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.

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.

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."

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.