Physics presentation: motion

How many of you use a GPS navigation device while driving? Or a navigation application on a smartphone? (Movie link: "Night Driving.")

Not too sure what the point of this setup is (other than sheer overkill), but yes, it is all kinds of awesome. (Movie link: "crossroads (what to do).")

While it's perfectly fine to let a dedicated navigation device or smartphone application tell us how to drive, let's see how (one-dimensional) motion can be described in physics.

We'll provide a brief overview of how graphs are used to describe motion...

...as well as equations. Here the emphasis is not on a comprehensive coverage of these conventions, but on the general, broad connections.

We'll use that scary word--calculus--but just to motivate the connections between different types of motion graphs.

We can connect these three key quantities--position, velocity, and acceleration--in a "calculus chain of pain," where we move to the left by differentiation, or move to the right by integrating. While we don't need to explicitly evaluate these operations in an algebra-based college physics course...

...these operations are embedded in our non-calculus "chain of pain," where we move to the left by finding slopes, or move to the right by calculating areas. Note that there are two types of slopes (tangent versus chord), corresponding to the two different types of velocities and accelerations (instantaneous versus average). I'm not a big fan of rote memorization, but if you're forced to memorize something for this class, memorize this chart. "Learn it, know it, live it."

Let's try to utilize the "chain of pain" in answering these types of questions:

The __________ gives the displacement of an object.
(A) chord slope of an x(t) graph.
(B) tangent slope of an x(t) graph.
(C) chord slope of a v_x(t) graph.
(D) tangent slope of a v_x(t) graph.
(E) area under a v_x(t) graph.
(F) area under an a_x(t) graph.
(G) (None of the above choices.)
(H) (Unsure/guessing/lost/help!)

The chord slope of a v_x(t) graph gives the __________ of an object.
(A) displacement.
(B) position.
(C) change in (instantaneous) velocity.
(D) (instantaneous) velocity.
(E) average velocity.
(F) (instantaneous) acceleration.
(G) average acceleration.
(H) (None of the above choices.)
(I) (Unsure/guessing/lost/help!)

Now let's consider the equations used to describe motion.

For the purposes of this course, let's limit our discussion to cases where the acceleration of motion is a constant value (and the starting position is x₀ = 0 m at t₀ = 0 s), and these equations are provided to you as is (although they follow from the application of calculus). You don't need to memorize these equations, as they'll be provided to you on the exam equation sheets, but you should know when/how to use these equations. (The "chain of pain" chart is not provided on the equation sheets.)

And one more equation--the quadratic formula, again provided on your exam equation sheets.

This "list of five" motion equations seems overwhelming, and the temptation is to use them willy-nilly, or by picking one arbitrarily without knowing whether/why it would happen to work. Which leads us to Will Ferrell in Elf (New Line Cinema, 2003), picking gum off of a railing. Sure, it may taste good, and it costs you nothing, but in physics as with found street gum, an equation you pick up without knowing or understanding where it came from is usually not going to turn out well. So later let's look at the most important part of solving physics problems--reading through and picking out the known/given/inferred quantities, identifying the remaining unknown quantities, and then this will help you determine just equation(s) you should be using for a particular situation.

(Hat tip to Rhett Allain, "Don’t Eat Candy You Find on the Ground," Wired Dot Physics, June 24, 2011 for the Elf reference.)

Physics clicker question: instantaneous speed vs. magnitude of instantaneous velocity

Physics 205A, Fall Semester 2008
Cuesta College, San Luis Obispo, CA

Students were asked the following clicker question (Classroom Performance System, einstruction.com) at the middle of their learning cycle, after discussing the distinction between average and instantaneous quantities.

In general, which statement(s) is/are possible?
(A) (Instantaneous) speed > magnitude of (instantaneous) velocity.
(B) (Instantaneous) speed < magnitude of (instantaneous) velocity.
(C) (Instantaneous) speed = magnitude of (instantaneous) velocity.
(D) (More than one of the above choices.)
(E) (I'm lost, and don't know how to answer this.)

Sections 70854, 70855
(A) : 1 student
(B) : 1 student
(C) : 16 students
(D) : 27 students
(E) : 0 students

This question was asked again after displaying the tallied results with the lack of consensus, with the following results. No comments were made by the instructor, in order to see if students were going to be able to discuss and determine the correct answer among themselves.

Sections 70854, 70855
(A) : 0 students
(B) : 0 student
(C) : 25 students
(D) : 20 students
(E) : 0 students

Correct answer: (C)

While not an overwhelming number of correct responses, at least an appreciable shift away from (D), which is an incorrect answer, a point brought up in the whole-class discussion after the second pass.

Instantaneous quantities are evaluated in the limit that the elapsed time interval approaches zero. From a previous discussion in class, average speeds can be greater than the magnitude of average velocities when the trajectory between the initial and final positions doubles back and forth on itself, but instantaneous speeds must be the same as the magnitudes of instantaneous velocities, because in the limit that the elapsed time interval approaches zero, there can be only straight-line travel with no "back-and-forth" between initial and final positions.

In the follow-up discussion, a student asked whether instantaneous speeds can still be greater than magnitudes of instantaneous velocities for something very small, say, like molecules. But even though the brownian motion of molecules would jostle it back and forth over small distances, the nature of the time interval for instantaneous quantities means that it should be made sufficiently small enough that there is no "back-and-forth" between initial and final positions.