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16 January 2017

Mail Time: Rigorous definitions of circuit properties in AP Physics 1

Buckeye native Matthew writes in with a question about circuits in AP Physics 1.  He’s referring to my summary post of the topics on the exam

Greg - Happy New Year!. As I am outlining the second semester of the year I am having difficulty finding information about

Non-rigorous definitions of voltage, current, resistance
Rigorous definitions of voltage, current, resistance

Any information you could provide me about finding the differences between non-rigorous or rigorous would be appreciated. 

I am thinking (hoping) that I already address this and just have not been exposed to the terms non-rigorous and rigorous when it comes to the definitions?!?!

Matthew, great timing -- I just worked on this difference with my AP class last week.  

"Rigorous" and "non-rigorous" definitions are my own personal terms, not anything to do with materials published by the College Board.

I start circuits on the very first day with the non-rigorous definitions:

Non-rigorous definitions of voltage, current, resistance
Voltage is provided by a battery.  Voltage is measured in units of volts.
Resistance is provided by a resistor, a lamp, or any electronic device.  The units of resistance are ohms (W).
Current relates to the amount of charge flowing through a resistor.  The units of current are amps.
Ohm’s law states that voltage is equal to current multiplied by resistance:  V = IR.

With just these facts, I can have students graph current and voltage to verify or discover the relationships in ohm's law; I can have students measure brightness of a bulb as a function of voltage and resistance to discover the power equation. And then we can do basic semi-quantitative questions with single resistor circuits, like "I replace a 10 ohm resistor with a 20 ohm resistor, by what factor has the current in the circuit changed?"  

Then we move on to circuits with series and parallel resistors, then to combinations of resistors, then to light bulbs, then to circuits with switches, using ammeters and voltmeters. I like to give circuit TIPERs, but make the students set up the situations experimentally to verify their prediction.

During these first couple of weeks, I never mention Kirchoff's laws -- rather, we have rules about current and voltage for parallel and series resistors which are a poor person's statement of Kirchoff.  (“Voltage across series resistors is different for each, but adds to the total.”)

Finally, once we've done all of this... everyone has a personal, intuitive understanding of what current and voltage are.  That understanding has been built on experience through problem solving, lab work, right and wrong answers.*  In eduspeak, this personal, intuitive understanding is referred to as an "operational definition."

*Never through analogy, though.  If students create their own analogies, great.  But direct experience without analogy has proven far more effective at building knowledge and avoiding misconceptions than any analogy I've ever tried.  Voltage and current aren't truly LIKE anything else.  

So, with that personal understanding built, it's time to introduce the rigorous definitions:

Rigorous definitions of voltage, current, resistance
Voltage is energy per charge.
Current is charge per time.
Power is energy per time.
Potential difference is a synonym for voltage.

Remember, your students aren't likely to come into the course with an operational definition of charge; and gaining the experience necessary to develop what charge truly means requires, I think, a full-on AP Physics 2/C treatment.  And "energy" is still a bit fuzzy in students' minds.  (These rigorous definitions can actually help students develop their operational definition of energy and charge, since they're so solid on voltage and current.)

I therefore tell the students to translate from rigorous language into our non-rigorous definitions.  When they see a problem like "rank these bulbs based on how much energy is gained by an electron passing through" they recognize that as asking about energy per charge; that just means "rank by voltage," which my class is well trained to do.

The last bit about circuits we do is to use Kirchoff's laws, and to make voltage vs. position-in-circuit graphs.  Here I use the terms "electric potential" and "electric potential difference" with impunity.  But by this time voltage is such an ingrained concept that the class has little difficulty anymore.

08 January 2017

What advice can I give a student with a C right now?

This showed up in the comment section from my August 2016 post in which I write a letter to my upcoming AP class.  It deserves a response in a full post, because I suspect that many physics teachers are confronting just this kind of problem this time of year.

A very concerned mother here. My very strong student pulled her first C in her life in the first semester of physics. We have tutors, spoken multiple times to the teacher and everyone says that she understands the materials, and almost always does badly on the test. When asked, she says that the test is so different she does not know what to do. As an engineer who had taken high levels of physics, I am really at a lost to help her. As an experienced teacher what advice can you give her. We need to make the next upcoming semester rock! Appreciate your kind assistance.

Now, remember that I have no direct contact with this specific student, so I can't give anything more than general advice. That said, I've seen this pattern many times -- historically outstanding student who gets As in history and biology, diligent, willing to work hard with support at home from subject matter experts... yet does not perform on physics tests.

The general advice starts with recognizing that there is no magic bullet. Neither this student's parents, her tutors, her teacher, or I can instantly create success. Physics skills are learned gradually, over time. They come quicker for some than for others.

That said: It's very, very hard for me to train even good physics teachers to back off and make students struggle without giving away answers. Students (and parents!) with good intentions often treat homework as a "just get the answer" exercise without engaging in the process.  Thus, in so many cases as you describe, the student's extensive support network is HURTING rather than helping. When tutors and expert parents get involved, students tend to ignore the part about "here's how to approach the problem" and think instead "thank goodness, I got the answer" -- no matter how good the tutoring might be.

So my fundamental advice is to let your daughter struggle. Give her loving emotional support, just as you would if she were on a softball team and kept striking out. When she asks questions, don't solve problems with her, don't help her figure out mistakes.  It's her homework, let her do it. Instead, advise her to think all the time about the process of getting answers, the general approach to different kinds of problems, even if she doesn't get the exact right answers. Help her keep focus on the big picture of all the things she's done well -- both in and out of physics class -- and don't engage with Chicken Little talk.

It's very likely that, by year's end, she'll start making connections and improve dramatically. I've had a number of students making Cs this time of year who ended up with 4s and 5s on the AP exam. Things often click after long-term exposure to physics.

It's also possible that she pulls a C for the year. That's okay, too. I have struck out every at-bat for four games in a row; I've earned Cs on tests and in classes. Those strikeouts and Cs no more define me than they should define your daughter. 

29 December 2016

Start Teaching Newton's 2nd Law Without Numbers or Equations

You've gone through a unit on motion; your students know the difference between velocity and acceleration.  (Or, at least some of them do, some of the time.) Now you're ready to introduce F = ma.  What do you do first?

I think most physics teachers, and certainly most textbooks, recognize the necessity of diving into free body diagrams right away.  Somehow, you must show the difference between an individual force and the NET force.  I concentrate on getting students to write out the object applying and experiencing the force; this helps avoid including fictitious forces (like "force of motion"), and it makes a future discussion of the third law child's play.

But, what do you do with those free body diagrams, other than make them?  

(1) Some books and teachers jump to a mathamatical treatment of F = ma.  Practice problems in which the free body is used to determine the value of the net force, use the second law to determine acceleration, then use kinematics to get something like the initial or final speed of an object, or its time in motion.  Then you can do the reverse -- use motion information to calculate net force, and then the amount of an individual force.

(2) Others go from the free body diagram to a semi-quantitative treatment of F = ma.  That is, show mathematically and experimentally that at constant mass, a larger net force yields a larger acceleration; for constant acceleration, a larger mass demands a larger net force.  Linear graphs can be created to verify the second law relationship.  

While I get to both (1) and (2), I don't start there.  I start merely with free body diagrams and the direction of motion.

But Greg, you say.  Free body diagrams have nothing to do with the direction of motion.  

Yes.  That's the point.

Before I do any work with the relationship F = ma, I ask every possible question I can think of about how the object is moving.  Here we're considering motion in a line only; circular and projectile motion are for later on.  

For example: This cart experiences a 3 N force to the left, and a 2 N force to the right. 

* Which way is the net force on the cart?  (Left, because the greater forces act to the left.)

* Which way is the cart's acceleration? (Left, because net force is always in the direction of acceleration, and we just said net force acts left.)

* Which way is the cart moving? (No clue.  Acceleration and motion aren't simply related.  The cart could be moving left and speeding up, or moving right and slowing down.)

* Could the cart be moving to the right?  (Sure -- if the cart is slowing down.  Note that the most common answer which is utterly unacceptable is "Yes, if another object applied another 2 N force to the right.")

* Could the cart be moving left at 1 m/s?  (Sure, as long as its speed a moment later is greater than 1 m/s.  NOT "Yes, as long as its mass is 1 kg.")

* Could the cart be moving left at a constant speed of 1 m/s?  (No way.  The cart experiences a net force, so the cart has an acceleration, so the cart's speed must change.)

It's useful to let students play with the phet simulation "force and motion basics."  In class, I have students do a series of experiments in which they predict the force necessary to cause an object to speed up or slow down.  We don't worry about the actual value of acceleration, just the directions of motion and acceleration.  

Once my students are rolling their eyes at these sorts of questions, answering with the same voice that my son uses when I remind him to wear a jacket to school on a cold day... well, then you're ready to move on to lessons (1) and (2) above.

21 December 2016

Followup: Three years later, what do conceptual students remember about circuits?

In 9th grade conceptual physics, we teach circuits without calculators.  Rather than asking "determine the voltage across each of these series resistors", we ask "estimate the voltage across each" and "rank the resistors by the voltage across each."  We don't allow direct calculation to answer these questions.

Rather, we expect a semiquantitative use of ohm's law, combined with instincts developed in laboratory.  I describe my class's Zen methods in this post.  

Those students who learned circuits conceptually now make up half my AP Physics 1 class.  Can the former conceptual students handle circuits problems in which actual computation is necessary?  Can they deal with more complex circuits than straight-up parallel and series resistors?  Can they describe their conceptual understanding in language appropriate to a college-level examination?  Yes and yes and yes.

In the freshman class, I hand students a page with circuits facts written on it.  (Scroll down on the linkned page to see the facts appropriate to circuits.)  By the second day of the unit, students are using the facts to predict voltages and currents for series circuits.  We do no lecture, no "going over" the facts.  Why not?  Because freshmen wouldn't pay attention anyway.  The class gets in the habit of reasoning based on facts, not of mimicking a teacher's steps.

Freshmen do very well with open-ended "here are some new facts, now figure out how to make predictions with them."  However, I learned the hard way that seniors generally do not.  They expect you to show them what to do, and get pissy if you expect them to use information you didn't "go over" -- even if that information is the first bold line on a sheet you handed them. 

Nevertheless, since half of my seniors had seen circuits in 9th grade conceptual physics, I thought I'd try the open-ended approach.  I was taking a twofold leap of faith:  (1) I hoped that the conceptual veterans would have enough familiarity that they weren't flummoxed by more complex circuit problems, or circuit problems requiring calculation; and (2) I hoped that there was enough comfort with the concepts and with the equipment that the conceptual veterans could provide leadership and advice to those who were completely new to circuits.  

This time -- thank goodness -- my faith was rewarded.  

I handed out the AP version of my circuits exercises, the version that includes series-parallel combinations.    Everyone worked in a relaxed manner and at a similar pace.  Information passed smoothly throughout the class -- when I gave advice to one student, I found that I rarely had to give the same advice to others.  

The conceptual veterans recalled rather quickly the subtleties of straightforward series and parallel resistors.  They easily helped the others make their predictions and set up their circuits.  The team atmosphere we built in the freshman class paid its dividends, as the conceptual veterans assumed -- without suggestion from me -- the roles of tutors and facilitators.  Even the students who had never seen circuits at all moved along at the same pace as most of the class.  Even the student who was new to circuits and was absent the first class picked up the process quickly.

Did anyone struggle now that we included calculation, now that we included combination circuits?  Not at all.  Sure, I had to show two of twenty students how to deal with the combination circuit.  The rest either figured it out for themselves, or were taught by one of the folks I helped directly.  

I'm on my fourth attempt at teaching AP Physics 1-level circuits.  And this is by far the smoothest introduction I've had.  I'm ready now, after a week of class, to discuss the deeper language and tougher situations that AP Physics 1 requires.  Most everyone can already accurately fill out a VIR chart for a simple circuit.  I can focus on the whys and hows.

In other words, teach eighth, ninth, or tenth graders about circuits, but conceptually.  The very basic three-week unit we created has paid off tremendously in my AP Physics 1 class, even though the unit was three years ago, even though we never used a calculator. 

And I remind myself how important the work I do with freshmen is.  I'm planting seeds with them... seeds that I usually don't get to see germinate.  But germinate they do.

14 December 2016

Quizzes to follow up AP Physics 1 problem solving: Try composing tweets.

Old-tymie physics questions would simply ask, "Calculate the horizontal distance block B travels after it leaves the table."  Such a question will be vanishingly rare in AP Physics 1.

Case in point: consider 2010 AP Physics B problem 1 part (d).  You have a block pushed by a compressed spring.  The block collides with another block, then falls off a table.  No analysis, no articulation of principles necessary... just perform the calculation.

Don't get me wrong, 2010 AP Physics B problem 1 is a fantastic question.  It combines in one simple situation the three canonical approaches to classical mechanics: force/kinematics, momentum, and energy.  I assigned this problem verbatim to my AP class last week.

Of course, I encourage collaboration in and out of class, as do most of us.  Thus, a significant fraction of the class got the approach right because someone pointed it out to them.  No, that's not "cheating," that's working together.  Students engaged the problem individually, most got stuck somewhere, and then through conversation and direct advice, they figured out what to do.  Awesome.

I will certainly grade this problem.  Presenting the solution clearly is an important skill to develop.  And by grading the problem, I provide incentive to engage in the collaborative process.  I can tell the difference between Fred, who just kinda blindly followed a friend's work, and Jim, who himself showed each step clearly.  At this point I don't care that Jim showed each step clearly because George told Jim how to do each step.  Jim wrote out his work, and so made progress toward personal understanding.

Nevertheless, I need to evaluate my students' personal understanding of the process.  I need to help my students evaluate for themselves what they understand and what they don't.  After all, the AP exam is not a collaborative exercise.  Everyone, by May, needs to be able to independently figure out how to approach this type of complex problem.

More to the point, my AP Physics 1 students must be able to do more than just perform the calculational procedures that lead to a correct answer.  The exam might ask, "Explain how you would calculate the distance block B travels after it leaves the table."  And the response can't be "I multiply 1/2 times 250 times 0.15 m squared, then plug into p=mv."

So I give a quiz.  What kind of quiz can you give based on this problem, Greg?  I'm glad you asked.

Sure, you can give the same problem and change the numbers.  That's okay.  It doesn't put students on the track toward answering AP Physics 1 verbal response questions, but it's a start.

You could also change the situation slightly... have the block initially slide down a ramp rather than be pushed by a compressed spring.  Or eliminate the collision.  Or put the table on Mars.

I've discussed in this post how I ask for annotated calculations in order to check for understanding.  An interesting quiz might present a full solution in numbers and ask the student to annotate the calculation to explain each step.

Even then, students have a hard time recognizing what parts of a solution are important to annotate.  They want to describe the arithmetic: "I divided both sides by 0.15."  Or, they say "I used p=mv.".  Um, I know -- you just wrote "p=mv," you don't need to tell me again.

Ask: "Explain in two tweets how to solve the problem."  I propose that students have a friend at our rival high school who needs help, saying via twitter that they don't know what to do.  You have to help.  You get to communicate in only two tweets -- that's two sets of 140 characters each.

The secret to teaching students to write is to clearly define an authentic audience.  They know without me saying anything that an online friend doesn't want to hear the poor annotations I've described above.  They want to hear simple articulations of principles:

Spring energy becomes A's KE. That gives A's speed and momentum before collision. P conservation gives the blocks' speed after collision. 1/

Now, blocks are a projectile. Vertical kmatics gives time, d=vt gives distance since horizontal v doesn't change once blocks leave table. 2/

And this explanation is a strong response to the AP Physics 1 question, "Explain how you would calculate the distance block B travels after it leaves the table."

08 December 2016

Momentum and kinetic energy when people push off each other

A mother and her son are initially at rest next to each other on an ice rink on which friction is negligible.  The mother’s mass is twice the son’s mass.  They push off of each other, causing them to glide apart. 

1. Is the magnitude of the two skaters' total momentum larger before or after the push?

Simplest answer: Momentum is conserved in a collision.  Total momentum is zero before the push because nothing moves.  So, afterward the total momentum must likewise be zero.

Deeper answer: How do we know momentum is conserved here?  Because no net external force acts.  The normal force and weight cancel because there is no vertical speed change.  The only horizontal forces are the forces of the skaters on each other -- these are internal to the two-skater system.

Corkscrew-thinking too-deep answer that is nonetheless fully correct:  J=Ft.  The forces of each skater on the other are equal due to Newton's third law.  The time of collision is the same for both skaters -- otherwise we wouldn't be in the same collision.  So impulse is the same for both.  Impulse is the change in momentum, meaning both skaters have the same amount of momentum after the push.  Since they move in opposite directions, the vector sum of the objects' momentums is zero.

Not fully correct: Since the skaters move in opposite directions with equal momentums, the total momentums subtract to zero.  (How do we know that the skaters have equal momentums after the push?  That requires the corkscrew thinking above.  Without reference to N3L and the impulse-momentum theorem, there's no evidence that the have equal and opposite momentums.)

2. Is the total kinetic energy of the two skaters larger before or after the push?

Simplest answer: Total kinetic energy is zero before the collision because there's no motion at all.  After the collision, both skaters move, so both have kinetic energy.  The system kinetic energy is the (scalar) sum of each skater's kinetic energy, which is not zero.  So larger KE after the push.

Deeper answer: Why is mechanical energy not conserved here?  After all, as in question 1, we can show that no net external force even acts, let alone does work, on the skaters.  Well, it's the internal energy from the skaters' muscles that is converted to kinetic energy.  The skaters' bodies lower their internal energy by converting ATP to ADP, in the process causing their arms to apply a force through a distance on the other skater.  That's mechanical work, which changes each skater's kinetic energy.

30 November 2016

Evidence that conceptual physics leads to strong understanding of college physics

This is the first year I've taught AP physics to students who have been through our recently revised conceptual physics program.  The results after a marking period are illuminating -- those who took our conceptual physics are definitively outperforming those who have taken a traditional calculational-style previous physics course, whether that course was here or at another school.

This result is independent of teacher -- the students that my colleague Alex taught in 9th grade are doing just as well as the students I taught.  And our former conceptual students aren't merely remembering content, either, as we didn't even discuss much of this AP material three years ago.  Furthermore, this correlation of top performers to veterans of 9th grade physics did not exist in the days when we taught calculational physics to 9th graders.

No, the difference is the discipline we instill of answering every physics question with a fact from our sheet, or with a carefully-taught methodology to make a calculation.  The top half of the class doesn't have to learn how to learn physics.  

My conclusion: a conceptual approach to first-year physics is better than a calculational approach, even for those who go on to study physics at the university level.  I take this as an article of faith, but many physics teachers, parents, administrators, and university folk would not agree.  Certainly more and more are on coming on board with a conceptual approach.  But lots of people -- professionals, even -- would say how we don't teach our students the math skills, or that we deny our students an understanding of "real" physics, or that we're too fluffy, or that we don't prepare our students for college.  These concerns aren't unique to 9th grade conceptual physics: they're some of the same critiques I've heard of the AP Physics 1 program.

The naysayers are wrong.  The evidence is clear to me.  Observationally, those who had conceptual physics behave in a more confident manner, are more comfortable discussing and explaining physics, and can adapt far more quickly to new and challenging physics concepts.

What about some non-observational evidence?*

* Obviously not scientific evidence, just evidence.  Teaching is a craft, not a science.  Others may have different evidence.  If so, please describe it in the comments.

I ranked the 20 students in my AP Physics 1 section by grade.  Numbers 1-8, 11, and 13 took 9th grade conceptual physics.

Numbers 9, 12, and 14-20 took a previous calculational physics class: two took our Regents-style calculational physics as juniors, and the rest had calculational physics at another school.  In all cases that I've asked about, that previous calculational class covered more topics than our conceptual class does.  Teaching physics isn't about topics covered, it's about the style of approach to the subject.) 

They've all seen physics before.  But the ones who have seen rigorous conceptual physics are doing best.

29 November 2016

Do you need a school-wide grading scale? No.

Occasionally I hear from other physics teachers that they have been asked to adhere to a school-wide, non-negotiable "grading scale."  They mean that some administrator has decided by fiat that 90% is an A, 80% is a B, with no flexibility.

Of course that's ridiculous.  Each teacher grades with her or his own idiosyncrasies.  As long as grading standards are clear, translucent, and applied across the board, no one should worry.  In fact, I have it on good authority from multiple college counselors that university admissions people are well aware that different schools, different departments, different courses, and even different teachers assign grades to somewhat different standards.  Admissions officers ain't stupid; they won't be pleased or tricked by an artificial standardization attempt.

But everyone has colleagues somewhere who advocate a form of grading in which "90%" has some sort of intrinsic meaning.  If you require 85% for an A, and your colleague in the history department requires 93% for an A, aren't his students at a disadvantage?  Well, no.  A family who makes 5 million yen is not more than a hundred times times better off than someone making 40,000 dollars...

Today I heard a number of conversations in which folks were advocating for a standardized school grading scale.  Here's my response.

Physics multiple choice questions do not ask for recall, or even for application of an algorithm.  They ask for application of facts and equations to new situations.  Similarly, free response items ask students to engage in creative problem solving in new situations.  We don't ask "In lab last week, how fast did the cart move?"  We ask, "You're asked to predict and then measure the speed of a cart on a bumpy track that you've never seen before but I'm just now describing.  Go."  Students are not expected to get 90% of the available points, any more than a soccer player is supposed to score on 90% of his shots.

On the AP exams, 65-70% is a 5, which the College Board has carefully correlated to an A in the best college classes.  On the New York Regents exam -- on which our conceptual program is based -- 85% is roughly equivalent to an A.  So we use these well-known and well-verified scales in physics. 

I'll never understand why people get so worked up about grading scales.  In English or History class, we can certainly call 90% an A, 80% a B.  But really, when you're grading an essay, what does an "89" mean, anyway?  And don't people manipulate grades through "bonus" questions, "extra credit", controlling the difficulty of questions, and controlling the grading rubric?

Our all-world quarterback's completion percentage was only about 75% this year... so he's a C quarterback, and the division 1 powerhouse football schools recruiting him should cut him loose.  And even our star baseball player in 2010 only hit 0.570 his senior year; he's an NC.  I don't know what these colleges and minor league teams were thinking, recruiting him.  His batting average is a failure.

Oh, think I'm being silly?  That's the same argument as "we must use the same grading scale in physics as in English or Spanish."

If the school demands that we adhere to a strict 90/80/70 scale, I will obediently give Ds and NCs to everyone in my honors physics class except one person, who will have a B.  We'll still have straight 4s on the AP Physics 1 exam, of course, but I suppose that's irrelevant.

(P.S. Funny thing is, even though I've heard some very loud voices in favor of a standardized grading scale, it seems that a great majority of faculty, even those outside the science department, in fact agree with me.  Most people I talk to quietly don't understand why there might even be a discussion.)

26 November 2016

Algebraic kinematics in CONCEPTUAL physics

Our discussion of motion in conceptual physics starts with a three-week unit.  The first week deals exclusively with position-time graphs.  The second week deals with velocity-time graphs.

The third introduces the definition of acceleration, along with three equations of motion.  

Acceleration is defined as how much an object's speed changes every second, and is measured in units of m/s per second.  The direction of acceleration is the same as the direction of movement when an object speeds up; the direction of acceleration is opposite the direction of movement when an object slows down.  We never use the words "positive" or "negative" to denote directions.  Instead, we use words like left, right, up, down, forward, backward.  In the absence of a clearly defined direction, students are directed (hah!) to talk about "toward the motion detector" and "away from the motion detector."

Now this is conceptual physics, so the descriptive analysis above forms the basis for the students' understanding.   Nevertheless, there are considerable quantitative and semi-quantitative elements to this course.  We do get numerical answers for distances, speeds, and accelerations; if nothing else, recognizing the reasonability of a numerical answer is a major physics skill.

I do not present an "equation for acceleration."  Most texts state my definition of acceleration as an equation: a = Δv /t.  I don't like that.  It's too easy to miss the delta, and to say that acceleration is speed divided by time.

Instead, I ask students to reason in their minds using the definition of acceleration.  "Since it took 5 s for the cart to slow down by 20 m/s, the cart slowed by 4 m/s every second.  Its acceleration is 4 m/s per second."  This works both pedagogically -- in every problem I'm reinforcing the operational definition of acceleration -- and physically, because we are always assuming constant acceleration.

However, I *do* present equations for distance traveled.  Three of them, as a matter of fact.

In conceptual physics, we restrict our linear kinematics problems to three situations:

(1) Objects moving at a steady speed
(2) Objects speeding up from rest
(3) Objects slowing down to rest

In case 1, the distance traveled is given by d = vt.
In cases 2 and 3, the distance traveled is given by EITHER d = (1/2)at2 OR d = v2/2a.

In previous units, we've taught our students about semi-quantitative reasoning (i.e. when resistance doubles at constant voltage, what happens to current?) and about predicting the shape of graphs (i.e. sketch a graph of resistance vs. current at constant voltage).  We also have been rigorous about beginning any direct calculation with a chart of values with units.  Here we have three new equations on which to practice these skills.  

Plus, we learn a new skill: picking the applicable equation out of a lineup.

"But that's not a skill, Greg, that's easy," you say.  Nuh-uh.  Students raised through the mathematical-industrial complex are rarely taught to consider the situations in which an equation might be valid or invalid; and though they're taught how to match variables with relevant values (in "word problems"), most have in no way mastered that skill.

Again and again conceptual physics students carefully calculate the distance an object traveled... by picking one of the three equations at random, while randomly assigning values from the problem statement to the variables in that equation.  Then the students are upset that I don't accept their answer.  They did the math right!  

So we work hard on identifying the right equation to use.  "Is it constant speed?  Then you must use d = vt.  No, you can't use d = (1/2)at2 , it doesn't matter that we know the time and the distance, you can't solve for acceleration when something moves at constant speed, and the problem asked for the speed, anyway."  "It's slowing down to rest and we know the acceleration.  Both d = (1/2)at2 and  d = v2/2a are valid.  Do we know the time it took or the initial speed?  Because then we'll know which of the equations is useful."

After this unit the students are only kinda okay at doing semi-quantitative and quantitative kinematics.  That's fine.  Because pretty soon we'll revisit these equations in new contexts.  Perhaps we've used Newton's second law to solve for an object's acceleration, and we want to know how far it goes in 4 seconds.  Even better, once we understand how to deal with velocity as a vector, we study projectile motion for objects shot horizontally off a cliff -- where d = vt is valid horizontally, and the other two equations are valid vertically.  

23 November 2016

Mail time: what system do I use for vertical springs?

A question from Hibisca, who attended my Atlanta summer institute last year:

I have a question about approaching energy with vertical mass-spring systems. I recently made the mistake of teaching my students to include gravitational potential energy in the analysis after choosing the new equilibrium position as the location where the spring hangs at rest with the mass attached. I have since mathematically proved to myself why PEg should be left out (and this Khan Academy videogoes through the step-by-step proof as well), but I have not found a clear conceptual explanation. If we analyze the block-spring-Earth system, there is no external work done on the system, mechanical energy is conserved, and it seems that PEg should be included. How do you explain this to your students? I assume relying on a mathematical proof is not sufficient.

This one's rather complicated. I always suggest, unless you're told otherwise, to treat a vertical spring as the spring-object-earth system. Then there are no forces at all external to the system, so mechanical energy is conserved.

The SYSTEM potential energy is just 1/2kx^2... where x is measured from the equilibrium position, not the position where the spring is unstretched.

See, you could include both a separate spring potential and a gravitational potential if you measure everything from the unstretched position. But that's really complicated. Why not fold all the potential energy into a single term? You can show mathematically -- and I suspect that Khan Academy does show -- that the 1/2kx^2 potential energy with x measured from equilibrium is entirely equivalent to both 1/2kx^2 and mgh terms measured from the unstretched position. But I don't think that's important. Just use the single term, and recognize that it's for the spring-object-earth combined system.