Re: [Phys-L] raising the game

[John Denker <jsd@av8n.com>]

On 06/29/2013 06:31 PM, Richard Tarara wrote:
> The point here is that the skills for LIFELONG LEARNING are really
> what are the key things to take from a formal education.

OK, so as they say in business meetings, what's the action item?
What concrete steps can we take that will move us toward that
goal?

I suggest that there is not any one big change that needs to
be made ... other than the not-very-concrete step of deciding
to do it.  Instead, there are thousands of small steps we can 
take.

One such step would be to take homework problems that don't
require much reasoning and upgrade them.  There are thousands
of such problems, and you don't need to convert them all at
once;  just upgrade them one by one, as you come across them.
Each one is only a small step, but over time the steps add up.

Here's an example chosen at random that illustrates what I
mean.  I flipped open a copy of _Matter and Interactions_ and 
my eyes fell upon the following exercise:

«10.P.35  There is an unstable particle called the "sigma-minus"
(Σ-), which can decay into a neutron and a negative pion (π-);
Σ- --> n + π-.  The mass of the Σ- is 1195 MeV/c^2, the mass of 
the neutron is 939 MeV/c^2, and the mass of the π- is 140 MeV/c^2.
Write equations that could be used to calculate the momentum
and energy of the neutron and the pion.  You ddo not need to
solve the equations, which would involve some messy algebra.
However, be clear in showing that you have enough equations 
that you could in principle solve for the unknown quantities in
your equations.»

I am reminded of the immortal Henny Youngman bit:

  -- Doctor, doctor, it hurts when I do /this/.
  ++ So don't do that.

My point is simple:  If the algebra is messy, don't let it
boss you around.  Find a way to simplify the algebra, or
to get the answer with no algebra at all.  Don't wimp out.
Learn to think like a physicist.  There are lots of ways
of simplifying problems like this.  Maybe such-and-such
term can be neglected entirely.  If not, maybe it can
be approximated to first order.  Maybe one of the particles
is non-relativistic.  Maybe you can leverage the computer 
against the algebra, so that the numerical answer guides
the algebraic simplifications and/or vice versa.  Maybe
you can use your intuition about ping-pong balls and bowling 
balls.  Look for connections between ideas!

We're talking about one algebraic equation in one unknown.
It's even a monotonic function of p!  On the scale of 
things, this does not count as tricky.

In this case, there are at least three ways of solving the 
problem:
  1) You can get a decent estimate just by /thinking/.
  2) You can get a more accurate estimate by expanding
   one of the square roots to first order.
  3) You can solve it numerically. 

=========

1) Before firing up the computer, you should /think/ about 
the problem a little bit.  Using a tiny bit of physical 
intuition, you can get a pretty decent approximate solution 
without doing very much algebra at all, let alone messy
algebra.  

You immediately know the two products will have equal and
opposite momenta.  You know from experience bouncing ping-
pong balls against bowling balls that the lighter particle 
is going to have the lion's share of the kinetic energy.
This is true in the non-relativistic case, where the energy
is ½ p•v, but would not be true in the relativistic case, 
where the energy is a whole p•v and both particles have the 
same velocity (v=c, hypothetically).  However, some trivial 
arithmetic shows that the reaction energy is only 117 MeV, 
which is nowhere near enough to drive the neutron into the 
hypothetical v=c regime.

Now in spacetime, in any particular frame, the total energy
is one component of the [energy,momentum] 4-vector, while
the momentum is another.  So they add (or rather subtract)
in quadrature.  We know from long experience with equilateral
right triangles that if the mass and the momentum were about
the same, the kinetic energy would be about 41% of the mass.
The reaction energy (which will mostly go into the kinetic
energy of the pion) is /more/ than 41% of the pion mass, so 
the momentum component must be /more/ than the mass.  OTOH 
probably not a whole lot more, not twice as much.  So a 
decent guess would be 1.5 m or maybe √2 m, i.e. somewhere 
around 200 or 210 MeV.

On the other hand, it could be 13% less than that, because
the pion has only 87% of the product mass, so a momentum
somewhere between 170 and 185 MeV doesn't seem too crazy.
OTOH this is just an estimate, and those numbers could
easily be off by 20% either way.  There's no way they
could be off by a factor of 2, unless I've made a gross
error somewhere.

====================

2) I leave this part as an exercise to the reader.  Note
 that it is safe to expand one of the square roots to
 first order, but not the other.  However, that's enough
 to greatly reduce the messiness of the algebra.

 This is only possible because we already have a decent 
 estimate of p (from part 1, or perhaps from part 3).

====================

3) A long time ago, back when the current crop of students 
were minus-thirty years old, an algebraic solution might 
have been the best way to understand what's going on.

However, nowadays, given that exercise 10.P.35 already requires
the students to come up with an equation, the sensible thing to
do is type the equation into a spreadsheet.  You could graph
the equation as a function of the unknown momentum p and find 
the root graphically, just by looking.

Or you could do a binary search, trial and error, to find a
good value for p.

Or (!) you could use the built-in "solver" function to find
an even better value for p, something like p=191.95046098 MeV.

Note that the first time I tried to solve this, I got the
wrong answer, because of a typo.  I knew it was the wrong
answer, because of the estimate I obtained by thinking, as
described in part (1) above.

If I hadn't detected the wrong answer that way, I would have
detected it a moment later, because I used the p-value to
directly evaluate E1 and E2, and I checked the  total product 
energy against the reactant energy.  Check the work!

=======================

Pedagogically speaking, it is like pulling teeth to get students
to check the work.  It is easy to understand why this is:  They
have been raised on a steady diet of game-show tests, where 
there is every incentive to come up with the quick answer rather
than the thoughtful answer.  There is no incentive -- and often
no opportunity -- to show the work, let alone check the work and
show the checks.  This is one of the eleventeen ways in which
multiple-guess tests are poisoning the system.

If you're serious about teaching people to think, you need to
have a grading scheme where getting the right answer does not
get all the points.  There have to be points for showing the
work, checking the work, and showing the checks.  

This connects to a truism in the programming business:  writing
the basic code is less than half the work.  You also have to
write the comments and other documentation.  If you are managing
a bunch of programmers (or just managing yourself), you have to
budget for this.