One of my favorite tasks for students to do is to create well-coordinated position, velocity, and acceleration vs time graphs for a bouncy ball, where careful attention is given to the moments of contact with the floor. I got this task from my high school physics teacher, but it’s in Arons’ book as well. The future physics teachers have this as a content standard in my class. There are many predictable obstacles, but the real meat I want them to get to is reasoning about the acceleration. This is so not easy for them-partially because they mostly know acceleration through special cases, and partially because they aren’t strong in thinking about vector kinematics. Mostly students say the acceleration is constant. It’s like the bounce isn’t even on their radar when considering acceleration. Part of what I like about it, is that students know enough to get started and the task itself is clear. My job when they ask to assess, is to keep them talking until they notice some inconsistency, and then to help them orient to that inconsistency and how they knew there was something wrong. Then I send them off.

In our physics department, every physics major has to serve as an undergraduate TA. Most of them get assignments in our algebra-based introductory physics course.   Because of the manner in which most of these students were taught (i.e., find an equation and substitute numbers), they can easily find themselves feeling a bit lost in my class, especially if they think they are supposed to be an expert of the content.

For example, here’s a question discussed in class. A bowling ball is dropped from a height of 45m, taking 3 seconds to hit the ground. How fast is it moving the very moment before it hits the ground? The problem is intended to draw out the following answers and arguments, which we hash out.

10 m/s, because all objects fall at the same rate

15 m/s because you can calculate the velocity as 45m/3s = 15 m/s

30 m/s because it gained 10 m/s in each of the 3 seconds

Other more idiosyncratic answers come up as well, but not with high frequency.

The first answer points to the ways in which students haven’t yet teased apart clearly the meaning of acceleration and velocity. The second answer points to the ways in which students haven’t yet teased apart clearly the meaning of average and instantaneous velocity. The third answers is consistent with the idea of constant acceleration. We hear arguments, and counter-arguments, and at some point I help clarify the right reasoning, and what’s both so tempting and subtly wrong about the other answers.

So, here is the way the TA solved it, before class began.

xf = (vf + vi)/2 * t + xi

0 = (vf + 0)/ 2 * 3 + 45

0 = 3/2 v + 45

-45 = 3/2 v

v = – 30 m/s

While the TA could solve this problem, they didn’t have a rich set of ideas for thinking about. It didn’t seem obvious that 30 m/s makes sense, because of the idea that its 10 m/s/s, or because final velocity sould be twice the average velocity (since it accelerated from rest). For other questions without numbers that we discussed, the TA seemed just likely as students to give answers inconsistent with the concept of acceleration. I’m perfectly OK with that, but my suspicion is that the TAs aren’t prepared for this. They aren’t prepared to be wrong about so many things or confused about so many things. I wonder how I can better position them as learners in the class–learners who just know somethings that the first-time students don’t, but not everything.

Of other interesting note is this. In my physics content course for future physics teachers, the students that have had me for a semester or two are pretty rock solid on having a repertoire of ways of think about kinematics problems, and also for avoiding common pitfalls. The others are pretty much falling for all the pitfalls. The difference is pretty striking. The thing that I like is that the range of expertise we have allows for peer-coaching, but also some, “Hey, it’s OK. We were making those exact same mistakes 4 months ago,” and, “Yeah, get used to it. Brian isn’t too into solving problems by putting numbers into equations.”

One way I’ve gotten some decent leverage in getting students to really initiate with goal-less problems in early kinematics problems is to ask, “If you had taken this trip with a GPS device, what summary trip information could it give you?”

As a class we generate a list of things the GPS device would or should be able to calculate like:

Time of Arrival

Duration of Trip

Final Location

Total Distance Traveled

Average speed (while moving)

Average speed (during entire trip)

I usually tack on a few things, like it could tell us how far we are from where we started (and in what direction). I also say that a good one should also be able to make a graph. I may or may not introduce average velocity.

My job at some point is to connect each of these to formal language and algebraic symbols used in our text. Unfortunately, our text is sloppy with clock readings vs. time intervals. It’s also sloppy with displacement and position. So it’s a little difficult.

I also think it’s cool to have “average speed while moving” be something that the physics text book doesn’t have, and that we’ll have to completely invent our own way of calculating it.

We don’t explicitly teach the notion of a velocity at an instant in our introductory algebra-based course, although the term instantaneous appears in the text every now and again. The concept is, of course, implicit in everything we do beyond chapter two. I have noticed this past year, the negative consequences for students’ understanding of velocity, acceleration, forces, and energy.

When I have mentioned to several colleagues this lacking in our introductory curriculum, a common question that arises is how to think about teaching velocity at an instant (in an intellectually honest way) without calculus.  In this post, I don’t intend to describe how to teach the concept, but to illustrate about how clear the ideas of constant, average, instantaneous, and change in velocity were in Galileo’s mind even without with the mathematical machinery of calculus. Here is a excerpt from a translation of Galileo’s Two New Sciences.

“When I think of a heavy body falling from rest, that is, starting with zero speed and gaining speed in proportion to the time from the beginning of the motion; such a motion as would, for instance, in eight beats of the pulse acquire eight degrees of speed; having at the end of the fourth beat acquired four degrees; at the end of the second, two; at the end of the first, one: and since time is divisible without limit, it follows from all these considerations that if the earlier speed of a body is less than its present speed in a constant ratio, then there is no degree of speed however small (or, one may say, no degree of slowness however great) with which we may not find this body travelling after starting from infinite slowness, i. e., from rest. So that if that speed which it had at the end of the fourth beat was such that, if kept uniform, the body would traverse two miles in an hour, and if keeping the speed which it had at the end of the second beat, it would traverse one mile an hour, we must infer that, as the instant of starting is more and more nearly approached, the body moves so slowly that, if it kept on moving at this rate, it would not traverse a mile in an hour, or in a day, or in a year or in a thousand years; indeed, it would not traverse a span in an even greater time; a phenomenon which baffles the imagination, while our senses show us that a heavy falling body suddenly acquires great speed.”

OK. So while Galileo didn’t have the mathematical machinery of calculus, he certainly had many of the ideas:

“Time is divisible without limit”

“There is no degree of speed however small with which we may not find after starting from infinite slowness”

“So that if that speed which it had at the fourth beat was such that, if kept uniform, the body would..”

“As the instant of starting is more and more approached, the body moves so slowly that, if kept moving at this rate, it would not…”

A key idea is the hypothetical, “So that if that speed which it had was such that, if kept uniform, the body would…” Another words, Galileo is specifically thinking about speed at an instant by considering how far it would go in a measure of time if the speed it had at that moment was not allowed to change. Of course, this idea is really the same as slope of the tangent line idea. We zoom in on a moment, fix the rate of change, hypothetically extend a line with that rate, and measure how far that line extends vertically in a fixed measure of horizontal change.

A Second Excerpt

Here Galileo describes ideas about the relationship between velocity at instant, displacement, and average velocity:

Let the line AI represent the lapse of time measured from the initial instant A; through A draw the straight line AF making any angle whatever; join the terminal points I and F; divide the time AI in half at C; draw CB parallel to IF. Let us consider CB as the maximum value of the velocity which increases from zero at the beginning, in simple proportionality to the intercepts on the triangle ABC of lines drawn parallel to BC; or what is the same thing, let us suppose the velocity to increase in proportion to the time; then I admit without question, in view of the preceding argument, that the space described by a body falling in the aforesaid manner will be equal to the space traversed by the same body during the same length of time travelling with a uniform speed equal to EC, the half of BC. Here Galileo has essentially constructed a velocity vs. time graph for an accelerated object (turned on its head), and is arguing that distance traveled during an interval of time is equal to the distance an object with uniform speed would cover if the speed was half the speed the accelerating object acquired at the end of that same interval. Not only did Galileo have in place many of the ideas for thinking about limits and rates of changes, but also the beginnings of integral calculus.

Hypothetical Velocities

The claim I want to make here is that Galileo made sense of instantaneous rates of change and accumulation via two different hypothetical constant velocities he had to imagine.

Instantaneous velocity at a given moment of time was construed as the distance an object would travel in a measure of time if the velocity it had at that moment were kept constant.

Accumulation was conceived as distance traveled by an object with a hypothetical constant velocity, here described as the velocity half of the velocity obtained at the end of the time interval.

And this is what makes instantaneous velocity so difficult to comprehend… it is a discussion of a hypothetical moving object and its relationship to a real moving object. Instantaneous velocity is thus a huge imaginative leap of faith–one in which we imagine a differently moving object and aim to establish some relationship between the imagined object and our real one. Instantaneous velocity requires a suspension of reality–an acknowledgement that you aren’t going to talk about what actually happened, but to make an explicit analogy between reality and an imagined one.

If I’m right, that instantaneous and average velocity are merely analogies to imagined hypothetical objects moving at constant velocity, then it seems that pinning down the meaning of constant velocity becomes even more important. Galileo, refined his definition of uniform motion over time:

### Definition

By steady or uniform motion, I mean one in which the distances traversed by the moving particle during any equal intervals of time, are themselves equal.

### Caution

We must add to the old definition (which defined steady motion simply as one in which equal distances are traversed in equal times) the word “any,” meaning by this, all equal intervals of time; for it may happen that the moving body will traverse equal distances during some equal intervals of time and yet the distances traversed during some small portion of these time-intervals may not be equal, even though the time-intervals be equal.

A question I’ve gotten a lot of leverage out the past two semesters is the following one:

You toss your keys straight up to a friend, who is 30m above you leaning out over a balcony. They keys leave your hand with a speed of 25 m/s. Will it get to your friend?

Sure this is a standard boring question. What makes it work is how the show is run. We start off by listing our best guesses about whether it makes it up and the top height they think it gets to: Their answers this semester ranged between 19m and 40m.

In my class, I actually work out this first answer for them (because I’m supposed to model a sample problem), but I ask for their help along the way.

First, I draw a motion map showing how the speed changes at 1s intervals, and we talk about the speed going from 25m/s to 15m/s to 5 m/s, etc, and how the time to the top is when v = 0 m/s. We talk about how much time it takes to get to 0 m/s if you are losing 10 m/s each second: it takes 2.5s to lose 25 m/s. We also talk about the average speed during the trip (12.5 m/s, half way in between 0 m/s and 25 m/s). This, of course, all builds on ideas we built up last week when talking about 1D acceleration problems.

The answer is immediately given as 12.5 m/s * 2.5s = 31.25m

The best guess this time was 32m, and kudos were given to that group.

Lot’s of students then want to talk about why it’s not 40 m (25m + 15m + 10m), and we get to talk about what constantly changing velocity means.

Because of class constraints, I typically re-derive the 31.25m in a way that is more typical of how they are expected to do it: Write down your knowns and unknowns and pick an equation or two to plug away with.

I then send them off to work on the next question. How fast are the keys moving by the time they reach your friend’s hand? Our guesses range between 1.25 m/s and 2.5 m/s.

The right answer is 5 m/s. And students are pretty surprised to find out that we all underestimated the speed. Every group got the right answer. Most students solved the problem by plugging away into equations. One group did so, but didn’t believe that 5 m/s was right, and so they took another approach, using two equations instead of one.

One group took this approach:

In the first second, the ball slows from 25 to 15, with an average velocity of 20 m/s. Thus in the first second, the ball covers 20 m. In the second second, the balls slows from 15 to 5, with an average velocity of 10 m/s. Thus in the second second, the ball covers 10m. That’s 30m covered, with a final speed of 5 m/s. That same group realized that for the first 2 seconds, the average speed was 15 m/s for 2 seconds, also giving 30m of travel.

Last semester, I had a group solve the problem by finding the speed of a ball dropped 1.25 m/s, arguing on the ground of symmetry that it had to be the same.

We ended the problem this semester by talking about the last 1/2 second, where the ball has an average speed of 2.5 m/s for 1/2 second, thus covering the final 1.25m, and why our guesses for the speed were so off.

Simple problem, but lots of places for intuition, lots of places for multiple approaches, and lots of opportunities to talk about velocity, distance, average velocity, and acceleration.

This week’s online pre-class question:

An object is dropped from a height of 45m and takes 3 seconds to hit the ground. Explain why someone might think the object’s speed just before hitting the ground is 15 m./s. Then explain why that can’t be correct.

Three responses representing very different places students can be:

“First of all, wow! That’s the exact answer I had in mind and that is because if it’s dropped from a height of 45 meters and it takes 3 seconds to hit the ground you would want to divide the 45 meters by the 3 s to speed per second (15 m/s), but that is if it was going at a constant speed. So you also have to keep in mind that it was dropped at rest/zero so the speed will increase slowly not constant. I’m still confused.”

“Someone might think it is that because they would divide distance (45m) by time (3 s) which would come out to be 15 m/s. But that would be the average speed.  To find the final speed you would take the initial speed (0 m/s) and add it to the acceleration (9.8 m/s^2) multiplied by the time (3 seconds). The final speed of the ball before it hits the ground would be 29.4 m/s.”

“Because most people would think just divided 45 into 3 to get 15m/s but we haven’t put in  our minds about the acceleration of gravity, which is 9.8m/s that can round up to 10m/s then if you was to times 10m/s by 3s you would get 30m not 45m.”

Here is Galileo (in the Two New Sciences) writing on some properties of naturally accelerated motion which are worth knowing:

“That the distances traversed during equal intervals of time by a body falling from rest, stand to one another in the same ratio as the odd numbers beginning with unity

Amid deepening consultation with Galileo, I am inclined to think that non-calculus-based physics would benefit from a framing in terms of integer sequences and series rather than the changing of d’s to Δ’s.

Acceleration as Simple but Vast Idea

So take the definition of average acceleration: a = Δv /Δt. Simple right? Well, yes, but not really. In order to really understand this definition, you’re going to have to explored a vast array of ideas, including ideas like

If an object is slowing down, its acceleration is in the opposite direction of its velocity

If an object is speeding up, its acceleration is in same direction as velocity

If an object is turning, there will be (a component of) acceleration in the direction of the turn.

It takes time to speed up or slow down. Some things speed up quickly and others speed up slowly and this has to do with an object’s acceleration.

Given a constant acceleration, speeds change linearly –an object gains (or loses) the same amount of speed in any equal interval of time.

A greater change in velocity in the same amount of time indicates a greater acceleration

The same change in velocity in less amount of time indicates a greater acceleration.

A greater acceleration will result in a greater change in speed in an equal amount of time.

A greater acceleration will result in an object taking less time to change its speed.

When an object is accelerating, the distances covered are not equal for equal times; objects cover more ground during times in which its moving faster and less when it’s moving slower.

Also, if you’re human, you’re going to have to become aware of and then wary of, a variety of other possible problematic ideas, like

Faster objects have more acceleration

A greater increase in speed means more acceleration

No velocity means no acceleration

Objects moving with same acceleration move in identical ways.

And we haven’t even begun to worry about having procedures for determining or estimating velocities at particular times or procedures for subtracting two velocities. We haven’t concerned ourselves with when it might be appropriate to model a situation with constant acceleration. We haven’t concerned ourselves with the difference between average and instantaneous velocity, or with strategies for selecting convenient intervals of time for carrying out one’s work. We haven’t talked about graphs and other representations. It turns out that acceleration is a high density idea.

Some Place Generative to Start:

Changing topics a little bit, one of the questions I have been thinking about is this: Given that there are so many ideas packed into definitions such as acceleration, which ideas are most generative? That is, which ideas serve as a good starting point for generating the entire set of ideas? Given such a good starting point, are there other ideas that come along for the ride? And I don’t mean logically generative–like you could derive certain ideas from others. I mean generative from a human learning perspective. What ideas serve as productive anchors or as productive leaping off points… So, now, I’m think, “Isn’t it odd to juxtapose the words anchor and leaping off point?” Like, one implies, “keeps you grounded somewhere.” The other implies “strong base from which to leave.” Those are totally different metaphors for generative starting place.

I also think about this a lot: Does the generative starting point need to be correct? or like a baby-version of correct? If I go with the anchor analogy, then yes, the generative starting point should be correct. It’s like “home base”–the place you are tethered too so you don’t get lost. It’s a trustworthy place to ground your thinking. But if I use the leap-pad analogy, then the most generative starting point can actually be a place you never return. It’s the place that launches you to the next place, which may be quite different, and possibly even wrong. I think we tend to operate under the tacit assumption that the starting points should be “anchors.” I think we have a hard time thinking about what a generative (but incorrect) launch pad would look like. I know I do. But still, I keep returning to the idea, because it has so many implications for how we might think about teaching, learning, and assessing progress.