I wrote:
>>
>> Asking what is "the" acceleration at the peak of a
>> parabolic arc is at best ambiguous.
>> -- The scalar acceleration is zero. The speed
>> is locally constant at this point.
>> -- The vector acceleration is of course just g.
On 11/19/2003 07:27 PM, Dan Crowe wrote:
> (d/dt)|v| is undefined at the peak:
> The limit from the rising side (-g) differs from
> the limit from the descending side (+g).
> Thus, the scalar acceleration is undefined
> at the peak.
Re-read the problem that I posed and answered. I
explicitly stated that I was analyzing a parabolic
arc.
1) Since the path is a _parabolic arc_, the scalar
acceleration does not switch discontinuously from
-g to +g. It goes smoothly through zero.
2) Even if I had not restricted the analysis to
parabolic arcs, assuming otherwise would be
unphysical. The degenerate case of a vertical
trajectory exists only on a set of measure zero.
(And it's not properly called a parabola.)
Usually in physics an exception that happens
infinitely rarely can be ignored.
If you can throw something with exactly zero
horizontal velocity you are infinitely more
skillful than I am.
3) Even on this set of measure zero, the undefinedness
is a removable singularity. One could verrry
reasonably take zero to be the "principal value"
and use it as the answer in this pathological case,
if anybody really cared.
In general, as I said in a previous thread: In
physics it is usually a good policy when faced with
a pathological case to solve the corresponding
non-pathological case and then (if necessary) pass
to the limit.