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Re: [Phys-l] quadratic uncertainty



I would say that there is NO good answer for either "x" or for it's uncertainty.

The function is a parabola. The "best" parabola (i.e. a = 1, b = -2.08, c = 1.08) has a minimum of -0.0016 at x = 1.04. It has zeros at 1.00 and 1.08.

While changing "a" has very little effect, changing "b" and "c" have dramatic effects.
* With (a,b,c) = (1.000, -2.09, 1.07), the zeros move out as far as ~0.9 and ~1.19.
* It is easy to move both zeros to 1.04
* However, with many values of (a,b,c) there are no solutions.

FOR THE CASES WHERE THERE IS A SOLUTION, then x is apparently 1.08 (+0.1/-0.04) and 1.00 (+0.04/-0.1).
WHEN ALL CASES ARE CONSIDERED, there is often no solution art all (at least no real solution). Since x ranges from 1.00 out to "undefined" or from 1.08 to "undefined" you can't write an answer in the form x +/- "something".

The analogy would be " You are tossing the ball up trying to *just barely* hit the ceiling. When does it hit? (where "x" is the time, "a" is acceleration/2 ...)." *IF* it hits, it hits right around 1 second after you throw it up. But often you fail and it doesn't hit at all. You can't give a "typical" value when the results are (1.02, 0.93, 0.99, null, null, .... ).


Tim



-----Original Message-----
From: phys-l-bounces@carnot.physics.buffalo.edu [mailto:phys-l-bounces@carnot.physics.buffalo.edu] On Behalf Of John Denker
Sent: Wednesday, August 25, 2010 10:09 PM
To: Forum for Physics Educators
Subject: [Phys-l] quadratic uncertainty

Here's a little puzzle with some seasonal relevance:

We need to find a good value for x
/and for the uncertainty associated with x/
given that:
a x^2 + b x + c = 0 [1]
a = 1 ± .0001
b = -2.08 ± .01
c = 1.08 ± .01

This was mentioned in connection with the annual "sig figs"
donnybrook on the chemistry list. There are a thousand people
on that list, and so far nobody has come up with a solution.
One person came kinda close, but no cigar.

I think it's safe to say that the problem is more interesting
than it might at first appear. The interest-to-difficulty
ratio is pretty good IMHO. After all, it's just a quadratic,
so there's a limit to how hard it can be.

The point of the exercise is to propagate the uncertainty
from the inputs (a,b,c) to the result (x). A lot of people
talk the talk about propagation of uncertainty but have not
much experience actually walking the walk, especially when
it comes to calculations that require more than two or three
steps.

Any method of solution you can think of is fair game.

So ... any takers?

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