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.)

Online reading assignment: motion

Physics 205A, fall semester 2013
Cuesta College, San Luis Obispo, CA

Students have a weekly online reading assignment (hosted by SurveyMonkey.com), where they answer questions based on reading their textbook, material covered in previous lectures, opinion questions, and/or asking (anonymous) questions or making (anonymous) comments. Full credit is given for completing the online reading assignment before next week's lecture, regardless if whether their answers are correct/incorrect. Selected results/questions/comments are addressed by the instructor at the start of the following lecture.

The following questions were asked on the reading textbook chapters (Giambattista/Richardson/Richardson, Physics, 2/e, Chs. 2.2-2.5) and previewing a flipped class presentation on (constant acceleration) motion.

Selected/edited responses are given below.

Describe something you found interesting from the assigned textbook reading or presentation preview, and explain why this was personally interesting for you.

"I thought the chain of pain was pretty interesting, because you really do experience it everyday, and I like the 'learn it, know it, live it' quote you put along with it."

"The calculus chain of pain was interesting to me because I feel like the calculus approach to physics is a little easier to understand than the non-calculus chain of pain. That one legitimately hurt my brain."

"How the connection between this part of physics and calculus are so closely related."

Describe something you found confusing from the assigned textbook reading or presentation preview, and explain why this was personally confusing for you.

"The calculus relations, I'm sure if it was explained I would understand, but I haven't seen too much calculus."

"Knowing exactly what scenarios require what plan of attack--as in when to go for the area, slope, etc. I'm just having a difficult time picturing scenarios in my head."

"The constant acceleration equations are a bit confusing. But in time, after applying them to problems, I think they might make more sense."

Briefly describe the difference(s) between a chord slope and a tangent slope on a graph.

"A chord slope goes through two different points through a line on a graph, while a tangent slope only goes through one."

"Chord slope gives you an average over time and a tangent slope give you an instantaneous value."

The __________ gives the displacement of an object.

chord slope of an x(t) graph.	***** [5]
tangent slope of an x(t) graph.	**** [4]
chord slope of a vx(t) graph.	***** [5]
tangent slope of a vx(t) graph.	******** [8]
area under a vx(t) graph.	*************************** [27]
area under an ax(t) graph.	[0]
(None of the above choices.)	[1]
(Unsure/guessing/lost/help!)	***** [6]

The chord slope of a v_x(t) graph gives the __________ of an object.

displacement.	** [2]
position.	** [2]
change in (instantaneous) velocity.	**** [4]
(instantaneous) velocity.	**** [4]
average velocity.	************************ [24]
(instantaneous) acceleration.	** [2]
average acceleration.	************ [12]
(None of the above choices.)	[0]
(Unsure/guessing/lost/help!)	****** [6]

Ask the instructor an anonymous question, or make a comment. Selected questions/comments may be discussed in class.

"I'd like more clarification on tangent and chord slopes." (I'll make sure to bring up that point in class. Or set of two points, that is.)

"Can we do some problems--please, all the subtle differences in the equations get confusing." (Sure. But be careful of what you ask for.)

"Please go over the (calculus) equations in this section. For those of us who haven't taken calculus, or at least for me, it looks like gibberish. Thanks!" (Even after taking calculus, those equations still looks like gibberish to me.)

"Is there a better way to find the displacement of ∆x than counting the amount of boxes underneath the line on a graph?" (You could directly integrate the functional equation of the graph, or you could break up and calculate the area underneath as rectangles and/or triangles. Maybe counting boxes is not so bad, after all.)

"Do you have to answer all of the questions? Even the ones without stars next to them?" (The starred questions are mandatory; as long as you answer substantively for most of the unstarred questions, you'll get most or all of the credit for completing the assignment.)

"Is there any good way besides memorization to go about learning all these slopes and what they provide?" (Uh, yes. It's called calculus.)

Online reading assignment: motion

Physics 205A, fall semester 2012
Cuesta College, San Luis Obispo, CA

Students have a weekly online reading assignment (hosted by SurveyMonkey.com), where they answer questions based on reading their textbook, material covered in previous lectures, opinion questions, and/or asking (anonymous) questions or making (anonymous) comments. Full credit is given for completing the online reading assignment before next week's lecture, regardless if whether their answers are correct/incorrect. Selected results/questions/comments are addressed by the instructor at the start of the following lecture.

The following questions were asked on reading textbook chapters (Giambattista/Richardson/Richardson, Physics, 2/e, Chs. 2.2-2.5) and previewing a presentation on constant acceleration motion.

Selected/edited responses are given below.

Describe something you found interesting from the assigned textbook reading or presentation preview, and explain why this was personally interesting for you.

"I'm learning more about the formulas and enjoy the connections with higher level calculus. The graphs are pretty."

"Honestly, I am having a hard time finding something interesting. This is very dry, unfortunately."

"I found your 'learn it, know it, live it' motto interesting and amusing!"

Describe something you found confusing from the assigned textbook reading or presentation preview, and explain why this was personally confusing for you.

"Some of the equations I couldn't wrap my head around. I need them explained to me in a different way than the book explains them."

"Nothing was too confusing in the reading mainly because I have taken calculus and a physics class in high school so the material is very familiar."

"The formulas were a little confusing since I never took calculus."

"The chain of pain is confusing."

Briefly describe the difference(s) between a chord slope and a tangent slope on a graph.

"A chord slope passes through a curve at two points and is the average slope between them. A tangent slope runs along a point on a curve, but does not pass through it and is the slope of the curve at that point."

"Honestly, I don't remember reading about those and couldn't tell you what the difference is."

Ask the instructor an anonymous question, or make a comment. Selected questions/comments may be discussed in class.

"How much calculus must we know for this course?"

"I could really really use your help on this reading material in class. It's way too deep for me to teach myself and just have it brushed over in class."

"I'm glad you simplify all this stuff in class. I would be so lost otherwise."

"Are the surveys for Monday's lecture due at 12:00 AM on Sunday night? Because I got an error message at 11:00 PM last Sunday."

Physics quiz question: diagonally-launched projectile

Physics 205A Quiz 3, fall semester 2008
Cuesta College, San Luis Obispo, CA

Cf. Giambattista/Richardson/Richardson, Physics, 1/e, Comprehensive Problem 3.74(c)

A ball is kicked off the edge of a cliff with an initial speed of 18.0 m/s, 10.0° above the horizontal. Neglect air resistance. Choose up to be the +y direction. How long would it take for the ball to fall to the ground 22.0 m below?
(A) 1.82 s.
(B) 2.12 s.
(C) 2.24 s.
(D) 2.46 s.

Correct answer (highlight to unhide): (D)

The following quantities are given (or assumed to be known):

(x₀ = 0),
(y₀ = 0),
(t₀ = 0 s),
y = –22.0 m,
a_y = –9.80 m/s²,

where the initial horizontal and vertical velocity components of the initial velocity vector are:

v_0x = (18.0 m/s)·cos(+10°) = +17.7 m/s,
v_0y = (18.0 m/s)·sin(+10°) = +3.13 m/s.

So in the equations for projectile motion, the following quantities are unknown, or are to be explicitly solved for:

x = v_0x·t,

v_y - v_0y = a_y·t,

y = (1/2)·(v_y + v_0y)·t,

y = v_0y·t + (1/2)·a_y·(t)²,

v_y² - v_0y² = 2·a_y·y.

With the unknown quantity t to be solved for appearing in the fourth equation, with all other quantities given (or assumed to be known), then it becomes a quadratic equation:

y = v_0y·t + (1/2)·a_y·(t)²,

0 = –y + v_0y·t + (1/2)·a_y·(t)²,

where the quadratic formula terms are "c" = –(–22.0 m) = +22.0 m; "b" = +3.13 m/s, and "a" = "–4.90 m/s², resulting in the roots:

t = –1.82 s, +2.46 s,

of which the positive root (response (D)) is the sole realistic answer, given the initial conditions.

(Response (B) is the time t = sqrt(2·y/a_y) it would take for the ball to fall to the ground if it were released from rest; response (A) is the time it would take for the ball to fall to the ground if it were thrown at an angle of 10.0° below the horizontal; while response (C) is merely –(y)/a_y).

Student responses
Sections 70854, 70855
(A) : 7 students
(B) : 11 students
(C) : 13 students
(D) : 14 students

Success level: 37%
Discrimination index (Aubrecht & Aubrecht, 1983): 0.73