Predicting the future-part 1
So it’s time I got back to writing about physics. And what better way than blogging about how we made the huge N2 synthesis.
It starts with this question:
What causes CAPM (Constant Acceleration)?
We brainstorm out what we think might affect the acceleration of an object, and they lock on pretty quickly to force and mass. I give the kids a bunch of stuff (more on that later) and ask them to design experiments to test how force and mass affect the affect the acceleration.
With the limited set of supplies I give the kids, they design 3 basic experiments.
- Old PSSC carts and springs. I love these old PSSC carts with roller skate wheels. Add to that a set of foil wrapped bricks (a wonderful investment of a half day’s work) and you’ve got a great set of masses to explore the world of dynamics.Students attach PASCO springs to the carts, and then stretch the springs a set length to create a constant force, and then measure the acceleration. While I love the PASCO springs, they are expensive, and easily damaged. If anyone has a cheaper alternative, I’d love to hear about it. If not, I suggest you make some sort of jig when doing this out of pegboard so that the students can multiple springs to the carts without overlapping the springs and causing them to get tangled.Once the students got their data for constant mass, they decided to attach a single spring and collect data for a varying mass (1, 2 and 3 bricks on the cart).
- Fan Cart and batteries
- Super Fan Cart. Last year, we picked up one of Pasco’s super fan carts. Here, the challenge was to vary the force, but it was more difficult because you didn’t know for sure how the lo, medium and high settings varied in terms of fan output. So students decided to vary the force by turning the fan, so that less of the airstream was pointed in direction of travel away from the direction of travel, etc (by the way, this is a great way to wrestle with some non-intuitive geometry ideas like the fact that doesn’t happen at .
They too then set a constant force at , and then varied the mass.
Here students varied the force the air exerts on the fan by changing the number of batteries in a typical pasco fan cart. They recorded data for 1, 2, 3 and 4 batteries (replacing batteries with aluminum slugs of equal mass), and then kept the force constant while varying the mass of the cart by adding brass masses.
Students are getting good enough at designing these experiments that the get to taking data pretty quickly, and they’re equally facile using the motion sensor and video physics to record the motion of the objects. Pretty soon they start graphing their acceleration vs force data, and it ends up looking something like this.
Right away, students seize upon the similarities and differences in these graphs. And they quickly become puzzled by one thing—none of these graphs pass through the origin. When they think about it, they see this contradicts their previous physics knowledge, which says that if the acceleration is zero, the velocity must be constant, and this requires balanced forces . It doesn’t take too long for students to realize that they aren’t plotting the net force here, they’re only plotting one of the forces acting on the cart, and so there must be another force acting in the opposite direction. And soon, someone realizes that it’s the frictional force of the ground/track on the cart.
So if we subtract this Frictional force off the applied force and re-plot , we get:
This leads to a pretty clear conclusion: acceleration must be directly proportional to the net force exerted on the object.
Next, students analyze the acceleration vs mass data, and get something like this:
Of course, this is one of those times when the students wish they had more data, but they don’t. So what to do. Most just jump to drawing a line through the data points like this.
The students are quick to see some logical inconsistencies with the predictions from this data. For example, the fact that there’s a horizontal intercept to the fit would seem to indicate that there’s a particular mass where the acceleration of the object would be zero, and beyond that, the acceleration of the object would be opposite the direction of the force, which can’t be.
I then ask the students what the data should do, and they see that while it should get closer and closer the line as mass increases, and as mass decreases, the acceleration should tend to get larger and larger. We then discuss the idea of asymptotes, and I ask them to play with their calculators and try to find functions that behave this way. Pretty quickly they seize upon the inverse function, , and so maybe, we should look at our data like this
But the challenge is how do we know whether this is true, without data at the tails to tell us? This is where I ask them to tell me what a graph of 1/acceleration versus mass would look like, if the inverse relationship were true. This takes some thinking but soon they predict something like this.
So time to test it out. My kids plot their data of 1/acceleration vs mass, and lo and behold they don’t get the above graph. They get this:
This graph really requires some unpacking. So I begin by asking them to write “big acceleration” and “small acceleration.” It takes a second, but they see that because this is an inverse graph, small accelerations are higher on the vertical axis, and as you get closer to zero on the vertical axis, the acceleration increases. By now, the kids have figured out that if you have a really small mass, you should have a really huge acceleration (for a constant force), so why does having a zero mass seem to imply the acceleration isn’t huge (infinite)? Soon, they figure out that what they’ve plotted is the mass they’ve added (in bricks, or masses), and they haven’t accounted for the mass of the cart. And just like that, they see that the horizontal intercept is the mass of the cart, and if they take this into account, and add this mass back in, they get this:
And now we have two big ideas:
and . This suggests we can write a formula somehting like
But there’s one nagging point. Some students measured their forces in batteries, others in springs and others still in fan units, and none of these are Newtons, our favorite unit. Mass measurements included both bricks and grams. How can we can we speak the same language?
And suddenly we have a need for that boring unit stuff that you typically do in the beginning of physics courses. All we need to do to convert batteries to springs is measure the force of 1 spring unit in Newtons using a spring scale, and 1 battery of force in Newtons, and we can convert between the three. Awesome. And this lesson is a million times more real that all those problems kids typically get to convert the speed of light into furlongs per fortnight, since google won’t ever be able to convert batteries of force to Newtons.
I used to painstakingly pull all of this out of kids in a multi-day lab follow-up ‘discussion’ led by me, that mostly consisted of me asking questions. Now, since I’ve embraced modeling, almost all of this came up in our 30 minute student-led discussion and in smaller discussions with individual groups. And at the end, I was able to ask—how could you use this to predict the future? And everyone was transfixed with trying to figure this out. But I’ll save this for my next blog post.