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Re: [Phys-l] "Looking up" results on a graph



On 01/12/2012 01:48 PM, Folkerts, Timothy J wrote:
Students (especially at high school or lower college levels) often have
a hard time deciding which variable to plot along the bottom of a graph.
A very simple memory aid suddenly occurred to me today -- "looking up"
the answer on the graph.

If I know the volume of water and want to find the mass, I would plot
the volume along the bottom and literally "look up" the mass by tracing
upward from the volume to the line representing the mass.
If I do some experiments to determine speed as a function of distance
fallen, then I would plot the distance fallen along the bottom so I can
"look up" the result later.

That probably makes the students happy in the short run, but
I'm not sure it's healthy for them in the long run. Students
like simple, definite guidelines, but in this case no such
thing is possible. The simple guidelines are not good, and
the good guidelines are not simple.

There is a fundamental reason why it is hard for students
(or anybody else) to decide which variable to plot in which
direction. The reason is: In most cases, it doesn't matter!

For example: A plot of pressure versus volume conveys exactly
the same information as a plot of volume versus pressure. Using
a standard indicator diagram, given the volume you should be able
to find the pressure, and given the pressure you should be able
to find the volume ... *without* needing to re-plot the data.

Here's another example, a very practical example, that illustrates
the point even more emphatically. Consider the _power curve_ that
a pilot needs to understand in order to fly the airplane with any
reasonable proficiency. The concept is introduced here:
http://www.av8n.com/how/htm/energy.html#sec-power-curve-intro
and discussed further here:
http://www.av8n.com/how/htm/aoa.html#sec-power-curve-more

Note that the axes on such a plot are airspeed and rate of climb
... but the power curve is not a *function* of airspeed or rate
of climb. If it were a function, then (by definition) there would
be only one ordinate for each abscissa. The physics here is that
both the airspeed and the rate of climb are functions of the angle
of attack. We have a parametric representation, with angle of
attack being the key parameter, as you see here:
http://www.av8n.com/how/htm/aoa.html#fig-power-curve-alpha

In the long run, we want students to be able to understand such
a plot. We want them to be able to interpret such a plot if
they see it, and indeed construct such a plot if need be.

My worry is that any rules about looking "up" will have to be
unlearned, sooner or later ... probably sooner.

For more about the evil of axes, including several more examples,
see
http://www.av8n.com/physics/axes.htm

My suggestion: In the introductory class, allow them to plot
x horizontally and t vertically, or vice versa. I don't care
either way.