Is there a vi hidden in that t?

In a previous post, I asked readers to consider how a student might end up with the following equation to describe a block sliding down a ramp of height h: vf = vi + sqrt(2gh)

I suggested there were two student approaches–both observed two years ago in an algebra-based physics course. The two approaches that were identified in the comments were exactly what I observed in real life:

Scott suggested that a student might have correctly approached the problem using energy principles and then simplified the algebra wrong–turning the square root of a sum of squares into a sum. Sure enough, one student in office hours last year made this mistake.

Chris suggested that a student might have been started thinking from the idea that final velocity is equal the initial velocity plus the change (i.e., vf = vi + dv). Sure enough a different student took this approach: having seen lots of energy problems where vf = dv = sqrt(2gh), he thought it would be reasonable to just add this change to the initial velocity.

I actually like this second mistake. Why? I think it's a reasonable mistake to make, it reflects underlying practices that are sophisticated, and it provides a generative opportunity for learning. Hear me out on this.

Despite the fact that we'd like students to start from first principles (i.e., energy conservation), many intro physics students simply start problems by following some mindless routines we've taught that mostly involve plugging numbers into equations. This student, however, seemed to be thinking about how the world and the math might work, and he was trying to figure it out without an equation from the textbook. He was going to construct his own equation. More specifically, he was thinking about how quantities change, which is one of the most important things for physics students to think about. He wasn't treating math as stuff to plug numbers into; he was thinking about math as a way to think about and express quantities and how they change.

Second, rather than treating every problem as completely new and different, he was trying to draw on previous results to simplify his approach to a new problem. Now maybe this drives you insane--once again, maybe you want all of your students to start from first principles every time. But the point is, this student was looking for possible connections between different situations and problems, and trying to generalize results. This is something we do all the time and it is quite sophisticated. Sure, when things go wrong, we go back to first principles. But to me, what makes the student approach even more sophisticated is that the student came to office hours, knowing that the equation was wrong. In fact, he had gone back to derive the right equation separately from first principles, but he still couldn't figure out why his first approach didn't work. It made sense to him that the final velocity should equal initial velocity + the change. This student knew the right answer, but wanted to know why a different approach–one he had invented and made sense to him–didn't work.

Where we went from there?

What I ended up telling the student was this: He was right. The final velocity does equal the initial velocity plus the change, but that we had to carefully consider what the change should be. I asked the student to recall what he had learned earlier in the semester.

vf = vi + at.

We discussed what this equation meant in terms of changing quantities, which was easy for him--because he was already in the mindset of thinking of mathematics this way. From this equation, we concluded the more time the ball spent on the ramp, the more the final speed it would have by the time it reached the bottom. So I started asking him how we could get the ball to spend more or less time on the ramp. We discussed several possibilities:

We could make the ramp longer or shorter
We could make the ramp more or less steep.
We could make the ball start with more or less speed

I focused our conversation on this last one, and offered: "That's weird. The equation is vf = vi + at . Vi only appears in one place to represent how much initial speed the ball has, but the time t spent on the ramp is also influenced by the initial velocity vi. "

He pondered over this for a while, looking puzzled before and finally asking, "So is there a hidden vi in the t?"

I loved this question.

I'm not going to go into all the details. But from there we spent a long time discussing and working through various mathematics to show that the reason that dv could not equal sqrt(2gh) is because dv depended on t, which depended on vi, which made vf depend on vi in a weird (non-linear) way. This, I think, was a way better learning opportunity (for both of us) than what might have happened had I simply validated his approach using conservation of energy.