Re: [Phys-l] problems requiring LOTS of digits ... or Taylor expansions

[John Denker <jsd@av8n.com>]

On 03/29/2012 04:29 PM, Quist, Oren wrote:

> "Expand" the square root (from the quadratic formula).  This shows
> precisely what is happening.  The "big" terms cancel out - exactly -
> leaving the answer in the (quickly converging) series.  Sig figs
> takes on a whole new meaning in this problem after the expanding
> algebra is complete.

Here's another example in that category:  Use special relativity to 
find the classical kinetic energy, via the correspondence principle.
Write the total energy as E = mc^2 cosh(ρ) where ρ = rapidity = v/c.
Define the kinetic energy as E(ρ) - E(0).  Let m = 1 kg, accurate to 
0.01 %.  Let v = 10,000 m/s, accurate to 0.01%.  Calculate E(ρ) and 
round off to some number of digits.  Calculate E(0) and round off to 
some number of digits.  Then subtract to find the KE.  How many digits
do you need?  Do it again at a slower speed:  v = 1 m/s.  Now you
need even more digits if you want to get the right answer ... vastly 
more than would be allowed by the sig figs rules, and indeed more 
than are provided by IEEE double precision floating point.

Better plan:  Expand cosh(ρ) - cosh(0) to lowest order analytically 
to get an expression for the KE.  Re-express the KE in terms of the 
spatial part of the momentum i.e. px, where px c = mc^2 sinh(ρ).
Then plug in to this expression, with no worries about excessive
roundoff error.

Let's be clear about the physics:
 -- The zeroth-order term in the expansion is the rest energy, m c^2.  
 -- The first-order term is the spatial part of the momentum, px c.  
 -- The second-order term is the kinetic energy, 0.5 px^2/m.  

This result never ceases to amaze me.  It is a lesson in the grandeur 
and unity of physics:  Lowest-order expansions allow us to unite the 
trigonometry and geometry of spacetime with familiar bits of classical 
kinematics.

Unless you completely ruin it by using sig figs.


> Do students learn how to expand a function any more?  

Some of them do.  The rest certainly should.

The most-common first-order expansions can be taught without calculus.
Students can "discover" the idea as an empirical fact, whether or not
they have a systematic explanation of where it comes from.
  Ask: What's the square of 1.1?  Of 1.01?  Of 1.001?
  Conclude: (1+eps)^2 is well approximated by 1 + 2 eps.
  Conversely, the square root is well approximated by 1 + 1/2 eps
  Extend to any power, including negative powers.
  Also do sin, cos, sinh, cosh, exp(eps) and ln(1+eps).

> Will a computer do this for them??

Maxima can do it.
   http://www.ma.utexas.edu/maxima/maxima_29.html#IDX717



> One of my favorite problems, before I retired, was the "drop a stone
> down a well and listen for the splash."  Calculate the well depth
> from the time delay.   Following the normal sig figs rule eliminates
> the answer.  Students did not know what to do next!

This problem has one slight wrinkle that needs to be clarified.
As I understand it, the game is to be played under the following 
rules:
  a) The speed of sound is to be taken into account.
  b) Air resistance is to be neglected.

These rules can be justified on artificial pedagogical grounds:
The speed of sound makes the problem complicated enough to be
interesting, whereas air resistance would make it too complicated.

These rules would be hard to justify in terms of real physics. 
 -- For a shallow well, the air resistance correction is not worth 
  worrying about, but the speed of sound correction is not worth
  worrying about either ... unless you look very closely ... but
  then again, the problem only asked for an estimate.
 -- For a deeper well, at the point where the speed of sound 
  correction gets to be big (more than a few percent), the air 
  resistance correction is already /bigger/ ... assuming a
  reasonable-sized stone.

It's OK to play the game under artificial rules, but let's not
pretend this is really a real-world problem.  If we are not 
playing by real-world rules, I strongly recommend *telling* 
students what rules are being imposed.