For the seasons unit, I’ve done a fair amount of giving students data sets to graph, looking for patterns, similarities, difference. We have been doing so in order to build evidence for or against various claims about what could cause the seasons. I think we’ve learned a lot along the way.

Anyway, there are two observations that have driven student a fair amount of engagement, and I don’t want to forget them:

#1 McCurdo Station in Antarctica has the sun shining on it for 4 straight months, but its average temperature is still below freezing during that time. [If duration was only factor, then we’d expect McCrudo Station to be very hot]

#2 In June, Murfreesboro, TN is 20 degrees hotter than Quito, Ecuador. [Shouldn’t the equator always be hotter?]

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.

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.

Here is an introduction from a student project in my physics course, who investigated issues of symmetry in projectile motion:

“Why choose this subject to investigate? Out of all the options to research, why this one? The main and best reason I can give to explain why I chose this is just out necessity… In class, …through visualization [of motion diagrams]we began to realize a possible connection between the upward and downward segments of the path. We started to see the motion as reversing itself after the object reached the top of its path. It was from this, that we as a class began to form that idea that if you throw something up at a specific speed, then when it comes back down and gets to the same height that it must be moving at the same speed.”

And here is another introduction from a different student studying the same phenomena

“The purpose of our experiment was to determine if the speed of a ball being thrown up is equal to the final speed of same ball going down. The motivation for this experiment was in part based on Galileo’s own experiments with gravity. Galileo, an Italian physicist, determined that the force of gravity is constant and objects fall at a constant acceleration toward the earth. He determined this by dropping two cannonballs of different size off of the Tower of Pisa. The law of parabolic fall states, “The distance traveled by a falling body is proportional to the square of the time it takes to fall.”

There are likely many different things to see and ways of responding to these different introductions. But, these two different introductions tell the story about the difference between ideas and concepts. Kevin Pugh, an educational psychologist, writes here about ideas: “Ideas are possibilities that must be acted upon and tried out… Ideas are ways of being in the world… They are inseparable from human experience.” Writing about concepts, he states, “Concepts are established meanings (classics)…When intellectual products attain classic status, they become isolated from the conditions in which they had an original signiﬁcance and from their potential consequences for everyday experience. As a result, their importance is reflexively accepted, but not fully appreciated…”