Yesterday, in my physics class, we discussed this question
A ball is tossed vertically downward from the same height that an identical ball is also dropped. Which ball will gain more speed before hitting the ground? [Ignore friction, air resistance, etc]
Here were our ideas:
“Almost to Top Speed Already”: The answer may depend on the exact height and the speed. However, since the thrown ball might already be close to the maximum speed of falling, it might not change all that much, while the dropped ball will have a lot of speed to gain to get up to that speed. The dropped ball gains more speed”
“The Rich get Richer”: If you start with more speed, your going to accumulate more speed. So the thrown ball will gain more speed.
“Gravity is the Same”: While the thrown ball will have a greater final speed when it lands, the changes in speed will be the same because the acceleration due to gravity is the still the same constant acceleration. The two gain the same amount of speed.
“It takes Time to Speed Up”: Acceleration happens over time. The thrown ball spend very little time in the air because its moving fast, and the dropped ball takes more time because it started slow. The dropped ball having more time, gains more speed.
What do I think of these ideas?
I’m not going to lie. In the “speed limit” idea, I love their close attention to discriminating change of speed, initial speed, and final speed. The students are wrong about the final speed (and about how objects freely fall), but that shouldn’t detract us from seeing some underlying value in the reasoning they are doing. Students easily confuse velocity and change in velocity, but these students are keeping that distinction clear here. Their reasoning would be correct, if the world worked a different way.
I’m not going to lie. I also love the “rich get richer idea”, because lots of things in the world work like this. It is the conceptual underpinning of exponential growth. The thinking applied here certainly contradicts the idea that constant acceleration is a linear accumulation model. I noticed as an instructor that we’ve seen thinking consistent with this idea before. As an instructor, I was also thinking, “I’m going to want this kind of reasoning later”, which is why I helped students give it a name: “The rich get richer.” It happens to be wrong in this situation, but the thinking is so useful, I couldn’t let is slip by unattended to.
I’m not going to lie, I’m torn about the gravity is the same idea. For some students, I think it might have been just an unthoughtful knee-jerk reaction that “gravity is always the same.” However, there is good reasoning here, especially for those students who were actually reasoning through this, thereby making a connection between acceleration and change in velocity, and also distinguishing velocity and change in velocity. Earlier in the year, most students though “more velocity means more acceleration”. So, the fact that they were seeing acceleration as connected to change in velocity (not just having velocity) is progress. This idea, would be true, if acceleration accumulated over distance.
The last idea is correct (and subtle). In this case, I love it, because of the way it happened in class. A student, who had initially thought that two would gain the same, was listening to the another student explain why the faster one would probably gain more, when she suddenly bursted out loud with excitement to share her idea that it’s not “the faster you start, the more speed you’ll gain”, but it’s “the more time you have to speed up, the more speed you can gain”. That idea came spontaneously, and you could hear in her voice the certainty and joy she was experiencing about this new idea.
The question itself
I’ve been asking this question for a few years now. I actually don’t know where I got it from. It’s a great discussion question, which I suggest you save in your pocket until after students have spent a fair amount of time discussing acceleration and free-fall. It’s subtle and fun, and easily trips up members of the human race, even if they be physics majors, physics graduate students, or physics professors.
The question itself also seems to spontaneously generates new questions that students begin to discuss. For example, a group of students came to realize that the answer would be the same if the question had asked, “After 3 seconds of falling, how does the gain in speed compare?” because the two would gain the same amount of speed in the same time. In working through this, students were making contact with rabbit hole of ideas that Galileo struggled with. In fact, on this blog, I’ve talked about what the world would be like if constant acceleration meant dv/ds rather than dv/dt, that is, if velocity accumulated in equal amounts over equal distances.
We are all struggling with the same ideas: Galileo, my students, and yours truly.
Teach. Brian. Teach. 
I really like comparing and contrasting impulse and work. I’ll have students compare situations where two objects of different speed experience the same force for the same distance, on the one hand, and the same force for the same time, on the other hand. Of course, using momentum is king helps with impulse, because it make clear that force (and hence acceleration) is cumulative over time, just as the last student quote makes clear. I also think that a lot of the conversation above still holds even if it’s not a constant force, so that’s cool too.
My favorite part of this post, though, was how you explained why you enjoyed how the “rich get richer” was a part of the conversation. I’ve learned a lot from you, Brian, about how to make use of preconceptions of ideas, and this is a great example. Thanks!
Yeah, I agree impulse/work questions also help to draw out lots of important distinctions. I think I first encountered those kind of question in UW tutorials.
I think part of recognizing how to make use of students’ ideas (even when incorrect) is to keep in mind what’s at the horizon. I find that If I am too focused on the immediate question or concepts at hand, I can miss the possibility that what the student is doing might be really valuable. Of course, simultaneously keeping in mind the immediate situation and what’s at the horizon can be hard, especially in the moment. I do, however, find that blogging helps a lot. Sometimes, I won’t realize a connection “in the moment of class”, but I’ll realize it later. My hope is that next time, in class, I’ll be more ready to hear and see that connection.
“This idea, would be true, if acceleration accumulated over distance.”
– Galileo struggled with what would be the best way to define acceleration – change in speed over time, or change in speed over distance. I think (?) his rationale for finally deciding it should be time was that when a ball is dropped with an initial speed of 0, time will pass, so acceleration will happen, while distance won’t “pass” so there shouldn’t be acceleration… as I type that it sounds wrong. But the point is, I like answer “Gravity is the same”!
Yeah, and as Andy points out above, energy accumulates over distance. So if the question had been, which ball gains the most energy… the answer would be the same… but energy accumulating over distance has exactly the problem of distance doesn’t pass when v=0, and somehow the energy still changes. I wrote about this in an earlier post, where a student was troubled by this.
I wrote my response before finishing reading your post… and perhaps I knew about Galileo from your earlier post. (I also started a recipe today without reading all the way through it first, and – predictably – have mis-timed everything. The blunder, then, is not limited to just blog posts.)
My Honors kids (HS sophomores) were working on some projectile motion problems today. One of them asked a question that reminded me of this question, so I posed it for them. Really interesting to hear them discuss it. There were only two (voiced) competing ideas: the acceleration is the same so the ∆v is the same —vs. the acceleration is the same but the ∆t’s are different, so the ∆v’s are different. I was especially excited that they appealed to velocity graphs to try and convince others (it went something like… they have the same slopes and areas, but one is a triangle and one is a trapezoid, so they must have different bases (∆t’s) (something everyone already agreed with anyway) and so they must have different rises (∆v’s)).
Whether they thought it or not, no one voiced the first two ideas that you mentioned (though only 10 kids in the class).
I’m not surprised those first two ideas didn’t come up, especially if you’ve been covering motion concepts for a while. We had 2.5 days of kinematics, before I popped this question, because it was the last day before moving onto the forces. I’ll say I had never heard the “rich get richer” idea before; although I had heard the “reaching speed of gravity” idea.
Despite this, it sounds like this was still a great question for your class, both in terms of deeper learning about acceleration, change in velocity, and time, but also great how they used the disciplinary tools of your classroom (representations and discourse around graphing) to settle the issue. I love the “triangle” vs “trapezoid” argument.