Re: MATH PHYSICS

["John S. Denker" <jsd@MONMOUTH.COM>]

David Abineri wrote:
> If one assumes that a projectile encounters an air resistance
> proportional to velocity, one can write a differential equation like
> mr''=-mgj - kr' which can be solved for r using an integrating factor
> e^(kt/m).
>
> The final solution for r, however, does not admit an interpretation for
> k=0.  Why is it that one does not get the ideal case to come from this
> more general case when k=0?
>
> I hope that the question makes sense.

The question makes perfect sense.

The answer isn't nearly as bad as it looks.

The strategy in all such cases is to consider what happens for
small but nonzero k, and *then* see what happens as k->0.

Therefore:

1) Solve the diffeq in the usual way.  You don't need anything as
fancy as an integrating factor;  just substitute p = m r' and turn
the crank.

1.5) If there is any doubt that you've solved it correctly, use
a spreadsheet or whatever to graph the solution, for some small
but nonzero value of k.

2) Expand the exponential in a Taylor series.  Keep terms up to
third (!) order in k.

3) Collect terms.  Show that
        r' = g t + const + (terms proportional to k)
and note that the terms proportional to k wouldn't have been there
if we hadn't kept third-order terms in step (2).

4) Convince yourself that all higher terms in the Taylor series
would involve higher powers of k.  Take the limit as k->0 and
find
        r' = g t + const
as it should be.

Note that some arbitrary constant divided by k is just some
other arbitrary constant.

OK?