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Re: [Phys-l] Student Misconceptions



On 10/03/2011 05:06 PM, Anthony Lapinski wrote:

It is a conceptual question. I also do similar ones with numbers they
can't do in their head, and then it becomes more of a "math" problem.

Like... You must average 40 mph on a trip. When you've gone half the
distance, you've averaged only 30 mph. What must be your average for the
remainder of the trip? Answer is not 50 mph...

These tend to be the most difficult motion problems for my students, even
my "brightest" ones.

This problem is not as hard as it looks, if you think about
it the right way.

In fact you *can* do it in your head, in less time than it
takes to tell about it, if you remember what I said the other
day: In this situation it is the average /inverse/ velocity
that matters ... averaged over distance.

Here's an almost-equivalent formulation: Look at this as a
time-management problem: Suppose the trip covers 120 miles.
The time budget for the whole trip is 120/40 = 3 hours. The
first half takes 60/30 = 2 hours. How much time do you have
left? How much distance do you have left?

In case you are wondering why I picked a distance of 120 miles,
that's a convenient common multiple of the speeds that were
mentioned in the statement of the problem. I can handle round
numbers of hours in my head more easily than fractions.

Now that we know the answer, we can check it even more quickly
than we derived it:
Is the average of 1/30th and 1/60th equal to 1/40th?
Is the average of 1/3rd and 1/6th equal to 1/4th?
Is the sum of 1/3rd and 1/6th equal to a half? (*)
Is the sum of 2/3rds and 1/3rd equal to 1?
I reckon this checks out.

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

Has everybody noticed that from line (*) on down, this is the
same as the resistors-in-parallel law?

That is to say, the original speed problem involves averaging
the inverses, whereas resistors in parallel involves summing
the inverses.

After you've done a few thousand resistor-in-parallel problems,
you get pretty good at doing them in your head.

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

So we get back to the point that Robert Cohen made: Knowing
whether to average over time or average over distance is in
itself an important conceptual point.

To say the same thing in even simpler terms: Given the
equation
distance = rate * time [2]
if the students see it as only a way to calculate distance
and not as a way to calculate time, then they don't really
/understand/ this equation ... or equations in general.

To say the same thing in reverse, back on Fri, 13 Mar 1998 08:49:36
Rick Tarara reminded us:
(Of course, we all know that the three most important
equations in electricity/electronics are I=V/R, V=RI, and R=V/I ;-)

and all kidding aside, there *are* some people who function
at a sufficiently low level that they do indeed see those as
three different equations.

OTOH that low level is not what we are aiming for.

This is where math and physics part company from computer
programming. If you have written a subroutine that calculates
distance in accordance with equation [2] and you later want
to calculate the time, you have to write a new subroutine.
Computers are fast, but they are not very smart. In contrast,
humans should be able to re-arrange equation [2] to solve for
time (or rate) in less time than it takes to tell about it,
so there is no point in remembering all three forms of the d
istance equation, or the voltage equation, or the ideal-gas
equation, et cetera.

Remembering only one member of each family of equations reduces
the memorization workload by a factor of 3. The overhead of
learning the re-arrangement rule is not a problem, because it
gets amortized over gazillions of applications.