Re: [Phys-l] Percent KE retained in perfectly inelastic collision

[John Denker <jsd@av8n.com>]

On 12/16/2007 11:16 PM, Michael Edmiston wrote:

> The problem is a one-dimensional perfectly-inelastic "dead-on"
> collision.  Mass one (m1) comes in with initial velocity v1i and
> strikes stationary mass two (m2).  The two masses stick together
> and the combined mass (m1+m2) goes off with final velocity vf.
> The data for the problem are m1, m2, and v1i and a statement that
> the collision is perfectly inelastic.  Part-A of the problem asks
> them to solve for vf.  Part-B of the problem asks them to solve
> for the percentage of KE retained.
>
> I assumed students would get the percentage by KEf/KEi*100%.
> However, one student got the correct answer by vf/v1i*100%.  I
> first marked it wrong with the comment that the answer was
> accidentally correct.  But then I did a little algebra and
> discovered this method is correct in general. 

0) Hypothetically, it is possible that the kid (gasp) did
the homework and simply /remembered/ that there were simple
proportionality relationships among the key variables.

This hypothesis is dubious because if this never happened
in the previous 30 years, why should it happen now?

1) It is possible that the kid actually understood the physics.
Yes, I prefer to see it as a physics puzzle rather than as an
algebraic/numerological coincidence..  You can do the physics
in your head, no algebra required:

  Formulation #1:  KE = ½ m v^2     (true but boring)
  Formulation #2:  KE = ½ p^2 / m   (no improvement)
  Formulation #3:  KE = ½ p • v     ho, ho, ho!

Momentum is conserved.  Therefore KE scales like v.  QED.

That's all there is to it.


Tangential remark:  I wrote the ½ p • v version third for
dramatic effect, but really it should be the first thing you
think of, not the third.  For ultrarelativistic particles,
E = p • v.  That is the smart way to write it.  That version
doesn't involve the mass, which is important because for
ultrarelativistic particles the mass is irrelevant and/or zero
... yet we still want to be able to express the energy-versus-
momentum relationship.

>  I still think the student made a lucky guess,

2)  On the information given, we cannot rule out dumb luck.
Sometimes a student can use a completely unsound method to
arrive at the correct answer, such as simplifying the
expression 16/64ths by "cancelling" the sixes.

       16
      ---- = ----
       64

OTOH, not all guesses are mere dumb luck.

There's a second method of solution, a valid method that involves
an Ansatz.  This sorta seems like a guess, but the method is in
fact 100% rigorous and reliable.

In this case you can guess that the E versus v relationship
is linear, then try to quantify the correction terms (if
any):

 -- Consider the case m1 << m2.
 -- Consider the case m1 == m2.
 -- Consider the case m1 >> m2.

In all three cases, the correction terms are zero, i.e. the
hypothesis that the relationship is linear survives all
three checks.  I suppose there is a possibility that some
nonlinear relationship could survive these checks, but if
there's nothing more at stake than a few quiz points, I'm
going to guess "linear" and take my lumps if I'm proved
wrong.

This technique of "guessing" the general form of the solution
and then probing to find the adjustable parameters is useful
in a wide range of problems.  For instance, suppose I asked
you for the sum (from i=0 to N) of the expression 7 + i + i^3.
Now you pretty much know the answer is going to be a polynomial,
and calculus tells you the degree can't be bigger than 4. so
you can try the general 4th-degree polynomial and determine
the polynomials by checking at most 5 cases.  This trick is
used internally by symbolic-math programs such as Mathematica.

3) There is a third way to attack the problem.  Disintegrate
mass 2 into small subparticles.  Let mass 1 encounter them
one by one.  The overall effect is the same, whether mass 1
encounters mass 2 all at once or sweeps it up bit by bit.
So if E is linear in v in the case m1 >> m2, then you can
integrate in your head and convince yourself that the rule
holds for all m1 and m2.

*) There are probably other ways of attacking the problem.
But these three should be enough to make the point.

>  I doubt the
> student knew that.  I've never seen it before.  Has anybody on
> the list seen this?  Does this appear in any textbook you are
> aware of?

I don't know if I've seen that exact thing in books, but that's
only the tip of the iceberg anyway.

The larger idea was expressed by Kip Thorne in one of his books
(I'm not sure exactly where) (paraphrase):
   If a complex calculation leads to a simple answer,
   you should look for a cleverer way of analyzing the
   problem.      [1]

Even if it weren't in a book, when I was an undergrad, statement
[1] was pretty much the motto of the place.  You got reminded of
it all day every day for four years.

 -- Look for the conservation law.
 -- Look for the symmetry.
 -- Look for the scaling law.
 -- Look for the sum rule.
 -- Look for a way to apply PVW.
 -- Keep track of what's going on in phase space.
 -- If you've found a symmetry, that's nice, now go back and
  see if there's a higher symmetry (e.g. rotation group -->
  Lorentz group).
 -- Do the infinitesimal case and then integrate.
 -- Cough up a variational Ansatz that's in the right ballpark,
  then refine it.
 -- The same equations have the same solutions.
 -- If you found a method of solution, that's nice, now go back
  and find another method of solution, independent of the first
  method.
 -- etc. etc. etc.


IMHO this is the heart and soul of physics.  This is the essence
of what physics "is".  Everything else is just details.

We agree that on a shelf of 100 books, you'd be lucky to find even
one that mentions KE ∝ v in this context.  But that isn't what's
important.  The important thing is for the students to learn the
principles and the thinking skills to the point where they can
see in an instant that KE /must be/ proportional to v, even if
they haven't been told.

The Feynman Lectures emphasize the principles.  And I thought
PSSC took a rather principled approach.  But after that, there's
only slim pickings, AFAICT.  That's depressing.