PH 211
Fall Term, 2013
Oregon State University
Before Class 3
(A preparation for the class—nothing here is for credit, but it’s strongly recommended)
Here’s a re-cap of what ODAVEST stands for (and again, see the link on the second page of HW1-2 for a complete example,
a file called SampleHW.pdf):
Re-state the Objective in your own words.
List all the
Data and other known values you’re explicitly given—named usefully and conveniently.
List all
Assumptions you’re also going to make—unstated conditions which you declare for needed simplification.
Write the
Equation(s) you need to get to the objective (and be sure to use the given Data variables here).
Draw a
Visual representation of the physics of the situation (not just a picture—something from which you can
write a mathematical model or equation—a visual such as a graph or diagram).
Now:
Write a description of how to Solve the equation(s) for the objective; don’t actually do the math (again, see
the online example).
Finally:
Test your understanding of the dependency of the solution on each of the given data:
For each given data value (one at a time), if you were to increase it, predict how (and why) this would
affect the outcome--the value of the objective.
1.
Try applying the ODAVEST checklist to your quest for the lottery ticket. Use your own words—and write down
everything. Let this be a good first run-through of the process! Yes, this technique takes more time and space (paper)
than you might expect, but in the real world, nobody solves and also justifies anything in a couple of lines.
It takes care, time and effort to do it right—and that’s what gets results.
Now, a little more discussion of vectors in general (which, as you’ve probably surmised by now, figures heavily in your
above quest for the lottery ticket...).
How do vectors help us understand motion? It’s all about the DIFFERENCE (D) vector
As we’ve seen, to use math to fully describe the position of an object, we must use a vector, because it tells us both the
magnitude (distance) and direction of that position (measuring from agreed-upon zero points for both distance and angle).
Without both pieces of information, we have just a number—a scalar—not sufficient to fully describe an object’s position.
Scalar
Vector
vs.
(magnitude only)
(magnitude and direction)
Distance (d)
Position (x)
But motion is about change (the change of position, first and foremost).
Question: If an object starts at at an initial position (call it positioni), then it moves to a final position (for positionf), how do
we measure its change of position, Dposition?
Well, start with something simpler—a non-vector situation. How would you measure the change in an simple (scalar) quantity? Suppose you start the day with a certain amount of money in your pocket, say, $23.00 ( call this cashi). Then at the end
of the day, you have a different amount, say, $39.00 (call this cashf ). What was the change in your pocket money (Dcash)—
and how do you calculate it?
You simply subtract: $39.00 – $23.00 = $16.00
In other words: Dcash = cashf – cashi
Rearrange this (still saying the same thing): $23.00 + $16.00 = $39.00
In other words: cashi + Dcash = cashf
What you start with, plus whatever change that happened, must result in what you end up with. Common sense, right?
It’s the same for vectors (such as position): positioni + Dposition = positionf
But we must keep in mind what it means to do vector addition: This is how we represent positioni + Dposition = positionf :
positioni
Dposition
positionf
This is the key to understanding all motion in Newtonian physics—that change (D) vector.
In the special case where the motion is all along one line (an x-axis, for example), the diagram simplifies.
No need for any trig to figure things out—just add the magnitudes, using a simple ± sign to indicate direction.
xi
Dx
xf
Above, all three vectors would have + signs associated with them, since they’re all pointing in the +x direction. Not true
below:
xi
xf
Dx
Important note: This is the only circumstance when we can do this simple addition of the vector magnitudes, together with
their signs—when all three of the vectors (initial, final and D) are collinear along the axis! That means you can do this only
when combining x-axis vectors; or y-axis vectors.
But that’s the key to adding vectors in general. You can simply sum (using the above easy technique) the x-vectors (to get a
resultant x-vector) and y-vectors (to get a resultant y-vector); then you build the final result from those two resultant vector components (see again part E of the Math Review if this technique is still hazy for you).
This is true for all vector summing—including the “sum that shows the change:” vectori + Dvector = vectorf
Representing vector quantities as free-floating arrows is one very good form of visual representation (the “V” in ODAVEST),
but there are others that are also very useful for representing motion: Motion Diagrams and Time Graphs.
2.
Read in the textbook sections 1.4 - 1.8.
3.
Try Conceptual Questions 5 and 7 (page 29).
Try Exercises 3, 19 and 21 (pages 29 and 30).
Then try problem 43 (page 31). See if you can draw both a motion diagram and a set of x-t, v-t and a-t graphs.