Conservation of momentum compares a total value before and after a collision. It does not compare the momentum in one collision to the momentum in a different collision.
That was the fundamental error on the 2022 AP Physics 1 paragraph question. Well, the same concept showed up again on the 2023 AP Physics 1 exam, on problem 1. But this time, it was about conservation of energy. These questions are both checking a student's understanding of what "conservation" means in each context, in each system being considered.
The pumpkin pie I* baked has a mass of 2.0 kilograms. The total mass of pie is 2.0 kg whether it's cut into four 0.50 kg pieces, or eight 0.25 kg pieces, or... all the pieces are always gonna add to 2.0 kg. That's conservation of mass.
*well, that my wife Shari baked. And therefore it's probably a pumpkin cheesecake, not mere pie. Mmmm.
Conservation of mass does NOT mean that every pie in the universe has 2.0 kg of mass.
In the 2023 P1 problem 1, the maximum potential energy of a spring-cart system is 4 J. Part (a) of the problem asks for an explanation of why the maximum kinetic energy of this system is also 4 J. That's conservation of energy - the total PE + KE will be the same for any cart position. Maximum PE means zero KE. Then when PE is zero, all 4 J of PE has converted to KE.
But part (c) of this problem has a block dropped on the cart when the spring is at its maximum stretch. The maximum PE (and the maximum KE) of the cart-spring-block system still is 4 J. Why? The answer is emphatically not "conservation of energy."
In this case, we're asked to compare the maximum potential energy of two different systems: the cart-spring system from part (a), and the cart-spring-block system. We're not comparing the mass of my pie-cut-in-two-slices to the mass of my pie-cut-in-four-slices; we're comparing the mass of my pumpkin pie to that of someone else's lemon méringue pie. Two different pies.
Now, in this particular case, the pies both have 2.0 kg of mass the systems both have 4 J of total energy. The potential energy of both systems is due to the stretched spring, and thus is given by the formula (1/2)kx^2 and is not affected by the system mass. The spring is stretched the same maximum distance, so the system potential energy is 4 J either way.
Great points about using conservation laws correctly, Greg.
ReplyDeleteFor part (cii), how would you draw an energy bar chart representing the system of only the cart and the spring when the block now has 3/4 of the spring's energy but it's not in the system?
Ryan, before I answer, I just want to point out to readers that the actual 2023 P1 question asked for a graph of KE vs. PE for the spring-cart system. No energy bar chart was necessary. Your question is really fascinating, but was NOT part of the AP Physics 1 exam in 2023!
ReplyDeleteOkay, here's what I see. At the max extension, the spring-cart has 4 J of potential energy and no KE. At the equilibrium position, the spring-cart has 1 J of kinetic energy and no potential energy.
The question that I suspect Ryan is really after is, how do we account in an energy bar chart for the remaining 3 J? Yes, they have become kinetic energy of the block, but the block is external to the system, and thus can't be included on the bar chart for the spring-cart. Nor can these 3 J be chalked up to internal energy - again, because the block is not part of the system, so nothing involving the block can be "internal."
There must be an external force doing negative 3 J of work on the cart-spring system in order to reduce its mechanical energy from 4 J to 1 J. That conclusion comes straight from the energy bar chart and "bars + bars = bars", that is, conservation. But what is that force?
While the cart and block move from maximum spring extension to equilibrium, they both speed up together. Consider a free body diagram of just the block on top of the cart. The cart exerts a (probably frictional) force in the direction of motion, causing the block's acceleration. By Newton's third law, the block also exerts a force on the cart opposite the direction of motion.
And there you go. The cart experiences four forces. The normal force and the weight are perpendicular to the direction of motion, so do no work. The spring is part of the cart-spring system, so is not considered in the "external work" column in the energy bar chart. The force of the block on the cart is opposite the direction of motion, so does negative work. And since the block is not part of the cart-spring system, this work is represented in the "external work" column.
TLDR: The bar chart for the cart-spring system looks like:
* At maximum spring stretch: 4 bars potential energy, 0 bars kinetic energy.
* Work done by external forces: -3 bars, due to the force of the block on the cart.
* At equilibrium: 1 bar kinetic energy.
My one li'l gripe about this problem is that the answer comes out a little too close to, "Because this here formula says so," for my liking; an answer along the lines of, "dropping the second block on top of the first doesn't change how stretched the spring was" or "doesn't change the amplitude of the oscillation" is a stronger causal argument for there being the same total energy in the system before and after the second block arrives. Alternately stated, that means there is "zero change in the system's total energy." Which, if you're going to split hairs about using the words "conservation" in an introductory physics course, is a wee bit pedantic. I was just happy if they didn't copy the 2018 paragraph answer and say "the collision slows it down because kinetic energy's lost in an inelastic collision." Or similar.
ReplyDeleteWill, the first two answers you quoted each earn full credit. And I don't believe this question is in any way "splitting hairs" about language. Would we accept a student calling any old equal and opposite forces a N3L pair? It's the same thing. The question is expressly testing the misconception that I describe in the post.
ReplyDeleteNow, if you're worried about pedantic grading, please don't! The AP exam is graded by physicists, not robots. A student who first explained that the total energy depends on the unchanged spring stretch, THEN explained that conservation of energy keeps the graph identical at each position, earns credit. They talked about conservation of energy correctly. But a student who says that the total energy is the same in the new case than the old case "by conservation of energy" is not demonstrating understanding of the situation. And a student who quotes the old saw that "energy can neither be created nor destroyed" without applying said saw correctly in context is not communicating physics understanding.
Oh, I'm not concerned with the graders, per se; physics teachers are very good in general at spotting actual knowledge vs. rote recitation. It's more about the grading rubric they're working with. And I know that's not the part we like to talk abut---we'd much rather talk physics knowledge. But it's been the achilles heel of the AP1 exam. The 2018 FRQ5 is a good example. The first part is at least 2 mathematical steps, but only earns 1 point. And the paragraph earns 6 points, all of which are instantly lost if the student doesn't recognize the inelastic collision. On 2023 FRQ1 the problem is different, only slightly; if a student says the underlying principle is conservation of energy but does not clarify the difference between "same energy" and "energy is conserved," they're in trouble.
DeleteIf the rubrics were qualitative--e.g. 5 points for a completely correct answer, instead of parsed out point by point for each learning objective met--then I'd have more confidence in the process, because it would be more about the student showing what they know and less about trying to "hit" every point.
Will, I encourage you to come to the reading. I personally love to talk about the AP Physics 1 rubrics - I've been leading their design and implementation since 2015. The fundamental design goal of a rubric is to consistently award credit for good physics, and to consistently avoid awarding credit for poor physics. Most of the P1 rubrics have been quite successful on this account.
DeleteWhen you join us at the reading, you'll have the chance to see the elegance of how a rubric applies consistently and appropriately to hundreds of thousands of responses. And, you'll be able to talk directly to the people who design the rubrics. I think you'll find your concerns addressed appropriately - and that you don't have so many concerns once you see first-hand how the process works.