Mail time: using notation as stated in the problem stem

On an AP exam, are points awarded or not when a student doesn't use exactly the variable notation as stated in the problem (such as, M vs. m, or I vs Io)?

The answer is, it depends on the context.  There's no one universal approach, because each part of each problem is testing different skills, and the rubrics are developed independently by different leadership teams who might make slightly different decisions from year to year and problem to problem.  

When we develop each problem's rubric, the leadership spends a long time discussing just this question, looking at hundreds of samples to see the range of responses.  I'd say the line we try to draw is, we don't want to award a point where a student may have been ambiguous about communicating their physics understanding; but we don't want to be pedantic splitters of hairs.

To give a couple of examples from an AP question about a modified Atwood machine: on the free body diagrams, we didn't care about the specificity of the labels.  The objects were of different masses, m[a] and m[b].  But who cares whether they wrote "mg" or "m[a]g" on the free body!  They had communicated that each object experiences a downward force of the earth.  The fact that these forces were of different magnitudes wasn't relevant to the particular skill that was being tested in this part.  We accepted anything reasonable and unambiguous - Fg, mg, Fe (for force of earth), etc.  Failing to award points because "you didn't put the subscript on the m!" would have felt pedantic to those of us charged with creating the rubric, especially when each free body dot was even labeled "object a".  

But in the derivation in the very next part, it made a significant difference which mass the student was talking about!  Here, the problem was indeed checking to see whether the students understood, in "F=ma", which F, which m?  So here, we demanded the final answer have correct notation throughout.  I did not feel like a pedant when I failed to award points for no subscript on mg in the numerator of the final expression - such a result does not communicate a full understanding of the physics of the problem!

If you're not sure, ask yourself the question - are you splitting hairs, or are you demanding clear communication?  And if you're not sure, just pick one way and be consistent.  If you do corrections instead of handing back graded tests right away, then students won't notice or care - they'll learn from the correction that they needed, in that case, to use the notation given in the problem.  Which isn't an onerous ask.  :-)

Demonstrations with the visual accelerometer

I introduce acceleration as how much an object's speed changes every second. No "delta-v over delta-t"  for me - this equation obscures the physical meaning of acceleration, and will invariably be used by students as simply v/t.  But a couple problem sets in which students write the definition of acceleration as their starting point and reason from that fact work wonders.

That said, before I even dig in to the meaning of the magnitude of acceleration, I work on the direction of acceleration.  We use two simple facts:

1. When an object speeds up, the object's acceleration is in the direction of motion.

2. When an object slows down, the object's acceleration is opposite the direction of motion.

First question, then: I release a cart from rest.  Now, as the cart rolls down an incline, what is the direction of the cart's acceleration?  

Anthony
(by @Aldescery)

No, Anthony, don't shout out an answer.  Start by reading a fact word-for-word, and then tell me how that fact answers the question.  

"When an object speeds up, the object's acceleration is in the direction of motion.  This cart speeds up while moving down the incline.  So the cart's acceleration is also down the incline."

Perfect. The concept is still extremely abstract to students, though.  On this first day of acceleration, my students are still translating "acceleration" to "speed" in their puppy-physicist heads.  They are answering this question by rote.  I need to show them, demonstrate for them, that their rote reasoning led to a physically-verifiable prediction.

I use the PASCO "visual accelerometer", pictured at the top of the post.  This particular device has been discontinued by PASCO - they offer a new device that attaches to the smart cart.  If one of your colleagues is good with the Arduino, then I've been told it's trivial to hook up some LEDs and an acceleration sensor to make a similar device to mount on a cart.

So next, I actually release a cart from rest at the top of an incline.  The visual accelerometer mounted on the cart lights up - the lights pointing down the incline light up.  Students see that the cart's acceleration was, in fact, down the incline. 

Second question: I push the cart and let go (without the visual accelerometer mounted).  While the cart is moving up the incline, right now! - I snap my fingers after I've let go, while the cart still is moving up the incline - what is the direction of the cart's acceleration?

"When an object slows down, the object's acceleration is opposite the direction of motion.  This cart is moving up the incline while slowing down, so the acceleration is opposite the direction of movement - acceleration is down the incline."

(Careful with language here - I try from the very first instance to stamp out the two uses of language that lead to serious misconceptions.  I don't allow students to use "accelerate" as a verb; and I don't allow students to say that acceleration "moves" in a direction.)

And I do the demonstration.  The lights pointing down the incline light up... even as the cart moves up the incline!

Third question:  This time, I'm letting the cart go from rest, the cart moves down the incline, and I catch the cart in my hand.  While my hand is touching the cart - I snap my fingers as I catch the cart - which direction is the cart's acceleration?

"When an object slows down, the object's acceleration is opposite the direction of motion.  While you're touching the cart, the cart is slowing down, so the acceleration should be opposite the motion, up the incline."  This one is a bit harder, because students have to recognize that the cart slows down AND still is moving down the incline while I'm catching it.  But seeing me execute this motion helps them understand.  If I need to, I exaggerate the contact time, so that they can see the cart slow down for longer.

Of course, physics works - the down-the-incline lights are on until I touch the cart, at which point the up-the-incline lights instantly appear.  Nice.

Final question, which I generally save for another day once the class has practiced a good bit with acceleration concepts:  Now I push the cart up the incline and let go.  Right at the top - CLAP! - when the cart briefly stops, I want to know the direction of acceleration.

(I answer this one myself.)

The cart's speed is momentarily zero at the top of the incline.  If an object's speed is zero, its acceleration must also be zero.  So the lights on the device will flick off briefly when the cart reaches the top.

As I busy myself with mounting the visual accelerometer on the cart, I usually can hear some students grousing a bit.  "Um, you didn't use a fact," one might say.  Or, "why does acceleration have to be zero when speed is zero?"  Those who have had physics before or those with amazing intuition might complain that gravity is still acting down, so the acceleration has to always be down the incline.  The majority of the class sits and listens... they had agreed with me because I'm the teacher and I spoke confidently.  They're a bit uncomfortable that classmates are objecting.  

It doesn't matter exactly what students say.  I move on and do the experiment - experiment is always the arbiter of truth.  

And, of course, the lights emphatically do not flicker off.  Even if I make the incline really steep.  

I stay away from a force explanation for now... I ask a student to read the definition of acceleration.  Acceleration is how much an object's speed changes every second.  So, if the cart's acceleration were truly zero at the top of the incline, the cart's speed wouldn't change.  The cart has zero speed for an instant... with zero acceleration, the cart would stay at rest at the top!  And then I push the cart up the incline, grab the cart at the top to keep the cart at rest while the lights blink off.  The class sees what it means for the lights to blink off at zero acceleration - the cart's speed couldn't change!

Please don't think this demonstration is the One Weird Trick for understanding acceleration!  Yes, it helps a lot.  But getting students to stop conflating acceleration and velocity is a battle of attrition, demanding multiple methods of misconception-busting over the course of a full year.  If you can get even 25 of 30 students to get a question like "A ball is thrown upwards with speed 5 m/s, what is the magnitude of its acceleration at the peak of its flight?" correct on an end-of-year test, you are a physics teaching virtuoso.  

Newton's first law on a rafting trip

 Yesterday, my family and I took a rafting trip down the Colorado River.  

The trip began with the obligatory safety and instructional lecture.  Dylan the perky guide assured us in an enthusiastic voice that we were about to have FUNNNN!  And then when we didn't holler in ecstasy, he asked us again if we were about to have FUNNNN!  Thank goodness the parents with two enthusiastic ten year olds in tow gave the requisite response so that Dylan moved on.

In a spiel that reminded me of the boat operators on Disney's Jungle Cruise, Dylan went on to explain truly important safety information like what to do if you fall out of the raft, how not to accidentally whack your seatmate with an oar, and so on.  Next came navigational instructions.  After demonstrating proper paddling technique, he described the commands we might hear.

"When I say 'forward one', everyone paddles one stroke - only one stroke! - forward.  Everyone show me 'forward one'!"  Everyone dutifully mimed a single stroke.

"Now show me what you do when I say 'backward one'!"  The assembled masses pretended to paddle backward once.

"Sometimes I'll have you paddle 'full forward'!  What do you think that means?" Some shouts from the ten year olds suggesting that we should keep paddling forward.  "Good!"

"And finally, if I say "all stop," then hold on to your paddle, but keep it out of the water.  What do you think happens to the boat?"

My moment had arrived!  "The boat continues to move at a constant speed, due to Newton's first law!" I said loudly enough to get a Look from the fellow rafters, including, especially, from my own family members.  Dylan looked a bit hurt that I had anticipated his punchline.  But, he cheered up a wee bit when the ten year olds shouted that the boat would stop.  "No, the boat is on a river, it's gonna keep going! Dylan cheerily told everyone, while he looked sidelong at me.  Sometimes it's a social curse to know physics.  

Tune in next episode when I describe how I explained the phenomenon of concave acoustic mirrors to a flummoxed guide on our Savannah, Georgia Ghost Tour.