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On Tue, 28 Oct 1997, David Abineri wrote:yes, we like to think that d vs t graphs are 'smooth'
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.
in 'reality' velocity graphs are also 'smooth' to a high degree
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.
<snip>
Infinities and infinitessimals cause no end of trouble in physics. ButCertainly for teaching kinematics (which I think should be taught
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.