## Monday, July 18, 2011

### Office Hours: Anticipating Student Thinking

Teaching Situation

A student comes into office hours and wants to know why the final velocity of a block sliding down a frictionless ramp of height h is not equal to vfinal = vinitial + Sqrt(2gh)

Describe two different lines of thinking that could have led a student to honestly arrive at this question.

Describe how your discussion with the students would likely need to differ depending on which line of thinking the student had taken.

* Note: I actually observed this situation–two times in fact.

1. I'm not sure I understand the assignment, because there are at least two things wrong with the students' thinking. One is that rotational energy is ignored, and the second is that this is not the right result even for a body under free fall. Maybe there's something else as well, but I don't quite see how to disentangle those into two different lines of thinking.

In any case, I would start by asking the student why he or she thought that was the right answer, and the discussion would depend on the answer to that question.

2. Sorry, Chris, yes it's supposed to be a block sliding down a frictionless incline with some initial velocity, not a rolling ball. I've change the post to reflect that. And yes, the equation is not correct, even for free fall. The question is, "Why would a student think this should be the correct equation?"

Yes, I believe most of us would ask the student what they think, and use those responses as the basis for how to proceed. The homework is to try to anticipate what that thinking might be. The two real students who did come to office hours with this question, had two different lines of thinking. Those lines of thinking came to light exactly because the professor asked them what they were thinking. Here, it's an exercise, if you will. I don't think it's a useless exercise.

3. My first guess would be that he/she (to avoid having to write this I just call him/her "Pat") is just simply simplifying (vf^2 = vo^2 +2ah). So my first guess is that Pat is having some issues with algebra. Secondly, Pat is assuming that all accelerations are "g", not recognizing that this system isn't a free fall problem. I would ask Pat, "If I drop and object and let it slide down a very smooth surface, will they arrive the bottom at the same time?" Hopefully slowly progress with this line of questioning until Pat realizes that we need to use dynamics to determine the acceleration, that it will be smaller than "g".

4. Oh, I don't think it's useless, I was just confused. (I didn't actually catch your mistake about rolling until I started to write my original answer.)

So my first thought is that the student is using vf = vi + dv, but is making a bad assumption about the value for dv, and just using the formula for falling from rest. So I would expect the student to say he or she was using vf = vi + dv, and then I would ask where the value for dv came from. Hopefully that would bring the assumption to light, and we could talk about how the sqrt(2gh) comes from conservation of energy (with the assumption that vi = 0), and how that might be a better place to start for the entire problem.

I can't think of a second way to get there; I'm too old and have lost some of my student cleverness for doing things wrong. Maybe Scott's answer about bad algebra is right?

5. Together, you guys both got it.

One student approached the problem correctly using energy, but had a misunderstanding in algebra--simplifying the squareroot of sum of squares as a sum.

The second student was thinking something like this: I've done tons of energy problems, and it seems the final velocity always works out to be Sqrt(2gh). Now, I'm doing a problem where the there is some initial velocity, so it would make sense that the new final velocity would be equal to the initial velocity plus the change.