Re: Problem

["RAUBER, JOEL" <JOEL_RAUBER@SDSTATE.EDU>]

Since I posed the reply I'll try to answer for the purpose.  For general
curvilinear motion of an object it can be useful to break the instantaneous
acceleration vector (a) into components parallel and perpendicular to the
velocity.

The perpendicular component is something quite familiar v^2/r (centripetal
acceleration) where v is the speed and r is the radius of curvature of the
trajectory at the point in question.

The parallel component is as stated before the time rate of change of speed.

If you are like me, and pound into the ground that vectors involve the
concepts of size and direction; and if you describe acceleration as telling
you something about changes in velocity vectors; then it is natural IMO to
ask if there is something about acceleration that can isolate and seperate
the change in the magnitude of v from the change in direction of v.

The answer is yes! The parallel and perpendicular components of the
acceleration.  The perp. comp. tells you how the direction of v is changing,
the parallel comp tells you how the magnitude is changing.

This is in some ways better than x and y comp. of acceleration, as they are
coordinate independent statements.

I actually mention the above to my Calculus intro class (not the coordinate
independent stuff), but don't make a big deal of it, other than
qualitatively (pictures of curvy trajectories with tangent velocity vectors
and acceleration vectors and conceptual questions of is this object speeding
up, slowing down, moving in straight line (perp. comp. of acceleration = 0);
etc.

BTW many textbooks have diagrams showing or discussing these components.

e.g.
Crummet & Western pg 89
Tipler pg 126
Halliday & Resknick & Krane (Physics) pg 63
Reif (a PER text, "Understanding Basic Mechanics") pg 76
Serway pg 87

etc.

Joel Rauber

> -----Original Message-----
> From: Michael Edmiston [mailto:edmiston@BLUFFTON.EDU]
> Sent: Wednesday, September 19, 2001 8:31 AM
> To: PHYS-L@lists.nau.edu
> Subject: Re: Problem
>
>
> Joel Rauber said:  Without really answering the second question, AFIK
> d|v|dt, as long as the derivative is defined, it is the instantaneous
> component of acceleration parallel to the instantaneous
> velocity; which is
> how it represents something connected to acceleration.
>
> Okay... you can say this.  I'm not criticizing; I am just
> asking out of
> curiosity... do others say this?   It this a common concept?
> If so, how is
> it used; what purpose does it serve?
>
> I do find it useful to know if an accelerating object is in
> the process of
> speeding up or slowing down.  But I approach that by noting
> if the signs of
> the velocity and acceleration are the same or opposite.  If
> they are the
> same, the object is currently speeding up; if they are
> opposite the object
> is currently slowing down.  Any other definitions beyond v = dx/dt and
> a=dv/dt are not necessary.
>
>
> Michael D. Edmiston, Ph.D.              Phone/voice-mail:
>   419-358-3270
> Professor of Chemistry & Physics        FAX:
>   419-358-3323
> Chairman, Science Department            E-Mail
>   edmiston@bluffton.edu
> Bluffton College
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> Bluffton, OH 45817