Day 18: The SOS Diagram

So, after a day of learning that the slope of the momentum-time graph is the force, and another day when we figured out the area of the force-time graph gives you the change in the momentum, we were ready to put this model to use.

I know that lots of people use momentum bar charts, but I'm not 100% sold on them. I could be convinced, and I know that energy bar charts are A-MA-ZING. I wanted to play around with momentum bar charts that made the focus more on the motion of the objects and less on the blocks. 

So, following in the footsteps of the LOL diagram, I use the SOS diagram. The name will make more sense a little later. 

Remember, my students have done balanced forces yet. They only know that a force is a push or a pull and is the slope of the momentum-time graph. I start by introducing interaction diagrams.

First, we make our first interaction diagram together as a class. We use the setup in lab as our situation. (I took a picture of the setup and posted it at the beginning of this post.) We start by listing all the objects we care about and putting them in bubbles. Then we connect the objects that interact with lines. Usually, to interact with an object in a physics sense, that is, to put a force on it, you have to be touching it. There seems to be only one exception we have right now, and that would be the Earth. It seems to push everything around, even when we're not touching the Earth. I mean, that's why babies are FASCINATED by dropping objects from high chairs. "Look!" they seem to scream. "This object is being pushed around and NOTHING IS TOUCHING IT." We aren't usually as impressed nowadays that we're older, but we should be. Anyway, here's what the interaction diagram looks like:

At this point, what interaction or interactions are changing the motion of the object do we care about? Really only the spring on the cart. That's the one that changes the car's momentum. Wait! This diagram shows what changes a car's momentum! Cool! So I can draw the picture before that interaction and after that interaction and put a (simplified version of the) interaction diagram in the middle:

The first diagram represents pᵢ, the second diagram represents ∆p, and the third diagram represents the final momentum (there's no Unicode for subscript f. That makes me mad.)

So, we have our equation here, in picture form: Initial momentum + change in momentum = final momentum. Or, since change in momentum is the area of the force-time graph, and the simplest way to calculate that area would be a rectangle where the base was the time and the height was the average force for the situation, we can write it as...

Initial momentum + average force * change in time = Final momentum.

So why is it an SOS diagram? The O is for the system schema in the middle (which I call an interaction diagram, because 'system' is used differently by the AP, and I don't want to confuse my students), and the two S's are for sketch. It also is a cry for help when you're using the momentum model.

Day 16: Starting the Momentum Model

If you've been following me and read this tweet, you know that I'm going to be defining momentum in a different way this year. We're defining force as the slope of the momentum-time graph. This means we have to define momentum too, which is the hand-wavy part of this unit. Why even come up with this concept? Luckily, we define momentum as "how hard it is to stop something" and "mass times velocity" before anyone was too awake.

Then we went to lab and had a cart bounce off a force probe. We changed the mass of the cart and the spring on the force probe. We graphed momentum-time and force-time graphs. Here's what students noticed in lab:

  • The momentum of the cart stayed constant except when it is touching the force probe.
  • The force was negative because it pushed in the opposite direction of the initial motion of the cart.
  • The momentum of the cart started positive and went negative.
  • The force was the largest when the momentum of the cart was zero.
  • When we used the springs, the momentum after hitting the spring was the opposite of the momentum before hitting the spring. The two momenta had the same magnitudes.
  • When we used the clay, the momentum after hitting the spring was less than the momentum before hitting the spring.
  • We could never get the final momentum to be larger than the initial momentum except that one time when the plunger exploded and hit the force probe, but we don't think that counts.

Day 31: Our Four Tools

We summarized all four models today in the middle of class. It a good way to realize we have four tools in our tool belt and that we have choices in the ways we look at problems. We will be looking at projectiles and circular motion next.

Before summarizing, we analyzed a ballistic pendulum. A few groups stopped when they found a velocity, thinking that any velocity must be the velocity they want.

After summarizing, we looked at different types of collisions. When would the kinetic energy be conserved in a collision?

Day 30: Conservation of Momentum in Detail

I went beyond what I planned today, and I ended up with the demonstration you see above. If you take four carts and put them under the four feet of your track, and then crash two carts on top, the track won't move. It doesn't matter if you use magnets, or velcro, or even use the plunger for an explosion. If, though, you put one of the carts on the track upside down, so that now there is friction in your system, the track will move in the direction of the total momentum of your system before the collision. Momentum is conserved, as long as you make your system large enough!

So let me tell you how I got here. I started with this question I wrote, which uses some variables because my student need practice with them: An iron sphere of mass m and velocity vo is attracted to a magnet of equal mass that is initially at rest.  After a time t, the sphere and magnet are attached, and both are traveling at a velocity of vo/4. What was the average net force on the sphere-magnet system during this time?

I ended up with two answers, neatly summarized on these two whiteboards written by two different students:

The green marker student was sure the other solution was correct, and the black marker student thought both boards were correct, depending on how you thought about the problem. The black marker student looked at the interaction diagram and said there were no net external forces. Yes, there was a normal force and force of gravity on the system from the Earth, but those forces balanced. The green marker student hesitantly defended his old work, saying that he calculated the change in momentum of the system, and that must be the impulse on the system. But he didn't see any net external forces.

Then someone else in the class piped up. "What about friction?" she asked. "We don't know if this is a frictionless situation or not." After some debate as to whether we could include friction or not, we tried to figure out what would happen in a collision between two objects of the same mass when they stick together after the collision. I showed quickly that the final velocity after the collision would be half the initial velocity. I looked around the room, and it looked like the class didn't believe me. 

So I pulled out two carts and a track and a motion detector and in a minute we had a confirmation from the real world. We realized then there must have been an impulse on the system since its momentum must have changed. I said if we expanded the system, though, to include the Earth, the total momentum would be the same. That's when the carts under the track came out and we made the videos in this post.

After discussing this and other problems through whiteboarding, we started a lab with a ballistic pendulum. We'll see results on Monday.

Day 29: More on Systems Thinking and Momentum

After our assessment, we tried some problems and did a little whiteboarding. A few things I learned from my students:

  • When students figure out momentum conservation on their own, they keep rediscovering it, and don't use a shortcut. They will calculate impulse on one object. Then they'd use Newton's Third Law to reason that the impulse on the other object is equal and opposite. And then they'd calculate the velocity of the second object. They'll also write momentum conservation in lots of interesting ways, including m₁∆v₁ = -m₂∆v₂. 
  • On the other hand, they really understand why momentum conservation works. They see that there's no net external force on the system, so they're no impulse on the system. That's why the total momentum doesn't change. I think this is worth it, especially since the biggest drawback is that they solve problems slower, which really isn't a drawback.
  • When students get stuck, they seem to stop. When I ask them to draw an SOS diagram (Sketch-Interaction Diagram-Sketch), they start working and then figure out their own way to solve the problem. It seems that they still don't realize how powerful drawing the picture is. 

Day 28: What Happens to Impulse if You Make the System Bigger?

After teaching the impulse-momentum relationship yesterday, I sent them off last night to try a few problems, telling them only that my new type of diagram for showing momentum transfer was an SOS diagram. The two S's stand for sketch, since, in momentum problems, knowing the direction of the momenta is so important. The O is, again, a interaction diagram (or system schema), where any net (non-negligible) external force will cause an impulse. So, instead of writing the impulse-momentum equation as F∆t=m∆v, we wrote it as mvi + Impulse = mvf.

We had a problem in the homework of two colliding railway cars. Without prompting, my students saw two railway cars as one system, and could figure out that the momentum of the whole system would be the sum of the momenta of the constituent parts. After about ten minutes of solidifying their comments, we were off to lab.

The lab I wanted to do wasn't working, because I couldn't get three PASCO probes work on a USB hub. (I'll detail that lab later when I get it to work.) So, instead, I had students set up their own labs. We put motion detectors on each end of the track, and we learned how to track the velocity of each cart on the track. Then, we learned how to have the computer calculate the momentum of each cart and the total momentum of the system. Then I asked them to break the model. Can you find a situation where there was no net external force but the total momentum changed? They couldn't. Here are some pictures I took of their most creative tries of making a cart-cart system that would break the momentum model:




Day 26: Building the Momentum Transfer Model

We're finally ready to figure out what happens when you give an object a little push! Since last time we looked into the area of the force-position graph, this time, since the force hardly acts over any distance, we could look at the area of the force-time graph. A bigger force would do more, and so would more time that the force was applied. So, what does the area of the force-time graph tell us? We think it has something to do with the initial velocity of the cart, or the final velocity of the cart, or something.

So, we were off into lab. There were lots of different approaches. Some groups changed what surface was on the force probe. Other groups tried different initial velocities of the car. One group was sure that the average velocity predicts the area of the force-time graph. 

It was messy and it was great. By the time we wrapped up to look at the data, only two or three groups really felt comfortable with their data. But everybody had thought so much about the problem that it made sense why that data was best and why their conclusion should be believed. We defined impulse and momentum and started using our new model