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

[Rick Tarara <rtarara@SAINTMARYS.EDU>]

I guess I would argue that while Bob is correct mathematically, physically
you still can't have a changing velocity and a zero acceleration.  That is,
the instantaneous quantities (the derivatives) are abstract and (at least)
operationally are not 'real'.  A CHANGE requires a non-zero time interval.
That interval can be small, it can approach zero, but the derivatives just
tell us the LIMIT as the delta-t goes towards zero.  In the physical world,
we really can't perceive or measure a zero time interval.  Since the
definition of acceleration is delta-v/delta-t, if delta-v is non-zero, then
the acceleration can't be zero.  The _limit_ as delta-t goes towards zero
can be.

So, I would say that in an abstract, mathematical sense, Bob's examples are
OK, but in any practical, physical sense, Seth's statement still holds.  I'm
willing though to consider a 'real world' example that would support Bob's
mathematical argument--one that has physical significance for a = 0, delta-v
not = 0.

Rick
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rtarara@saintmarys.edu
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----- Original Message -----
From: "Bob Sciamanda" <trebor@WINBEAM.COM>

> Seth wrote:
> | 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).
>
> Consider the displacement y(t):
> (Think of it as a vertically tossed ball on a very strange planet!)
>
> y(t) =200t/3 - 5t^2 + (t^3)/6 - (t^4)/480
>
> Take three derivatives and plot y(t) , v(t) , a(t) and j(t) from t=0 to
> 40.
> Observe that the thing seems to "hang" momentarily at y=333.33:
>
> At t=20 the object has reached the top of its path. It then turns around
> and
> falls.
> But at t=20, the velocity, the acceleration and the jerk are ALL zero!
>
> It only keeps going (turns around) because the derivative of its jerk is
> non-zero (-1/20).  ALL other derivatives of y(t) are zero at t = 20.
> This should disabuse one of (sometimes spoken, sometimes implied)
> arguments
> that the acceleration (and/or jerk) cannot be zero at a turnaround point
> simply because the velocity is obviously changing.
>
> You can concoct for yourself more bizarre (but possible) motions in which
> the velocity keeps changing even though (at some time) it and all of its
> derivatives but the Nth one are zero, N being as large as you please.
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