I’ve elaborated some, in the hopes of clarifying, but here is the gist of the students’ question:
When you throw a ball up in the air, we know it loses kinetic energy, because it’s obviously slowing down. We also know we can describe that loss of kinetic energy in terms of the work being done on the ball due to the gravitational force from earth. But what about the rate at which energy is being lost? Since we also know we can determine the rate at which energy leaves the ball by considering the quantity F.v. , then we can say when the ball is moving fast it loses energy quickly, and when the ball is moving slowly it loses energy slowly. This also makes sense from a potential energy perspective, because when you are moving fast you cover more distance, so the potential energy term mgh also changes quickly. But right at the top, when the ball is not moving, it has zero kinetic energy; but also the rate at which energy is being transferred to / from the ball is zero.
Question: How does the ball go from having no kinetic energy to having some kinetic energy a moment later? Think about it. For the ball to have kinetic energy, it has to be moving; but for it to get moving there needs to be a flow of energy; but for there to be a flow of energy, it needs to be moving already.
Teach. Brian. Teach. 
That’s a good question. My immediate (and not satisfying) answer is that the second derivative of the kinetic energy is not zero.
BTW, I think you can recast this question in terms of position and velocity. At the top, the position is not changing and the velocity is zero. But for the velocity to become nonzero, the position must be changing, which its not. So casting the conundrum in terms of energy is not necessary.
Somehow, I think the ghost of Zeno of Elea is watching over my shoulder.
Now that I have caught up on my paradoxes, I understand better your last sentence here. Plenty to think about fo sure.
Those were exactly my initial thoughts as well, both about the 2nd derivative and then about position, velocity. One difference I see between the two, however, is that we don’t often think of velocity as the mechanism by which position changes, but we do think of work as a mechanism of energy transfer. So, the “causalness” of the chicken-or-the-egg problem seems less potent for the kinematics only version.
From the momentum-is-king perspective, this is no big deal. Energy doesn’t exist, it’s just a handy way of figuring out momentum transfers. The earth and ball are continuously exchanging momentum, in fact, the rate doesn’t change. The fact that at one point the ball is at rest is no big deal.
I also thought about this kind of thing, but from a slightly different perspective. I was thinking the rate of momentum transfer is a frame-independent quantity…(well at least for inertial frames) whereas rate of energy transfer is a frame dependent quantity… This also reminds me of this: A turn around point does no define a unique event (in the SR sense)… Every point along the trajectory could be describe das turn around for some frame of reference.
On the other hand, what I don’t like about the momentum answer is that it doesn’t try to sort out what’s possibly wrong with the argument on its own terms; it instead offers a different argument.
great point about the various frame approach. As far as “offering a different argument” I was trying to address that with the “energy doesn’t exist” line. It was a lot of fun arguing with my students about that last semester.
This is a beautiful question. Whatever the “right” answer from a physicist’s perspective (or a mathematician’s), it is precisely the sort of inquiry you hope to foster, right? Truly wonderful.
From a pedagogical perspective, I’m curious whether this question came up in class and what kind of discussion it generated in class as a result.
On the last day of class in my math content for elementary teachers course, a student strategy set me up for about 10-15 minutes of public reflection. I talked to them about the tension I was experiencing between (1) wanting to honor a really interesting and correct way of thinking, and (2) pursuing a different line of reasoning that would get us to a predetermined destination. We had some back-and-forth and they took a lot away from the experience.
What’s my point? That if you are so fascinated with your student’s question, I hope some of that interest got publicly aired in class. Your students would dig it.
Chris, yes, the question itself is exactly the kind of inquiry I’m looking to foster and support. The specific question here arose outside of class, after the semester was over. The student tracked me down to talk to me about it. We had a really interesting conversation, from which we mostly just better articulated the question than we did settle its answer, which is always nice.
However, the seeds of that this question did arise from discussion in my intro physics class… it came about because we used a particular representation in class to talk about energy, which led several students to have some earlier questions which generated this one.
I don’t want to brush the argument under the rug, but I would say something about how I can’t think of a way for this model to describe this situation.
Turning to other models, like momentum-transfer mentioned above, allows us to clearly describe and predict what will happen in this case.
Taking a crack at the argument on it’s own terms, the v, in that moment is really dx/dt, an infinitesimally small span of time. We can look at a graph and point to single instants, using the idea of a derivative, but can we do the same in physical spacetime? I’m thinking that you can only have an arbitrarily small time or position interval, but never truly consider a moment or a “point” location by itself.
My thoughts have wandered to both of these places as well… something insightful and worrisome about both possibilities.