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Re: speed and velocity



-----Original Message-----
From: H. Scott Wiley <hswiley@SWBELL.NET>
To: PHYS-L@LISTS.NAU.EDU <PHYS-L@LISTS.NAU.EDU>
Date: Wednesday, November 25, 1998 12:06 AM
Subject: Re: speed and velocity
. . .
When we compare two velocities or
decide when velocity is increasing or decreasing (a comparison), aren't
we really comparing "speeds"? I try to stress this to my students, but
it is easy to fall back to the standard "calculus rules" when it comes
to determining maxima, inc/dec intervals, etc. Am I wrong in my
thinking? How do you approach this subject
. . .
H. Scott Wiley

This is a worthwhile point, When we model an oscillator velocity as v(t)
= V*sin(wt) we are not giving a speed (we would write that as always
positive), we are giving a velocity, a one-dimensional vector, using the
sign to indicate direction. However, we can choose to interpret and plot
that simply as a signed algebraic quantity and observe algebraic maxima
and minima in the algebraic variations of the algebraic quantity thus
introduced.

This is a good lesson that mathematical models are our own constructs,
subject to our own definition and use. In this case we (perhaps
unfortunately) can use V*sin(wt) both as a model of a one dimensional
vector and simply as an algebraic quantity. As the value of v(t) falls
through zero, the sign change in the vector expression means a 180 deg
change in direction; the sign change in the algebraic quantity (eg.,
temperature) simply means continuing to fall to a lower value.

Note that introducing an offset and writing v(t) = V + V*sin(wt) gives
the (now unidirectional) velocity of a very different motion, but does
not significantly change the behavior of the algebraic quantity (except
for the offset itself).

Bob Sciamanda
Physics, Edinboro Univ of PA (ret)
trebor@velocity.net
http://www.velocity.net/~trebor