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[Phys-L] Re: A problem of motion and derivatives



For an object with v=0 and a=0 at t < 0, the imposition of a constant force
or acceleration at t=0 is a discontinuous event. (This includes dropping an
object, where the holding force has to go to zero instanteously.)

Of course, you can model these transitions as some sort of smooth function
with a large jerk whose duration approaches zero.

Al Bachman

From: Seth T Miller <stmiller@GOETHEANSTUDIES.ORG>
Reply-To: Forum for Physics Educators <PHYS-L@list1.ucc.nau.edu>
To: PHYS-L@LISTS.NAU.EDU
Subject: A problem of motion and derivatives
Date: Wed, 02 Nov 2005 22:16:29 -0800

Hello all. If you will excuse me for thinking philosophically for a moment
about motion, I would like to seek your help in disabusing me of any
unknown
assumptions or idiotic notions I am carrying with respect to the following
line of thinking:



Given: Position L and time t, instantaneous velocity v is given by the
derivative of position with respect to time: dL/dt.



The second derivative of L wrt t, dL/dt^2 (equivalently: the derivative of
v
wrt t, dv/dt) is instantaneous acceleration a.



The third derivative of L wrt t, dL/dt^3 (equivalently the derivative of a
wrt t, da/dt) is known as jerk - I don't know the proper symbol so I'll
call
it j for now.



Assume a situation where an object has v = 0, and a = 0 (any inertial
frame). If the object is to change its inertial state to have a non-zero
final velocity, then the velocity must undergo a change from 0 to some
arbitrary positive value. I recognize that I am assuming that changes in
velocity are smooth and not discontinuous.



If the object is to change its velocity it must have a non-zero value for
acceleration; i.e. to change one's velocity requires an acceleration
(either
by changing one's speed or direction). But if an object begins in a
non-accelerated frame where a = 0, in order to do this, the value for
acceleration must change from 0 to some non-zero value...



But this change in acceleration requires a non-zero value for jerk, and so
on ad infinitum.



This seems an unreasonable solution which most likely points to an error in
the analysis, which I hope can be explicitly pointed out to me.



Additionally there is one related problem: that of changes in motion and
whether any arbitrary path can be traveled in a way that keeps jerk to a
value of 0. If jerk = 0 then acceleration must be constant by definition.

But to move along an arbitrary path (the simplest example is circular
motion, say a car driving around in a circle), one cannot keep one's
velocity constant. When acceleration is constant for a car moving on a
circular path, the car can keep jerk =0 and end up driving in a tighter or
wider circle if it changes its direction and velocity in a harmonious way
so
that their combined contribution to acceleration = some constant. For
example by driving in a tighter circle, one must apply some amount of
braking to keep overall acceleration constant. Is this the case or am I
missing something obvious? Can jerk be kept to 0 in this instance? Can it
be kept to 0 on any arbitrary path between two points?



Any of your thoughts, comments, insights, or critiques are welcome!



Thanks-

-Seth Miller

East Bay Waldorf School
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