Re: transfer of momentum

[John Mallinckrodt <ajm@CSUPOMONA.EDU>]

Chuck writes:

> A MUCH better representation of Force is Newton's own version
>
>         F = delta p/delta t
>
> Even for those intro students who are being introduced to SR (or GR),
> it is VERY good pedagogy to use the momentum formalism in favor of
> the acceleration approach.

Hear, hear!  But I would offer a constructive criticism.  Newton
expressed and thought about his law as

         Impulse = change in momentum

As I mentioned recently, our students are not the only one's who have
trouble with the concept time derivatives of things in general, let
alone of things that are themselves time derivatives. I believe
Newton had very similar conceptual difficulties.   He performed his
proofs in terms of the effect of a series of impulses (except that he
uses the word "force" for what we would call "impulse") and rarely if
ever (as I read him) seems to separate out the "modern" concept of
force which would in any event to him have been something much more
like the "rate of delivery of impulse" or what John Denker thinks of
as "flow of momentum."

For example see his "Corollary 1" which appears immediately following
his exposition of the laws of motion.  He writes (according to
Cajori's revision of Motte's translation):

"A body acted on by two forces simultaneously, will describe the
diagonal of a paralellogram in the same time as it would describe the
sides by those forces separately."

He makes it absolutely clear in the subsequent discussion that he
envisions what we would call an impulse delivered instantaneously
when the particle is at one vertex of the paralellogram after which
the particle coasts along a straight line a) to an adjacent vertex if
acted on by one of the "forces", b) to the other adjacent vertex if
acted on by the other "force", and c) to the diagonally opposite
vertex if acted on by both.

We may forgive Newton for neglecting to specify that the particle
must start at rest for his analysis to hold.  Indeed, he might argue
that he had implicitly specified that by saying (in the discussion):

"If a body in a given time, by the force M impressed apart in the
place A, should with an uniform motion be carried from A to B, ..."

He may well have in mind that when he says it is "carried ... by the
force impressed ... in the place A", he is ruling out the possibility
that any of that "carrying" is being done by previously impressed
"forces," i.e., that the particle *was* at rest.

I recommend a little reading in the Principia to get a better sense
of how mightily Newton struggled with the same concepts that torment
our students.  For all of it's well-deserved acclaim, the Principia
can hardly be called a clear exposition of Newtonian dynamics even
allowing generously for language difficulties.

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