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First let me comment about consistency in symbols. If you are going
to use V0 as the initial velocity, then it seems to me you should be
using t0 as the initial time and in your first equation you should be
using t-t0 instead of just t. Similarly since you don't used vf for
v final, why would you tf for t final. I think there is something to
be said for consistency especially when you are working to make the
connection between concrete events and conceptual entities via the
use of symbols.
Regarding the pedagogy...a very important issue...I don't know of any
research that would point to one formalism over another...perhaps
others do. Having none, I would focus on what you want the students
to understand and use that as a guide.
So if you want to make the point that velocity values are dependent
on the frame of reference, you might want to emphasize changes in
velocity and write the equation as
v - v0 + a(t-t0). Or you might want to start with a simpler form and
evolve into a more complete form as the richer physical context might
demand.
There is always a problem with differing from the text, while the
students might not read it in depth unless you force them in some
way, they will of course take the equations as bible and simply as
the only way to calculate a number. In my experience they don't see
equations as sentences that relate concepts.
So here is an idea to get students to think about equations as more
than tools to calculate a number. Ask them the same question you
asked the list, early on, when you hit the first kinematic equation.
Perhaps the discussion will begin to change their perspective on what
an equation is. Just an idea that your note suggested.
joe
Joseph J. Bellina, Jr. Ph.D.
Professor of Physics
Saint Mary's College
Notre Dame, IN 46556
On Aug 9, 2007, at 11:18 PM, Folkerts, Timothy J wrote:
As, I prepare for this coming year, I am trying to decide what
symbols to use for topics from kinematics (which will naturally
extend through much of the semester).
For simplicity, let's look at the equation relating velocity,
acceleration and time.
This equation is commonly written as
v = v_0 + at
but there are many possible variations. For example, instead of "t"
sometimes you see "delta(t)" or "t_f - t_i" or "t_f - t_0".
Instead of "v" you might see "v(t)" or "v_f". Or you might see
velocities combined as "v_f - v_i" or "delta(v)" on the left side.
ARE THERE PEDAGOGICAL REASONS TO CHOOSE ONE OVER THE OTHERS?
For example:
* v(t) emphasize that the equation will work at various times
* Delta(v) and delta(t) emphasize that changes in v and t are
important, not the actual values.
* "t_f - t_i" emphasizes that you need the beginning and ending
time. (For example, in problems with different acceleration at
different parts of the problem, I find students simply plugging in
the final time, rather than the amount of time elapsed at that
particular acceleration).
I find the books are not consistent. For example, they seem to
start with "t_f - t_i" and shift to "t" notation with very little
discussion. Pedagogically, it would be nice to use the same
notation as the book, but not if 1) the book is not consistent
itself or 2) the book chose a very poor option.
(Of course, the same applies to the other equations - e.g. x = x_0
+ v_0 t + 1/2 at^2)
Tim F
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Forum for Physics Educators
Phys-l@carnot.physics.buffalo.edu
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_______________________________________________
Forum for Physics Educators
Phys-l@carnot.physics.buffalo.edu
https://carnot.physics.buffalo.edu/mailman/listinfo/phys-l