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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.

17 November 2016

Hints on reading body language: shoulder-to-shoulder spacing is generally a bad thing.

My wife and my students will tell you that I'm The Worst at recognizing faces.  Learning students' names is a process fraught with difficulty for me.  A published list of captioned photographs does not help me -- more than once I've been looking straight at a student's photograph and not realized that he's the one standing in front of me.

However, I'm quite good at body language.  I do recognize people by how they walk, how they move.  It's only too bad that these published photo books don't come with video.

I'd suggest that an intuitive feel for body language is an important skill for a physics teacher to develop.  My class is regularly shocked that I seem to know exactly what they're thinking, even when they haven't said a word.  "You're giving me a Look, Mr. Smith," I'll announce.  "The cart must have experienced a force to get it moving, so why didn't I put that on the free body?  Is that your concern?"  Sure, experienced physics teachers know this misconception is out there... but I pinpoint it to the particular student whose forehead wrinkles, whose mouth turns a bit frowny, and who generally looks like my very dear mother listening to Chance the Rapper.

How do I read body language?  How should you?  That I can't exactly explain.  It's not science -- it's art, or perhaps craft.  But it's such an important skill that it's worth trying to share some observations.

Today's thought on body language:  Break up knots of students standing or sitting shoulder-to-shoulder.

I often encounter this phenomenon in the laboratory.  A student who doesn't know exactly what to do sidles in between a working partnership... then more and more join until nine students are assembled around the same lab table.  

They're not working.

Maybe they're commiserating about how "no one" knows how the lab is supposed to work.  Maybe they're making fun of me or of each other.  Maybe they're gossiping.  I'm not sure.  But that gang of too-close-to-each-other students is never, ever productive.

What can you do?  Break it up by any means necessary.

The five-foot rule can be your friend here if they're supposed to be writing things to turn in.  Assigning partnerships, and limiting to three the number of students who are allowed to share data, can provide the original basis for your request for the knot to separate:

"Hey, Mr. Jones, don't forget the five-foot rule... could you please move to that vacant desk?  Thanks!"  "Folks, we don't need five people working with the same setup.  They'll be finished shortly.  Until then, Mr. King and Mr. Garrard, would you please work on the written assignment?  Then when the equipment is available you may use it."  Chances are, students will give you a sheepish look and move away -- problem solved, because everyone will just get to work, finding out that the physics task wasn't as hard as they tried to make each other believe.  

But don't accept any guff.  "Hey, why do I have to move, that's not fair!"  That's when I get upset.  I wasn't initially upset -- and I was careful to control the tone of my voice such that I was making a gentle request, not an angry demand.  But now, after that ridiculous whine, I'm upset.  "I made a reasonable request as to what you are to do in this classroom.  Either comply, or leave."  No need to get into an argument about whether the student was or wasn't goofing off... you won't win that one.  No need to make innumerable rules to enforce about behavior in lab.  It's not hard -- when the teacher politely asks a student to do something reasonable, the student should do so without backtalk.  I believe I was taught that in first grade.*  A high schooler who has difficulty with the concept of compliance has bigger problems than 'what is the speed of the block at the bottom of the incline.'

* I didn't actually learn it until third grade.  But nevertheless.

This battle is worth fighting.  Knots of students drain morale.  Even those who aren't in the knot might be working, but they're working resentfully.  ("Why do I have to do physics while those folks eff around?" they'll think to themselves.)  

So recognize the signs, and break up the critical mass of too-close students before it becomes a problem.  Read the body language.

(Any other body language tips?  Post in the comments.)

15 November 2016

Don't be afraid to ask an open-ended question; but make your students answer it.


A man of mass 80 kg stands on a scale in an elevator.

Think of all the questions that can be asked in this situation.  I'll give some typical questions... but look at the end to see my crazy open-ended versions.

The scale reads 750 N.  What is the magnitude and direction of the man's acceleration?

Good start.  Your student has to recognize that the acceleration is the NET force divided by the mass, not any old force.  And has to know that 80 kg means a weight of 800 N, making the net force (800 N - 750 N) = 50 N.  Acceleration is in the direction of the net force, so downward.

The biggest misconception here, though, isn't about the relationship between net force and acceleration -- it's the relationship between net force and motion.  Try this one:

The elevator moves upward and slows down.  Is the reading in the scale greater than, less than, or equal to 800 N?

Half of your class will say the scale reading is greater than the 800 N weight of the man, because they assume that net force and acceleration must always be in the direction of motion.  Aarrgh!  No, net force is in the direction of acceleration; when an object slows down, its acceleration is opposite the direction of its motion.

No matter how good you are at disabusing folks of this misconception, getting 100% of a class to answer this question correctly is a neigh-impossible task.

That said, questions that directly address the misconception and then require students to explain their reasoning are a bigly step toward busting the misconception.  Yes, your students will get this wrong; but fewer and fewer will get it wrong as the year goes on and as you ask similar questions again and again.

I phrase some of these questions in an even more open-ended style:

The scale reads 750 N.  Which way is the elevator moving?

Even better!  Now the students have to recognize and articulate for themselves that the question is unanswerable, and why it's unanswerable.  Acceptable answers include "we don't know because we only know the direction of acceleration, not motion" or "if the elevator is speeding up, it's moving upward."  The important point of this style of question is, the answer can't easily be gamed by the good test-takers taking clues from the phrasing.  Even including choices such as (A) up (B) down (C) it can't be determined gives a clue.  A physics student who has truly attained a masterful understanding of the relationship between force and motion will not have any difficulty with this question.

Now, this approach doesn't work if students habitually ask questions during a quiz, or if they come straight to the teacher for help on homework before answering.  "Wait, teacher, you didn't give us enough information to solve the problem, what are we supposed to do?"  Whatever you say, you're sunk; the student AND HIS CLASSMATES have just heard the game-the-question clue they wanted.  

So you must establish from the start of your class that these sorts of open-ended questions will be asked; and that they must be answered without any attempt to drag hints out of the teacher.  Don't tolerate whining like "that's not fair, it's a trick question."  There are no trick questions.  Success in physics comes when the class stops looking for the tricks and starts answering confidently with reference to physics facts.  Physics is, at its heart, about understanding the universe.  And Mother Nature is generally unsympathetic to complaints that the problems she poses are too tricky and open-ended.






26 October 2016

Mail Time: "Normal force", "Flex force," or what?

Elizabeth Pate posted a comment asking about the reactions of AP readers:

I have a question about FRQ scoring. I use modeling instruction in my classroom so a few vocabulary terms that I use are different from a traditional physics course. For example, my students would call the normal force a flex force and the net force an unbalanced force. I do teach them both terms but I find they usually prefer flex and unbalanced force to Fnet and Fn, will these cause a problem for graders reading their FRQ's?

Reading AP exams is kind of how I learn about new trends in physics education.  Last year I learned about a commercial device with two (visible) lasers and photoreceptors; when the beams are broken by a moving object, the device calculates speed.  I'd call it a visible photogate.  I forget what it's actually called, but enough students used this device on their lab problem that I found out about it.

So, "flex force," eh?  That makes sense.  Never heard it before.  How would I react to it as an AP reader?  It depends on the rubric, and how the problem is phrased.

When the test simply asks for a labeled free body, we are usually quite generous about those labels.  My quintessential example was a few years back when students had to label a buoyant force, the force of water on a cup.  We accepted "buoyancy."  Then we accepted the misspelled "boyance".  Then we accepted "bouncy," because it made sense in context.  A rumor was spread that someone accepted "Beyoncé," but that's unconfirmed.  :-)

So would I accept "Fflex" on a labeled free body?  Possibly, especially now that I've heard from you that "flex force" is modeling vocabulary for "force of a surface on an object, acting perpendicular to the surface."  

I always train the students to define their labels.  Don't just say "Fn" -- say "force of the road on the car" or "force of the scale on the boy."  Then even if someone misreads the label, or if the label is unintentionally ambiguous, there's no issue.  

For example: we've never accepted "G" as a bare label for the gravitational force, even though some texts and teachers may teach that.  Why not?  Because "G" has a well defined conventional meaning: it's the universal gravitation constant, 6 x 10^-11 N*m^2/kg^2.  

But a student who labels the diagram with "G" and then says "G: force of the earth on the car" earns full credit.  Oh, and he can deal well with Newton's Third Law, too, but that's for a different post.  :-)


14 October 2016

Mail Time: Why do we have to memorize facts in 9th grade conceptual physics?

In conceptual physics, I don't use a textbook.  Instead, the reference material for the class is contained in our "fact sheets."  These facts are handed out piecemeal to the class, about three to six sentences at a time as they're relevant to the current content.

You can see all the facts from the full conceptual course here; feel free to use these in your own class.

I ask students to learn the facts by heart.  We have occasional fill-in-the-blank quizzes in which they have to recall the important words in a fact; on homework, students are required to write these facts nearly word-for-word as the first step in responding to any physics problem (though they have access to notes for all homework). 

I got an email from Keri, who is using these fact sheets for the first time in her conceptual class.  She's encountered an unusual problem: her freshmen are complaining about having to memorize these facts.  How should she justify to students, parents, and administrators why a physics class requires remembering words?

Keri has three proposed responses: 

(1) Since she's giving students a formula sheet, they don't need to memorize formulas.  It's more important to memorize facts; and those facts won't be available during an exam.

(2) Paraphrasing the facts generally isn't enough, because it's too easy for a new student to miss something important in the paraphrase

(3) Facts should be instant-recall, not thought processing, so students can focus instead on the reasoning involved with each problem.  

Keri, that's a really interesting and unusual complaint. The vast majority of complaints that I get, and that other physics teachers report to me, are of the form "but I learned the facts and equations, I can spit them back, why aren't I getting an A?" Now, your students are saying, hey, don't make me memorize anything, THAT'S too hard, too! I suppose they're suggesting that we all just sit here and watch videos for a year? My cynical mind and experience as a baseball umpire draws the conclusion that people will kvetch about teachers regardless of what we do. Everyone thinks they can do a better job than we can, everyone's a critic. 

I've never been asked this particular question, but it deserves a good answer. You've given an excellent three-pronged argument. I'll elaborate on each prong.

 I love your answer #1. I'd add that memorizing facts is what allows students to deal with problems that aren't merely recitation of facts. How is anyone supposed to interpret a position-time graph if they can't remember that the steepness (or slope) determines the object's speed? 

In terms of paraphrasing, note that our daily quizzes don't say "Write fact #3 about velocity-time graphs word for word." They say, "on a velocity-time graph, the speed is determined by _____." If you can't tell me that's the vertical axis, you don't understand velocity-time graphs.  Now, I generally accept "y-axis" or something that's pretty danged close. My colleague Curtis, who taught me about this style of quiz, insists on word-for-word terminology pretty much because he wants the class using the same language as each other, and he wants quick recall not processing (which is your point #3 above). 

With ninth graders especially, we start by asking them to copy these facts by hand into a notebook. Then they can use their personally handwritten notes on some of the quizzes. For example, we'll give a quiz with notes the day after they get the facts. Then, after they've used the facts for a day or two, we give a later quiz without notes. There's so much repetition in our class -- via quizzes, writing on homework, writing on in-class exercises -- that student draw confidence and comfort from the rote knowledge of the facts that they develop.

And finally, remember that a lot of the whining you're hearing comes from a position of ignorance. You as a physicist know when a substitute word is truly a synonym, and when a substitute word changes the meaning. In the example above, "y-axis" and "vertical axis" have the same meaning, even to a first year student. But my students have written, "on a velocity-time graph, the speed is determined by the velocity." "Oh, come on, the vertical axis is velocity, so you know I meant that velocity is the vertical axis!" No, sorry. That doesn't make sense. I've chosen my words very carefully on those fact sheets so that learning the facts leads to understanding. 

Physics is already a difficult subject -- it becomes EASIER, not harder, when students learn the facts by rote. There's gotta be trust in you as the teacher, just as we trust the musician who tells her students to practice scales. You, not your students, not your parents, are the physics teaching expert. When your students have a physics degree and a job in your school teaching physics, then they can decide what is a correct fact of physics. That was your final point in the email: "because I said so!"

(Of course, feel free to hide behind me, too -- "Hey, it's not me, these were written by this AP physics reader who's published five books and a blog. Feel free to take your complaints to him." :-) )

Good luck, Keri, and to all using these fact sheets.  They work.  

01 October 2016

Teaching the qualitative-quantitative translation: Why our students use common sense instead of calculation.

Today's question: Planet X has three times the free-fall acceleration of Earth. 

(a)          A ball is thrown vertically upward with the same initial velocity on Earth and on X.  How does the maximum height reached by the ball on X compare to the maximum height on Earth? 

(b)          Next, a ball is thrown vertically upward on X with three times the initial velocity of an identical ball on earth.  How does the maximum height reached by the ball on X compare to the maximum height on Earth?

The newbie physics student generally doesn't want to learn how to approach a physics problem.  He or she wants to get to an answer.  Usually the unsuccessful path to the answer takes one of two forms:

(1) Common sense.  "Obviously, more gravity means smaller max height, and three times more gravity means three times less height.  And in (b), three times more speed cancels out the three times more gravity, so the same height for each."

(2) Trying to find the one weird trick. "In class, I remember you asked a question like this.  Since the equation has a square in it, three times the gravity means nine times less height.  And in (b), the square terms cancel to give the same height."

(If you've been teaching for more than a month, you've seen these sorts of answers.  Let me know if you haven't seen one, and I'll buy a beer for you in anticipation of when you do.)

Why do our students say these things rather than just do the calculation?

In many of our students' minds, good, smart boys and girls know the answer.  The thought of "figuring out" the answer from first principles isn't part of their skill set.  You don't "figure out" the 3rd person plural present active indicative of cupio; you remember that -io verbs take -iunt in this form.  And if you don't remember, you should, 'cause you've been taught that.  So take a guess, knowing that -nt is a typical 3rd person ending.  You'll at least get close.  

There's your common sense approach in (1) above.  Since of course smart students should know the answer, they take a reasonable guess based on their instincts and previous experience.  Those instincts have been good in previous classes, especially math class; so guesses like this should work in physics, too.

The next step for students is to try to mimic what they see in class, what they read in the textbook.  When they recognize that common sense approaches don't work, they despair -- "oh, physics is impossible, every question has a trick to it."  So find the trick.  Note the language used in response (2) above, referring to "the equation".  WHAT EQUATION?  I want to shout.  Shouting is useless, of course... 

The student looking for one weird trick doesn't remember that in class I showed how to determine the correct relevant equation, then to solve it for the height of the ball in terms of the other variables.  

The student does remember that I got the answer.


The student assumes that I already knew the answer, and only went through the calculations for form's sake, like a prosecutor painstakingly presenting evidence that Jack Ruby shot Lee Harvey Oswald.


The student takes away, then, that HE is supposed to know the answer, too.  He shows his work only because I insist on it for silly teachery reasons.


How do we train students to use quantitative reasoning when answering these sorts of questions?

Understand that we're fighting a war of attrition.  There's no one weird trick for teachers that will suddenly cause enlightenment.  Chip away at the class, getting one student at a time to use quantitative reasoning properly.

Begin the battle by modeling good qualitative-quantitative translation skills.  Model an organized approach, in which you solve for the desired quantity in variables.  Use numbers too, not only variables -- early in the year, most of your class will not yet be comfortable looking at variables with squares and square roots, but they're easily able to compare 9 meters to 3 meters.  As the year goes on, you can help everyone transition to using variables only.  

Next, do an experiment to demonstrate your results.  Okay, you can't travel to planet X for the problem I've posed here.  But you can show that doubling a block's initial speed will quadruple its stopping distance on a track.  You can show that dropping from twice the height does not double the time a ball is in the air.  Emphasize that we're making physical predictions, not merely doing abstract mathematics.

And finally, demand to see a quantitative approach.  Look how I've rephrased these same questions below, to emphasize that we're looking for a multiplicative factor.  

For the students who still don't do the calculations -- yes, that means about 1/4 of my class -- make them do the problems again with even more explicit instruction, like "Pretend the initial speed is 10 m/s.  Use kinematics to calculate the height on X and on Earth."  Eventually, they'll get it.  Just be patient yet persistent.


The question, rephrased: Planet X has three times the free-fall acceleration of Earth. 

(a)          A ball is thrown vertically upward with the same initial velocity on Earth and on X.  How does the maximum height reached by the ball on X compare to the maximum height on Earth?  Justify your answer with both words and kinematics calculations.  Then your answer should state “The ball goes ____ times higher on X.”


(b)          Now, a ball is thrown vertically upward on X with three times the initial velocity of an identical ball on earth.  How does the maximum height reached by the ball on X compare to the maximum height on Earth?  Justify your answer with both words and kinematics calculations.  Then your answer should state “The ball goes ____ times higher on X.”