscopic_ interval. But as a matter of terminology,
whenever somebody says XXX is constant _at a point_ you
may assume that is shorthand for saying XXX is _locally_
constant, which in turn is shorthand for saying XXX is
constant to first order in the infinitesimal neighborhood
of that point.

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

I liked Bob's fourth-order equation of motion.

Our intuition for harmonic oscillators is that the
zero-velocity points are out of phase (indeed maximally
out of phase) with the zero-acceleration points.
OTOH it's good to be reminded that not everything
is a harmonic oscillator, and other phasings can
occur.

Bob didn't explain the physical context of his
equation of motion, and apparently some readers
dismissed it as unduly contrived, perhaps thinking
that if you tried to set up a situation where the
zero-velocity point _exactly_ coincided with the
zero-acceleration point, you could only succeed
on a set of measure zero.

But in fact it's not really as unphysical as all
that. There are situations e.g. self-organized
criticality where a whole bunch of low-order
derivatives vanish simultaneously for good physical
reasons.

The prototypical self-organized critical system is
sand dribbling on a sand-pile. In the long run it's
always right on the edge of instability ... you never
know if the next sand-grain will cause a small
avalanche or a large avalanche. You might think
that if positive stability is possible and negative
stability is possible, you would only find neutral
stability on a set of measure zero ... but think
again: in the large-N limit, a neutrally stable
sand-pile is exactly what you get.