Re: Problem

[John Mallinckrodt <ajmallinckro@CSUPOMONA.EDU>]

On Wed, 19 Sep 2001, Michael Edmiston wrote:

> 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 use it regularly.  I think it is very useful in many
problems--especially conceptual ones--to understand that the
component of acceleration parallel to the velocity affects *only*
the magnitude of the velocity (the speed) and that the component
perpendicular to the velocity affects *only* the direction of the
velocity.

> 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.

But this only works in 1-d.  Velocity and acceleration are vectors
and, as has been mentioned here recently, do not have signs except
as a convenient short hand for direction in the highly restricted
case of a 1-d problem.  To know whether an object is speeding up
or slowing down you simply look at the angle between the a and v
vectors.  If it is acute (or obtuse), the object is speeding up
(or slowing down).  If it is 90 degrees, the object is purely
deflecting.  This can also be determined, of course, by
considering the dot product of a and v whose sign effectively
distinguishes between the three cases.

John Mallinckrodt      mailto:ajm@csupomona.edu
Cal Poly Pomona          http://www.csupomona.edu/~ajm