Mail Time: Waves on a snakey demonstration

Deidre Higgins, of Corbin Kentucky and veteran of my 2009 APSI at Morehead State University, writes:

I was trying to do some lesson planning over the break to get caught up & I was wondering if you could help me with a wave demo. I have in my notes from this summer something about showing interference using paper cups set up two "tiles" away and starting with an amplitude of one "tile," but that's about all I have. (Apparently I need to improve my note-taking skills!) I was wondering if you could give me a brief description of how to do this demo--I remember it being really good to show interference, I just don't remember the details.

Happy to help, Deidre. After I’ve shown the class all the relevant definitions for waves (i.e. amplitude, transverse wave, longitudinal wave, etc.), the class takes a field trip to the tiled floor of the hallway. I bring along a “snakey,” which is like a smaller-diameter slinky with the coils very close together. One student holds an end of the slinky on the floor, while I hold the other on the floor about 5 yards away. The setup is similar to what you see in the picture above, but with the slinky aligned parallel to and on top of a line between tiles. And my kiddies are somewhat older.

I place a line of paper cups on each side of the snakey, parallel to the snakey but about 1.5 tiles away from the snakey’s resting location. I send a wave pulse along the string. If this wave pulse has an amplitude equal to one tile, then the cups aren’t knocked over. (A two-tile-amplitude pulse knocks over all of the cups.)

But when I have the student holding the other end of the snakey send a one-tile-amplitude pulse to me AT THE SAME TIME that I send a pulse to him, the wave pulses interfere in the middle – knocking down just a couple of cups. You see, this is a manifestation of constructive interference. In the middle, the amplitudes of the wave pulses add to form, briefly, a pulse of amplitude two tiles, knocking down the cups.

There’s more you can do with the snakey – move the cups closer and demonstrate DESTRUCTIVE interference, show what a longitudinal wave looks like, show harmonics of standing waves on the snakey, and more. Play with the setup, see what else you and your students are interested in.

GCJ

What do YOU want me to post about?

Christmas/Chanukkah/Saturnalia break means I can procrastinate my grading for weeks.  I have some other writing to do (I'm working on the next edition of the Everything Kids Football Book, buy yours now!), but I certainly would like to update this blog in between gulps of Cheese Nog.

What do you want to hear about?  Do you have questions about how I teach a particular topic?  Questions about an assignment that you give, or that I give?  Have an idea you want to share with me?  Send it in!  Post in the comments section, or send an email to me at greg_jacobs@woodberry.org. 

One-off assignments, and avoiding politics

The underlying point of the article "Read Any Good Science Lately?" (in The Physics Teacher, March 2009) is that (a) assigning reading beyond the textbook can be beneficial and inspirational, and (b) such reading assignments must be carefully constructed lest students treat them with a "read because teacher told me to" attitude which serves as an obstacle to comprehension and enjoyment. Patricia Blanton is right-on here. I tend to refer to such assignments outside the traditional realm of physics problem solving as "one-off" tasks, in the spirit of Scott Adams' Dilbert.*  I'm concerned that Ms. Blanton subverts the usefulness of her idea with educational and political baggage.

In a general (non-AP) class especially, reading or research assignments can provide a welcome break from the routine of problem solving. From the students' perspective, they might be pleased to earn a grade using skills with which they might be more comfortable -- reading and writing have been part of their lives for years, while physics problem solving is often a new and intimidating skill. From the teacher's perspective, the "one-off" assignment allows us to show our students aspects of physics beyond equations and experiments.

Now, let's be brutally honest about that last sentence. As soon as we get beyond problem solving and laboratory work, most of what we do will be colored politically in some way. Ms. Blanton unveils her pet causes in her article through the assignments she suggests: women in science, environmental effects of nuclear power, and global economic inequity. Your own political issues may be less obvious, but still present: for example, my bugaboos about pseudoscience and belief-without-evidence are apparent to all as soon as I assign reading or writing on these topics. I don't care how purely academic you make such an assignment, your personal opinion is on raw display.

That's not necessarily a bad thing. But it must be handled carefully.

It doesn't matter how much your students like or respect you, they are still teenagers who question everything, and who have an innate anti-authority streak. And they don't like to be told what to think. If it's obvious from the start where the teacher stands on an issue, the typical teenager has one of two reactions: (1) My writing better support what the teacher thinks so I can get a good grade; or, (2) I'll find a way to show that teacher how wrong he is. Either way, you don't get the effect you want out of the assignment.

What's the solution? I don’t know that there is one. I’d say, keep your cards close to your chest at first, and you’re likely to avoid the two major obnoxious teenager reactions.

If you come out, guns a’blazin’ about how nuclear waste will lead to the death of all mankind, or about how nuclear power is the only possible savior of industrial society, your class is done listening. Their research is virtually guaranteed to be inauthentic.

But if, instead, you make a conscious effort to conceal your own position… well, then you might provoke some serious thought. Provide a counterargument to anyone in the class who makes a statement of opinion – and do so WITHOUT SMIRKING, even if you have to say something you consider personally outrageous or offensive. You are making it clear to your students that the only route to success on this assignment is reasoned logic.

Then, if your own position is truly supported by sound evidence, students will find and quote that evidence. You can subsequently argue with your students about the quality of evidence rather than about your (or their) pet opinions.

Now, don’t get me wrong here – I DON’T think that physics class is an appropriate place to be doing social studies research, or addressing highly charged political issues. I do think you can do successful reading and writing projects that involve physics, as long as you stick to the science and avoid the politics.  If you choose to attempt these one-off tasks for whatever reason, tread carefully on highly charged opinions, or risk sabotaging the entire point to the exercise.

* Scott Adams, author of the "Dilbert" comic strip, theorizes that a successful corporate manager minimizes the "one-off" tasks required of employees in order to keep up production and morale. A "one-off" task is anything not directly related to the enterprise at hand. For example, an engineer sketching a machine component is on-task; the engineer filling out a timesheet allocating his efforts among different departmental accounts is "one-off." For physics teachers, grading papers or setting up demonstrations is on-task; discussing minutia about dress code enforcement is one-off. Adams cautions that "one-off" tasks are often useful, or even important. He does not suggest that such tasks be eliminated altogether; rather, he suggests they be severely limited, so that most of an employee's time is on-task, and so that any one-off task receives the employee's full attention.

Jacobs Physics: Upcoming Public Appearance!

Folks, I will be running the "experienced teachers" section of the one-day AP Physics workshop on Monday, January 4, at Georgia Perimeter College in Dunwoody, GA.  (That's essentially in Atlanta.)  I'm flying into Atlanta on January 3, running the workshop from 8-3 on Jan. 4, and then flying home. 

Sign up for the workshop via the College Board website.  Or, if you're in the area on Sunday afternoon, let me know, and we'll find a place to watch the NFL playoffs and talk physics.

Other public appearances in the far future include at least three AP Summer Institutes this year:

* Morehead State University, Kentucky, July 12-16;
* North Carolina State University, July 19-23;
* Manhattan College, July 26-30

Tell your friends!  Get your tickets now, while they last!  No shoving in line, please.

First contact with E&M

In AP physics B, it's time to start electricity. The portion of the course dealing with electricity and magnetism is a non-negligible 25% of the exam.  This is also the most difficult section of the course, because it deals with such abstract concepts as "charge" and "field".  I still do as many quantitative demonstrations as I can, especially with magnetic forces and electric circuits.  But when it comes down to it, it's not straightforward to give my students an easy mental picture of, say, a +0.2 μC charge at rest in a 200 N/C electric field.

On the first day that I broach the subject of electricity, therefore, I try to make the explicit connection between electricity, magnetism, and MECHANICS.  I point out that the whole point of the E&M unit is to apply the mechanics we've learned to a new and strange regime, that of charges.  It requires a considerable leap of faith and reasoning to deal with a problem as simple as, "a proton at rest experiences a force of 10^-17 N to the right.  What is its speed after 10^-15 s?"  Because it's a proton, and because the numbers are "so small," suddenly the concepts of Fnet=ma and kinematics become impossible.

So my very first quantitative demonstration is done with the PASCO e/m device, pictured above.  This is an expensive but worthwhile item.  I put in a capital request a few years ago, and the money came through.  Purchasing one of these would be good use of grant money.  Or, if you can't find the money to buy one of your own, check with the local college's physics department -- most colleges have large numbers of these lying around for use in the freshman lab, and you may be able to borrow one for a week.

Now, this device can do many awesome things.  You can deflect electrons electrically with charged plates; or, you can use a magnetic field to bend electrons in a circle.  The relevant voltages and magnetic fields are either printed right on the device, or are read clearly off of the power supplies.  Certainly this would be a good demonstration in the magnetism section, to show that you can in fact predict the magnetic field given the radius of electrons' circular motion, or to predict the speed of electrons in a circle, or to verify the equation for electrical potential energy, qV.  But at the begninning of the unit, all these concepts are completely foreign.  I do something much, much simpler.

All I tell the class is that I have electrons that have been given a kinetic energy of 4 x 10^-17 J.  (Sure, I figured that out using qV... but I say NOTHING about that to the class.)  Then, I tell the class that these electrons experience a magnetic force that acts as a centripetal force, and that is equal to qvB.  I tell them the value of this magnetic field thing labeled as B, which equals 7.8 x 10,^-4 T. 

The question:  what is the radius of the electrons' circular motion?

We follow a Newton's Second Law approach, drawing a free body diagram, setting the net force equal to ma, where acceleration is v²/r.  We solve in variables first to get r=mv/qB. We use the definition of kinetic energy (and the looked-up mass of the electron) to calculate the speed of the electrons.  We look up the charge of the electron.  Plugging in, we find that the radius of the circular motion should be in the neighborhood of 6 or 7 cm. 

Then I turn on the device and hit the lights.  We see a dim greenish circle inside the big globe, a circle of diameter that we measure to be... between 12 and 14 cm. 

Follow-up questions that I ask right there:  1. If I increase the magnetic field thingy, what should happen to the circle of electrons?  Well, since B is in the denominator, the radius should get smaller.  (Then I increase the current to the Helmholz coils, increasing the magnetic field, and the circle gets tighter.)  2. If instead I give the electrons more kinetic energy, what should happen to the circle of electrons?  Well, increasing the kinetic energy increases the electrons' speed.  Since v is in the numerator, the circle's radius should get bigger.  (Then I increase the voltage on the electron gun, increasing the electrons' speed, and expanding the circle.)

This demonstration provides a nice "gee, wow" effect, while hammering the point:  once an electricity or magnetism problem can be phrased in terms of forces or energies, then it becomes a mechanics problem -- and we already know how to deal with those.

Department of the Obvious: Test Corrections Work

Don't know whether you saw the December 2009 edition of The Physics Teacher. This generally excellent magazine has given me untold helpful hints, lab ideas, and physics concepts to think about in the context of teaching high school (and low undergraduate) physics. In fact, I have a co-written article being published in next month's edition, about the USAYPT, the organization that made the mistake of appointing me President. (Check us out at http://www.usaypt.org/!)

This month, TPT explains how they use "Assessment Corrections" as a teaching tool. Great idea, obviously. What bugs me about this article is certainly not that I think anyone "stole" the idea of corrections from me! Of course they didn't. In fact, I "stole" the idea from Haverford professor Lyle Roelofs -- pictured above --  who inflicted test corrections on us in Advanced Quantum Physics in 1994.  He offered half credit back on the test if we corrected out mistakes.  A classmate astutely commented, "Lyle, you know we're going to do the corrections, because without them our grades are lousy, but with them the grades are good.  So even though corrections aren't required, you're insidiously getting us to do them."  Lyle just smiled.

Anyway.  What bugged me about this article was the conceit that it was determining, through the use of a scientifically valid theory, that assessment corrections are useful, and that corrections help students learn.  The article is full of phrases like "formative assessment"  and "metacognition."  AARRGH! 

Okay, okay, okay.  I know that college professors, especially those the education departments, need data to take new ideas seriously.  And that makes some sense - even I ask other teachers “so does it work?  How can you tell?” When other teachers tell me about their pet new idea, whether I think it awesome or terrible on the surface. The authors are communicating clearly and well, with their intended audience in mind.

But look, readers, I don't care what your "theoretical basis for assessment corrections" is, or whether you even have one.  Does anyone ever ask Roger Federer for the "theoretical basis" for his forehand?  Does anyone ask Albert Pujols for the "theoretical basis" of his swing?  No, these folks just do what works.  They're probably happy to share what they know about what works for them, but what works for them may or may not work for another professional. 

Physics may be a peer reviewed science, but physics teaching is far, far closer to art than science.   Good artists may do things in a similar manner, but they don't need peer-reviewed evidence to know they're doing something right.  All anyone -- INCLUDING ME -- can tell you about a physics teaching method is, "it worked for me, it worked for lots of other people, here's how I do it, now try it if you'd like."

Test corrections work for me.  Test corrections apparently work for a lot of people who write PER articles, too -- you can read the TPT article for useful examples of how other teachers have made the corrections assignment.  Corrections worked for Lyle Roelofs.  They have worked for a number of attendees at my summer institutes.  They will probably work for you.

Thermal expansion – quantitative demonstration

It’s easy to discuss thermal expansion in terms of students’ experience. They see expansion slats in sidewalks all the time. They may have noticed highway bridge expansion joints (and if they haven’t, they will probably be able to see one within 24 hours if they’re observant). Evidence for the existence of thermal expansion abounds.

However, creating measurable expansion in the classroom is a challenge. The coefficient of linear expansion for virtually all materials is on the order of 10^-5 K^-1. So even with a temperature increase of 100 K, an object will expand by about 0.1% of its length.

I can think of three experimental ways to create a reasonable thermal expansion demonstration:

1. Create an enormous temperature increase. Not practical, though, because even a 1000 K temperature change only causes a 1% length expansion.

2. Use a very, very long object, so that the 0.1% increase is big enough to see. Well, if you heat a 10 m long rod by 100 K, it will expand by about a centimeter. I don’t know about you, but I don’t have a 10 m rod handy, nor do I have room for it in my classroom, nor do I have a way of heating it relatively uniformly.  Ugh.

3. Get a measuring device that can measure itty bitty length changes.

I’ve gotten method number 3 to work in my classroom. Yesterday I used one of those metal rings with a wooden handle that come with commercial thermal expansion kits. I used a micrometer (pictured), which has can measure plus or minus 5 thousandths of a millimeter, to measure the WIDTH of the ring. You see, linear expansion happens in all directions. The width of, say, a rod will expand by the same percentage as will the length. I choose to measure the width of the ring because my micrometer can measure that easily.

The width at room temperature was 4.30 mm. I heated the ring for a minute or two in a Bunsen burner. I used an infrared non-contact thermometer – pictured to the right, available for no more than $30 – to find that the ring’s temperature rose to about 220 degrees Celsius, an increase of about 200 degrees over room temperature. Thus, an order of magnitude estimate of the ring’s expansion is that ΔL/L = (10^-5)(200) = 0.2%. Since I don’t know what my ring is made of, I can merely guess that the expansion should be in the neighborhood of a few tenths of a percent. I measured a new length of 4.33 mm, an increase of about 0.7%, which works for me.

Live PV diagrams!

It’s time to teach the ideal gas law, heat engines, and PV diagrams in AP physics. A lot of AP teachers are a bit intimidated by these topics. They’re more abstract than mechanics, and are farther divorced from our experience than, say, electricity and magnetism or waves and optics. I hope this next series of posts can help out.

Certainly your students have studied the ideal gas law in chemistry. But chances are, all they did (in their minds) was plug numbers into the equation PV=nRT. A major first step in teaching this unit is to give your class a firm understanding of the physical meaning of each of these variables.

You may or may not have seen Pasco’s heat engine / gas law apparatus. It consists of a low-friction piston that attaches to a metal cylinder – see the pasco.com picture above. I use it to demonstrate the ideal gas law and PV diagrams with live data collection.

The three variables to measure are pressure, volume, and temperature of the gas in the cylinder. I attach a Vernier pressure probe to one of the ports on the front to measure, um, pressure. Temperature can be taken care of with a Vernier temperature probe inserted into the hole in the stopper on top of the metal cylinder. It’s only volume measurement that’s truly tricky.

If you can measure the height of the piston, then the volume of the gas under the piston and in the cylinder can be calculated. Pasco provides an instruction packet that suggests the use of a rotary motion sensor or smart pulley to get the piston’s position. I don’t do that, though it should work fine.

Instead, I mount a motion detector above the piston. By measuring the height of the detector above the piston’s lowest point, I can set the Logger Pro software to calculate the volume of the gas automatically from the motion detector reading. Thus, I’m collecting volume, temperature, and pressure data as many as 20 times per second.

In the picture to the right, you can see me using this apparatus to demonstrate PV diagrams at last summer’s AP Summer Institute at the University of Georgia. (Thanks to Laura Englebert, a physics teacher from the Atlanta area, for sending me the pictures. Woo-hoo!) I told Logger Pro to graph pressure on the y-axis and volume on the x-axis. Then, I slowly raised the piston, taking care not to let my hand get in the way of the motion detector. The graph showed a nicely hyperbolic curve – an isothermal process. But then I let the piston compress the gas rapidly. When gas compresses (or expands) quickly enough that there’s not enough time for heat to flow into or out of the gas, the process is adiabatic. Adiabatic compression on a PV diagram should jump to a higher isotherm, because the temperature goes up. Sure enough, while the process happens too fast to define the adiabatic curve, you can see that the graph ends up at a higher product of PV.

If you’re a bit lost in that last paragraph, don’t worry, it will make more sense once you get a chance to study the four major types of thermodynamic process that are tested on the AP exam – isothermal, adiabatic, isobaric, and isovolumetric. The 5 Steps book (now in a new and much-edited edition!) gives a good, short, readable treatment of these processes.

But note anyway that ANY portion of the ideal gas law can be tested experimentally! The linear relationship between pressure and temperature at constant volume? Plunge the metal gas cylinder into boiling water while keeping the piston from expanding. The linear relationship between volume and temperature at constant pressure? Do the same thing, but instead allow the piston to rise. And the experiment I previously described shows the inverse relationship between pressure and volume at constant temperature! Cool, eh?

Mailbag -- thermodynamics sign convention?

From Jonathan Kirby, an Atlantan:

"I have a quick question for you. We are just getting into thermodynamics, and I was wondering if I should teach "ΔU = Q-W" where W is the work done BY the system, or if I should teach "ΔU = Q+W" where W is the work done ON the system. Which way would be better (if either) for the AP Test?"

Answer:
The AP test changed to the ΔU = Q+W route in about 2002. Don't even mention the other way, unless your textbook does, in which case, good luck. :-)

(When I've used such a textbook, I've just repeated the correct definition over and over, and prayed.)

GCJ

Follow-up to multiple choice test corrections

Those of you who have attended my workshops know that, in Jacobs Physics, test corrections are one of the two most important components of the course. Sometimes, though, even the test corrections need correction.

Instead of assigning another round of “correction corrections,” I tend to just give the whole class a quiz when I find consistent misunderstandings. For example, consider the two multiple choice questions below. These were originally AAPT Physics Bowl questions, I believe…

1. A 2 kg object initially moving with a constant velocity is subjected to a force of magnitude F in the direction of motion. A graph of F as a function of time t is shown. What is the increase, if any, in the velocity of the object during the time the force is applied?

(A) 0 m/s

(B) 2.0 m/s

(C) 3.0 m/s

(D) 4.0 m/s

(E) 6.0 m/s

2. A deliveryman moves 10 cartons from the sidewalk, along a 10-meter ramp to a loading dock, which is 1.5 meters above the sidewalk. If each carton has a mass of 25 kg, what is the total work done by the deliveryman on the cartons to move them to the loading dock?

(A) 2500 J

(B) 3750 J

(C) 10 000 J

(D) 25 000 J

(E) 37 500 J

Many students showed an iffy grasp of these two questions on their test corrections. So, I posted to our class folder early last night. I noted that we would take a follow-up quiz today on these problems. I wrote the quiz to address specifically the mistakes that I had repeatedly seen on the first attempt at corrections. Here’s the quiz:

1. (a) What’s wrong with the statement “Work is done both up and to the right in order to move the boxes up the incline?”

(b) What is the direction of the force necessary to carry one box up the incline at constant speed? Justify your answer. Your justification should include a free body diagram.

2. (a) Explain why the average force during the time interval t = 1 s to t = 5 s is NOT 1.0 N.

(b) How do you get impulse from this graph WITHOUT trying to find an average force?

Going over a test

I know it's happened to you and it's frustrated you.  You give back a test, you discuss one of the more frequently missed questions, hoping for a teachable moment.  But half the class is rooting through the rest of the test, sitting back with a vacant expression, or simply absent mentally.  What to do?

One option, which I've discussed before, is to allow corrections for half credit.  Then there's no need for you to take too much class time to go over the test -- it's the students' job to figure out what they missed, and to convince you they understand now.  A related idea is to announce a "fundamentals quiz" over commonly missed concepts from the test.  Either way, the students are forced to think about the test beyond just "what did I get?"

Of course, test corrections are time- and manpower-intensive. You have to give time in or out of class to get the corrections done, you have to grade them as thoroughly as you would a test.  I only do corrections in my AP class -- I find the general class moves slowly enough that those who missed important points will pick them up soon. 

So how do I go over a test in general physics?  Well, keeping my comments brief and to the point helps.  But the key little trick is to HOLD THE TESTS IN MY HAND while I go over them. 

Here I'm playing with the students' minds.  They desperately want their tests back, but only so they can see the grade.  Once they see that grade, their mind is done for a while, and they don't want to think about physics.  So I use the grade as a carrot.  I dangle the papers with the grades on them right in front of the class.  Not obviously or obnoxiously, of course, but they are never sure when I'm going to shut up and hand out the tests.  And, they're nervous about what they did right or wrong.

So they listen.  And ask questions.  They want to hear what I say, so they can figure out whether they were right or wrong.  The same discussion AFTER I give the tests back would be fruitless.

How do I know this technique works?  Well, I don't for sure.  But I do note that folks occasionally note to their friends whether they did or didn't make the mistakes that I discussed... so they must have paid some attention.

GCJ

Introduction to vectors in general physics

Folks are often surprised that, in AP physics, my "vectors unit" consists of one problem, in which we show that a rope at an angle is equivalent to two ropes, one vertical and one horizontal.  We use sines and cosines to calculate the vertical and horizontal components of the rope's tension.  That's it.  And that's enough -- my class developes the skills necessary to break all kinds of vectors into components, and to add vector quantities, just as well now as when I used to do a week-long unit on vector math.

In general physics, I do a bit more mathematical prep work as we move into forces at angles.  But not much.

By now, my general physics students can deal with Newton's second law, as long all forces are either horizontal or vertical.  For example, we do problems with cars slamming on the brakes, and slowing due to friction.  Next, I want them to deal with forces at angles, like the lawnmower that's pushed along its handle. 

We start the process of breaking vectors into components with displacement vectors, not force vectors.  Why?  Because students are familiar with the concept of cardinal directions and shortcuts:  going 3 miles north and 4 miles west is the same thing as going 5 miles northwest, as long as you choose the angle correctly.  Everyone gets that.  If I can show them the mathematics in the context of displacement, then they can transfer those mathematics to force vectors.

Take a look at the assignment below.  Note that Will Collier is a real student, whom I taught in 2001-02.  He really does run a transport unit in Iraq, or at least he did last year.  The rest of the exercise is merely convenient science fiction.  I've done similar exercises before using maps of New York City, Berlin, and South Africa -- pick something of interest to one of your students!

Instructions for vector assignment:

Imagine that a new type of pilotless aircraft is developed. The plane is reliable and safe, yet inefficient; due to the vagaries of its design, this plane can fly ONLY along the cardinal directions: north, south, east, or west.

Will Collier, Woodberry class of 2002, runs a military transport unit. We will imagine that he has commandeered a large number of these aircraft to deliver goods throughout Iraq from a base in Baghdad. Your job as Lt. Collier’s unit navigator is to give directions to other Iraqi cities such that these special aircraft can fly there.

So, for each of the ten destinations you choose for Lt. Collier’s supplies, draw a displacement vector to the town.

Then, for each displacement vector, on unlined paper, using half a page per town, do the following:

• sketch the displacement vector and its components on x and y axes.

• state the magnitude and direction of the displacement vector.

• calculate the x and y components of the displacement vector, showing all work carefully as instructed in class

• write out, in words: "To fly to [whatever town], fly 200 km north, then 300 km west." Of course, fill in the correct distances and directions.

Follow these instructions (which are exactly what I said in class) carefully. I expect thorough, neat work. This should not take an exceptionally long time; work quickly but carefully. You should find this assignment to be the easiest all year. If you do not, come see me ASAP!

Conservation of momentum in the English Premier League

So, I watched the Sunderland-Liverpool match in the English Premier League on Saturday morning.  (I like to write my student comments in front of sports on TV.)  Just a few minutes into the game, Sunderland scored a strange goal.  Watch -- there's a real physics purpose here...

[Edit:  Looks like the EPL removed this video from youtube.  I'm sure you can find it somewhere... it's fantastic.]

[To summarize:  A Sunderland striker executed a shot on goal from about 15 yards out.  The shot hit a red beach ball-type object and deflected at an angle; an easy save for the Liverpool goalkeeper turned into the deciding goal in the1-0 match.]

The commentators originally called the unusual red object on the pitch a "balloon."  I wrote some sports-related commentary on this event on my sports blog,  "Nachoman's Baseball."  But imagine the physics possibilities here...

The assignment, which I will use as an independent experiment in the spring in AP physics, is:  Determine the mass of the balloon / beach ball / whatever that caused the goal. 

GCJ

Clicker activity -- basics of Newton's Second Law

My wife has been gone for four days. She's hiking with the sophomores on their Outward Bound trip to North Carolina. That means, however, that I'm in charge of six-year-old Milo all by myself. I quite enjoy occasional solo time with the boy. But what to do on Saturday morning, when I had to teach a physics class some more about Newton's Second Law?

I prepared a clicker exercise. Milo loves using the clickers, and he loves being part of a class with juniors and seniors. In turn, the juniors and seniors are welcoming and friendly to Milo -- even moreso due to the "Milo Questions."

I had Milo write a set of multiple choice questions about himself. I promised to use these as part of the in-class activity. Thus, the overall set of questions for the day's clicker exercise consisted of two Newton's Second Law questions, followed by one Milo Question, followed by two more second law questions, etc. The class (including Milo!) was divided randomly into teams; the team that got Milo was excited, because they knew that they had the Milo questions in the bag. As always, each team could collaborate and submit a single answer to each question. They earned one point for a correct answer, and one bonus point for each group who did NOT get the correct answer.

The actual set of questions is below. Feel free to use them. They may sound really easy, but remember how difficult it is to remember and assimilate even the most basic facts about the second law. It takes an amazingly huge number of repetitions before we can break down the most common misconceptions like "motion requires a force" and "acceleration tells which direction something is moving."

(I gave a "fundamentals quiz" about some of these same ideas a few days later. I'll try to post that soon.)

GCJ

1. A bucket whose mass is 10 kg hangs by a rope in which there is 63 N of tension. What is the weight of the bucket?
(A) 100 N
(B) 10 N
(C) 10 kg
(D) 100 kg
(E) 63 N
(F) 63 kg
(G) 73 N
(H) 73 kg
(I) 163 N
(J) 37 N


2. A bucket whose mass is 10 kg hangs by a rope in which there is 63 N of tension. What is the net force on the bucket?
(A) 37 N
(B) 163 N
(C) 100 N
(D) 63 N
(E) The answer depends on which way the bucket is moving.


3. What color is Milo’s house?
(A) Green
(B) Blue
(C) Purple
(D) Yellow
(E) Grey
(F) Brown
(G) White


4. A bucket whose mass is 10 kg hangs by a rope in which there is 63 N of tension. What is the magnitude [i.e. the amount] of the bucket’s acceleration?
(A) 6.3 m/s2
(B) 0.63 m/s2
(C) 3.7 m/s2
(D) 0.37 m/s2
(E) 10 m/s2
(F) 1.0 m/s2

5. A bucket whose mass is 10 kg hangs by a rope in which there is 63 N of tension. What is the direction of the bucket’s acceleration?
(A) Up
(B) Down
(C) The direction of acceleration is unknown


6. What does Milo do after seated meal?
(A) Go out back
(B) Go home
(C) Come here
(D) Go to bed


7. A bucket whose mass is 10 kg hangs by a rope in which there is 63 N of tension. What is the direction of the bucket’s velocity?
(A) Up
(B) Down
(C) The direction of velocity is unknown


8. So how could it possible for the bucket to move upward, then?
(A) The bucket must be slowing down
(B) The bucket must be moving at constant speed
(C) The bucket must be speeding up
(D) The tension has to increase to more than 100 N


9. How many bunnies does Milo have?
(A) 0
(B) 7
(C) 2
(D) 1

Centripetal vs. Centrifugal Force: Golf Cart

A golf cart is moving in a straight line. I want the cart to move in a circle. Should I push or pull the cart TOWARD the center of the circle, or AWAY FROM the center of the circle?

Of course, this is the central (ha!) question of the circular motion unit. Students have preconceived notions of "centrifugal force," as well as mistaken ideas about force in the direction of motion. It's nice to begin the circular motion unit with this central question, followed by a demonstration that shows unambiguously and memorably that force toward the center of the circle is required.

Since I live on campus, about 0.5 miles from my classroom, I drive a golf cart to work. This morning I blocked off about 10 spaces in the little parking lot next to the science dungeon. I tied a sturdy rope to the corner of my cart. With the class watching, I drove the cart forward. A physically strong student pulled on the rope in a direction perpendicular to the cart's velocity. Sure enough, the cart's path arced slightly.

Next, I had THREE students tug on the rope. This time the cart's path described a "tighter" circle. We will use this qualitative observation on Monday, when we write and use the equation for centripetal acceleration.

And finally, I turned to a student who originally answered that we should pull the cart AWAY from the center of the circle. I asked him to do so, but he smiled and politely declined. Woo-hoo -- he gets it.

GCJ

A quick thought for a busy October

October at Woodberry Forest is "death month" -- two parents' weekends (one for upperclassmen, one for lowerclassmen), grades and comments due, and teaching the heart of a front-loaded physics program lead to exhaustion. I actually asked IN to duty last night so I would stay awake long enough to grade my first general physics test.

So while I don't even have time for postseason baseball in October -- who does, when the games go on for 4.5 hours -- I can at least post a quick exchange from last Saturday's general physics class.

I brought out the classroom response system, the "clickers," because it was 8:00 on a Saturday morning. I gave everyone a velocity-time graph, and asked questions about it: "During which time interval(s) is the car speeding up?" "During which time interval(s) is the car traveling south?" I divided everyone into teams of two, giving each group a clicker.

Elliott asked, as someone all-too-often does: "So, are you going to count this for a grade?"

My stock response, which this class heard for the first time: "Why? Are you going to take it more or less seriously depending on my answer?" Elliott wisely kept his mouth shut.

There's a moral to that story, but I'm too exhausted this morning to figure out what it is. Good luck surviving your own October.

GCJ

An equilibrium quiz

Here's a classic question for the end of the equilibrium unit. I say "classic" because although I got the picture from Giancoli's text, I first encountered the question in Dave Ledden's physics class when I was a senior in high school.

A bear sling, as shown above, is used in some national parks for placing backpackers’ food out of reach of the federal bears. Is it possible to pull the rope hard enough so that it doesn’t sag at all? (Obviously, justify your answer in a couple of sentences… just “yes” or “no” doesn’t cut it :-) )


A good explanation points out that with no sag, no upward force would counteract the bag's weight; thus the bag would not be in equilibrium, and the bag must fall. An even better explanation would explain that the vertical components of the rope's tension provide the upward force to counteract the bag's weight; in order for vertical components of tension to exist, the rope must pull somewhat upward, and so must sag.

An outstanding explanation shows that the vertical components are each Tsinθ, where θ is the angle of the rope measured from the horizontal. If the rope approaches purely horizontal, the angle goes to zero. The vertical equilibrium statement is 2(Tsinθ) = mg. As θ goes to zero, sin θ also goes to zero... meaning that the equilibrium statement for a horizontal rope is mg=0. Impossible!

Position-time graphs assignment and quiz

I used the Jerold Touger book for my general physics class for several years. I don't use it anymore -- the writing was way too complicated for my students to read, though I appreciated some of the conceptual treatments and problems. I still use some Touger problems for homework.

Consider the position-time graph in red, representing the motion of a fish. Touger's problem asks a number of questions about average velocity and instantaneous velocity at various times. I assign the problem... and then I give a quiz based on the problem. I drew the five lines in black on the graph, and scanned this graph into the quiz. Here is the quiz, which is based almost verbatim on Touger's posed questions:

A marine biologist shooting video of a rare species of fish uses the video data to produce the graph of the fish’s position x plotted against clock reading t in the graph above.

Velocity is the slope of a position-time graph. For each question, indicate which line on the graph above you should take the slope of in order to answer the question. Mark your choice as:
(A) Line #1
(B) Line #2
(C) Line #3
(D) Line #4
(E) Line #5

1.What is the fish’s average velocity between t = 0 and t = 8 s?

2.What is the fish’s average velocity between t = 8 s and t = 10 s?

3.What is the fish’s average velocity between t = 0 and t = 10 s?

4.What is the fish’s instantaneous velocity at t = 8 s?

5.Estimate the fish’s instantaneous velocity at t = 5 s.

Mailbag: Forces on a Table

From Michael Herrin, from Chestatee High School in Gainesville, GA:

I’m having trouble working a problem. It’s from the Cutnell and Johnson’s book chapter 4, and I am curious if you could help point me in the right direction.

Mr. Herrin actually pointed me to a problem in the 5th edition, but I found a slightly different version in the newer edition. In the diagram shown, all pulleys are massless and the surfaces are frictionless. Find the tension in the rope and the acceleration of the masses.

The problem solving process that I teach for Newton’s second law problems is:

1. Draw a free body diagram.
2. Break angled forces into components, if necessary.
3. Write (up-down=ma) and (left-right=ma).[1]

This is a two-body problem; therefore, we start with not one but TWO free body diagrams: Look here -->

Note the tricksiness of the diagram for the 3 kg mass. The rope is pulling up TWICE on this mass: once from the left side of the pulley, once from the right side. So we put two tensions on the diagram.

Jacobs’ law of tensions says “One Rope Equals One Tension.” That’s why I didn’t label the tensions T₁ and T₂, but just T. The tension will be the same throughout.

Now, write Newton’s second law twice, once for each block. The direction of acceleration will be to the right for the 10 kg block, and down for the 3 kg block. (We can see that by imagining releasing the masses from rest, and seeing which way the blocks speed up.)

T = (10 kg)a and (3 kg)g - (T+T) = (3 kg)a

Now, the PHYSICS IS DONE: I have two equations and two unknowns, a and T. Everything else is a known value. All that’s remaining is mathematics.

I think it’s easiest to solve by addition – multiply the first equation by 2, and add the equations together. This cancels out the tension terms; solve to get a = 1.3 m/s²? . Plugging back into either equation, I get T = 13 N. That’s reasonable: the tension is less than the weight of either mass, but is on the same order of magnitude.

Now, ask your students: what would happen to the acceleration if we were to give the 10 kg mass an initial shove to the left. Would the acceleration be greater than, less than, or equal to 1.3 m/s²? Post YOUR answer in the comment section.

GCJ



[1] Well, okay, sometimes we write (right-left=ma), not (left-right=ma). How to know which is which? Determine the direction of acceleration, and start with that direction.

The Milk Problem

Finally, yesterday, I got to teach. On the first day, as discussed previously, I get into real physics: equilibrium situations in AP, position-time graphs in general.

I found a new problem appropriate for the first night’s assignment. I want to give students a sense of how physics will be different from math class, but I’m not yet ready to assign conceptual questions about the current topics – we haven’t gotten far enough. I use problems that require serious reasoning, especially those which lend themselves to order-of-magnitude estimates. The following is based on a problem from the Young and Freedman text:

Milk is often sold by the gallon in plastic containers. You are to solve by calculation and reasoning, not through research.

(a) Estimate the number of gallons of milk that are purchased in the United States each year. (Obviously, your answer should include both verbal and mathematical reasoning.)
(b) What approximate weight of plastic does this represent? Compare this weight to something with which you are familiar.

What an excellent question! Even if students WANTED to try to answer through library or google research, that’s a daunting task… as I found out.

My own reasoning: with 300 million people in the USA, figure about a gallon per week for a family of four. That’s around 70 million gallons per week, times 52 weeks, or somewhere near 4 billion gallons of milk per year sold in the US.

A bunch of googling produced

this article from the US General Accounting Office, suggesting that about 7 billion gallons of milk per year were sold in 2001. Hey! I’m well within a factor of 10, which is the goal of such a problem, anyway.

As for the weight of plastic… I originally guessed about 20 g from an empty milk jug, based on my experience with my hanging weights. This gives 108 kg of plastic for the billion gallons sold each year, or about 100,000 tons.

My chemistry colleague, the Atlanta Cracker, Mr. Paul Vickers weighed an empty half-gallon milk carton, getting 47 g. Woodberry Forest Librarian Phoebe Warmack turned up a claim that nowadays gallon milk jugs are less than 60 g. So my estimate was definitely good enough, because it gives the same ~100,000 tons of plastic as does a weight of 50 g or so.

And this is the whole point of the exercise. Not only do I want my students to gain their first exposure to “Fermi problems” and order-of-magnitude estimation, I also want to make a preemptive strike against the arguments I inevitably hear in class: “You said the answer was 5.9 N, but I got 5.8 N. What did I do wrong?” Or, as always, “isn’t g 9.8, not 10?” Hopefully they will see that a 2% difference is meaningless when making everyday measurements.

Mailbag: Should you guess on multiple choice problems?

From Joey Konieczny, from Georgia:

Another question I had was about how you grade your multiple choice questions. The AP exam takes off 1/4 points for every wrong answer, correct? Do you do the same? Do you encourage your students to not answer your exam questions if they are not confident in their answer?

Yes, the AP multiple choice exam is graded exactly like the SAT: with 5 choices per item, students get 1 raw point per correct answer, and lose ¼ raw point per incorrect answer as a correction for guessing. And yes, I do the same on my in-class tests, because they are given as authentic AP practice exams.

However, I encourage my students to answer EVERY SINGLE multiple choice item, regardless of how confident they may be. Why? Look at the mathematics, to begin with.

Imagine a 100 question multiple choice test answered completely randomly, with absolutely no hint as to the correct answer. A person who leaves everything blank gets zero raw score. The person who answers every question will, on average, get 20 right for 20 raw points; but this person will get 80 wrong, for -20 raw points, leading to an overall raw score of: zero. Exactly the same as if the test were blank.

My point here is that, contrary to the myth that some idiot test-prep corporations propagate, there’s no harm in random guessing. On average, random guessing scores exactly the same as leaving blank. Of course, there’s really no point to truly random guessing: if someone runs out of time with ten questions left, I do not advise randomly filling in bubbles.

However, this analysis shows that there is, in fact, benefit to guessing that is in any way not random. Now, some will give the advice that if you can’t surely eliminate a couple of answers, then you shouldn’t guess. I disagree. My students – and, since you’re reading this blog, yours too – develop pretty good instincts about physics problems by the end of our year-long course. Even if they can’t bet their life on the wrongness of an answer choice, my students are certainly more than 20% likely to pick out the correct answer on virtually every multiple choice problem I’ve ever assigned. So they should always guess! In the best-case scenario, these non-random guesses add a few raw points to the overall raw score. And in the worst case – THIS IS THE CRUX OF MY ARGUMENT – these random guesses do no harm.

My rule is, if you read a question, mark an answer.

Now, it’s actually a rather tough process to convince students to guess, because they’ve heard for so long about guessing penalties. The most convincing argument to encourage students to guess on multiple choice questions came from my former student, Bret Holbrook, who truly understood the methods to my madness. He reminded his classmates that I make students do a correction for every multiple choice question that isn’t right, whether or not an incorrect answer was marked. So, he reasoned, he might as well guess. That gave him at least a 20% chance of avoiding the hard work inherent in the correction that he’d have to do anyway if he left the question blank.

You don't want homework to look like this --> .

I only assign about two problems per night, while math classes usually assign 10-30 problems per night. Yet, my physics assignment should take about the same amount of time to finish as a night of math homework. Not only are the physics questions more involved than their math counterparts, physics problems also require enormously more communication.

Without guidance from me, my students’ problem sets will look a lot like the picture to the right. Even with repeated written guidance, oral guidance, examples, threats, and groveling from me, it takes a long time to establish my expectations for problem presentation.

One trick that has helped tremendously is to require all homework to be done on UNLINED paper, with merely one or two problems per page. (One problem per page in AP physics; no more than two per page in general.) Notebook paper seems to constrain a response: all fractions are written on two lines, diagrams are rarely drawn; and if they are, they often fit nicely alongside the lines to the detriment of the communicative power of the diagram. Students schooled in resource conservation and basic economics put as many problems on a single page as they can fit.

A well presented problem usually includes three elements: diagrams, words, and mathematics. In fact, when I’m pressed for time, I will often grade problems cursorily by merely looking for these three elements and the answer. The blank canvas of the unlined paper, along with the mandate to fill up that canvas, seems to free up my students so they are more likely to include all three of the parts of carefully done homework.

It would seem that the cost associated with the benefit of homework on unlined paper would be the necessity for students to purchase expensive paper, along with the associated environmental cost of recycling or trashing the mounds of used homework assignments. I get around this issue by issuing biannual pleas to the school for usable unlined paper that would otherwise be dumped. Sure enough, my class’s needs have been more than met for a decade.

The library has given me their recycle pile, made up mainly of abandoned print jobs that were printed only on one side. (It’s obviously fine for homework to appear on the back of an old English essay, though students are initially hesitant not to present work on a perfectly clean sheet. What do I care about the flip side?) I make a point of keeping the recycling pile in the science department copy room neat, so I can just take it to my room for student use.

But perhaps the most awesome donations I’ve received come from the admissions and development offices. Whenever someone leaves, or whenever a telephone number changes, they must print new stationery. Their attics were full of stacks out-of-date sheets saying the equivalent of “From the desk of President Herbert Hoover.” The folks in these offices now make a point of sending old boxes of stationery to my room, where students love to use the high quality paper.

No one tries to fit lots of work into the corner of a page anymore. On the rare occasion at the beginning of the year when someone does so out of habit, I just remind him what he has to pay for really nice unlined paper.

GCJ

Mailbag: Dead Rats

I’m back from Yosemite, where I probably COULD have posted to this blog if I had wanted to – I saw not one but numerous people talking on cell phones while they hiked the trails. Nevertheless, I went somewhat old school, and stayed out of touch. I’ll be back to Woodberry on Tuesday, and frequent posts should begin again as the start of school approaches for me.

For now, chew on this question, asked by a participant in one of my AP Summer Institutes:

“I was re-reading some of the materials you shared, particularly your homework submission guidelines. There is one thing I don't get: What is a dead rat?”

-- Samuel Holiday, Pinson, Alabama

I learned the term a decade ago from Arkansas professor and AP reader Gay Stewart, though it wasn’t her invention, I don’t think.

Say you find a dead rat in a pickle barrel you're selling. Well, if you remove the dead rat before the customer sees it, you can still sell the pickle barrel, though possibly at a discount. If you don’t remove the dead rat, you're in big trouble.

In physics, a "dead rat" is a bloody ridiculous answer: a commercial airplane that has a mass of 10^2 kg... a car moving 10,000 m/s... not just an incorrect answer, but one that could and should be ruled out based on any kind of physical sense.

An unidentified dead rat causes a student big trouble to the tune of an enormous loss of points, regardless of the reason for the error. But if that student points out the dead rat and how he knows an answer is a dead rat, then he'll lose very little credit.

GCJ

AP physics – the first assignment

The last post discussed what goes on in my class on the first day of school. But what about the first night’s assignment?

I’ve not covered enough material by the end of day 1 that would justify assigning equilibrium problems. And, assigning math review is worse than useless – it’s harmful. I need problems for the first night that are doable without any in-class content, but that have useful pedagogical aims. (If I can’t justify homework as truly advancing students’ understanding, then I can’t assign it. There will be no busywork in my class.)

Consider what makes physics homework different from the homework that high school students are used to. A properly presented homework problem includes words, equations, diagrams, and a brief “comparison” in which the student shows an understanding of the physical meaning of a numerical answer. On the other hand, students habitually “solve” math problems by cramming a few numbers into the corner of a piece of notebook paper. The first day’s assignment should show clearly the level of thought, effort, and communication that will be regularly required.

So… here are two problems that I have assigned on the first night. The first is adapted nearly verbatim from a problem in Giancoli 5th edition.


1. An average family of four uses roughly 1200 liters – about 300 gallons – of water per day. How much depth would a lake lose per year if it uniformly covered an area of 50 square kilometers and supplied a local town with a population of 40,000 people? (Your comparison should discuss the size and/or depth of the lake compared to bodies of water you may be familiar with.)

2. Which is faster – your hair’s growth rate, or the speed of continental drift? (Your answer – in words with mathematical justification – is your comparison.)


Then, on day 2, here are two questions from the multiple choice quiz I give at the beginning of class. Note that the choices to the first question are local references… I encourage you to use this quiz with descriptions relevant to your class.

• How big is a 50 km2 lake?
(A) It would cover Woodberry and Orange, together
(B) About half the size of Woodberry’s campus
(C) About as big as the lakes on the golf course
(D) Would cover Woodberry and Charlottesville, together
(E) About one-tenth the size of one of the Great Lakes

• Which is faster, your hair’s growth rate, or the speed of continental drift?
(A) continental drift
(B) hair growth
(C) they essentially are the same


While everyone is working on these problems, I collect the homework. It is easy to flip through the papers to find all of the crazy answers to problem 1. I’ve had people tell me the lake’s depth dropped by as little as .8 mm, and as much as 87 million kilometers. As soon as I've gone over the quiz, I pick one egregiously silly depth and discuss why that can’t possibly be a correct answer. Thus, even before I graded anything, the homework problems have served a purpose: they’ve provoked a discussion of physical reasonability, and shown better than the best-written essay the difference between physics and math.

GCJ

First day of school -- DO PHYSICS

Think about the experience of high school students on the first day of school. They will likely attend four to six academic classes, each for somewhere between 40 and 90 minutes. What will happen in those classes?

Most teachers will take care of administrative minutia. Pass out and read the syllabus, hand out and sign for textbooks, go over rules of the class, how grades are assigned, and blah b-b-blah blah blah. Perhaps a few perceptive teachers might undertake a short discussion about the class’s overall goals, like “What was the most important event in American History?” But I would hazard that in most classes, the actual content covered on the first day is minimal, passive, and non-essential.

Physics can be different.

I teach juniors and seniors only, in general and in AP physics. Presumably the 16-18 year olds in my classes can read; so I send them the syllabus ahead of time via email, and make them read it. Presumably my upperclassmen have learned how to behave in a high school class; so I consider it unnecessary and condescending to discuss a list of class rules such as “respect one another” or “no chewing gum.” (How would YOU feel if you attended a conference which started with a litany of restrictive, prescriptive rules behind which is the underlying assumption that you will do all of these naughty things but for the recitation of said rules?)

Within fifteen minutes of my students’ arrival on the first day of AP, I dive into physics. We define a force as a push or a pull, measured on a scale; I write the definition of an object in equilibrium, and show how to solve equilibrium problems. By the end of the first day, the class is ready for the following quiz (which leads off day 2):

The box pictured above moves at constant speed to the left. Which of the following is correct?

(A) The situation is impossible. Since more forces act right, the block must move to the right.
(B) T3 > T1 + T2
(C) T3 < T2 + T2
(D) T3 = T1 + T2
(E) A relationship between the forces cannot be determined.

And then on day 2, I show with a quantitative demonstration how to deal with a force that acts at an angle. We’re off and running, such that the problems on the SECOND NIGHT OF CLASS are at the AP-level.

The same principle applies to general physics – on the very first day we are making position-time graphs with the motion detector, such that the second night’s problems can involve serious graphical kinematics.

And since most other teachers are talking about the penalties for late work while I’m holding an active class complete with demonstrations, I instantly capture attention. I do think that, in general, physics is more entertaining than most other subjects. But if nothing else, on day one I’ve made students FEEL like my class is special.

Mailbag: Writing in physics class

The Nachograndpa, a.k.a. mathematics professor Dr. Barton Jacobs, writes:

"I now have a question that came up with your Mother regarding her English 101 class, where they're supposed to learn how to write for a variety of categories of writing. I said that beyond Freshman Calculus, all math courses require putting together an argument that entails as much in the way of English words as it does symbols. I had vaguely remembered that as early as Freshman Physics this is also true. Is that right?"

Yes, but not in the same ritualized manner as, say, a geometric proof. We do enormous amounts of writing in my class, and I don't mean just lab reports. A typical question on an assignment might be,

"A ball is dropped from a table. The height of the table is now doubled. What happens to the time it takes for the ball to hit the ground?

___ the time increases, but does not double

___ the time doubles

___ the time more than doubles

Now justify your answer."

The "Justify your answer" questions generally require verbal discussions with reference to formulae. I have the dangdest time getting my students not to BS, but to use no more than two sentences with reference to the relevant equation.

The correct answer is that the time increases, but does not double. The justification: "The kinematics equation can be solved for time to get t=root (2x/g). Since we doubled the quantity under the square root, the time to fall increases by a factor of root 2 rather than 2." That's a fully sufficient explanation! But most folks want to go on for paragraphs... why?

Usually because they're not quite sure, and their experience in other classes is that if you write enough with sophisticated-sounding vocabulary and phrasology, the instructor will essentially throw up her hands in despair and award credit. Problem is, on the AP exam at least, incorrect statements interspersed with correct statements will always lose credit; correct but irrelevant statements cannot earn credit. The logic has to flow properly; getting it to do so is as much a writing skill as a physics skill.