Re: Derivatives

[Chuck Britton <britton@odie.ncssm.edu>]

> 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.
		yes, we like to think that d vs t graphs are 'smooth'
> > 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.
	in 'reality' velocity graphs are also 'smooth' to a high degree
	    ^^^^^^^ (and ALWAYS be wary of this word!)
>
		<snip>

and "Donald E. Simanek" <dsimanek@eagle.lhup.edu> responded

> 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.
	Certainly for teaching kinematics (which I think should be taught
	in math class as in the UK?) we need to use 'standard' calculus
	ideas. But I will wager that when a 'correct' quantum/gravity
	theory arises, it will recognize that time and space are quantized
	and that the calculus of infinities and infinitesimals is not adequate
	for that next level of physics.

	But for getting kinematics across we need to follow the KISS principle.

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Chuck Britton  					Education is what is left when
britton@odie.ncssm.edu  			you have forgotten everything
North Carolina School of Science & Math		you learned in school.
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