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Re: Derivatives





On Tue, 28 Oct 1997, David Abineri wrote:

I have always pointed out in my Physics classes that a Displacement vs
Time graph must always be differentiable if it is to represent a "real"
situation since one cannot change instantaneously from one velocity to
another.

Today, the question of the differentiability of the Velocity vs Time
graph came into question. It seems to me that this may be non
differentiable, that is to say that acceleration may instantaneously
change from one value to another. HOWEVER, it is difficult for my high
school students to come to grips with this at the intuitive level. I
have shown such graphs to them BUT they seem to be applying their
knowledge of velocity to acceleration incorrectly. I have also said that
an object released from rest suddenly goes from zero acceleration to
9.8m/s/s but I am not sure I have convinced them yet.


You haven't convinced me that you can *really* remove the force from that
body at rest *instantaneously* (dt=0). To hold such a body at rest,
something must apply an upward force. Let's say that the body applying
that force touches the body at rest. If so, it deforms that body slightly,
and to remove the force, the bodies move out of contact, through a very
tiny distance, requiring a very small, but finite, time.

Infinities and infinitessimals cause no end of trouble in physics. But
remember that in the real world calculus itself is only an approximation,
for it presumes one can take a limit as anything goes to
zero--continuously. But in the real world you take that limit and bump up
against quanta--which involve discontinuities on the small scale. So you
really can't take those limits as "variable goes to zero", but only "fake
limits" as "variable gets very small, but not small enough to expose
atomic or quantum discontinuities." This works fine for macroscopic
physics, but takes a hekuva lot of time to get across to students (and is
it worth it?) Occasionally a student will (after such a discussion) say
that "Since we have to "cheat" on the calculus, then calculus isn't
strictly applicable to problems of the real world, so all physics derived
using calculus is suspect and we shouldn't use calculus at all." Then you
have to have more discussion to overcome *that* misconception.

Why is it important to you to impress upon students that the acceleration
can change discontinuously? Why is it important to them to address this
question, when there are so many other less troublesome matters of physics
which must be gotten across in a limited time? Did a student initiate the
question? None of my students ever have.

Does anyone know of some nice way(s) of getting this point across?

Thanks for any help on this one.

David Abineri


--
David Abineri dabineri@dot-net.net