Day 39: The Mulvey Method

The Mulvey Method helps students see what happens how something like the force of gravity changes when other quantities change. It's great for questions where all the answers look like F, 2F, 4F, F/4, etc. It's named after Hillary Mulvey, Brown '14, who came up with this technique.

Let's try it on an easy relationship.  If the kinetic energy of an object is doubled, what happens to its velocity? Start by writing the equation twice that describes the relationship you're studying, once under a column labeled NEW and once under a column labeled OLD. Then, under the OLD column, plug 1 in for everything, and I mean everything, that isn't the dependent variable in the situation. So your work should look like this:

Yes, even replace the 1/2 with a 1. You're looking for the ratio between quantities before the change and after the change, so we set all the quantities, including constants, equal to 1. 

Then, it's time to figure out the values for the NEW column. The kinetic energy doubles, so it goes from 1 to 2. The 1/2 factor didn't change, it was "multiplied by 1", so it's a 1 in the next line, and, since the mass didn't change, it also was multiplied by 1. So that leaves me with:

Now you can solve for v to be the square root of 2, and so the velocity is multiplied by the square root of 2. Now this technique isn't the most mathematically rigorous, but I've found it can help students figure out what these types of questions are asking, and, over time, they start seeing the relationships without having to resort to the Mulvey Method. I also found that when I try to do it a more mathematically rigorous way at the beginning, my students get confused by all the variables, but when I show why the Mulvey Method works in March or April, it makes perfect sense.

We also got to play with PHeT's Solar System simulation, using a lab I think I've done since 1998 in some form or another, starting with Interactive Physics software on the old fruit-colored iMacs. 

Day 37: Circular Motion & Orbits

Thinking about circular motion is hard because we have to think in two dimensions. Those of us who were attached to positive and negative direction now need to think in terms of left, right, up, and down. It's harder than you'd think.

We played a new game today, a twist on the mistake game. I had groups of two put up two whiteboards, one with a mistake they thought was a good mistake. It turns about the best board discussions were around how to convince someone of their mistake. It also was great when students made two different mistakes on the two boards. I think next time, I'll ask the groups of two to make a mistake in at least one of the two boards.

We played with the Greek waiter's tray (see above), which is always fun. I make my students go through a three-stage training with the tray: first, a soft object that's grippy like a foam apple, then a roll of tape, which seems to slide off pretty easily if they're not careful, and finally, for stage three certification, a cup of water. 

We had a little time today to start talking about what would happen if you launched an object horizontally faster and faster from a little ways off the surface of the Earth. I got to draw Newton's Mountain on the board, which is one of my favorite visuals.

Day 36: What Force Makes Objects Move in a Circle?

After our assessment on momentum and projectiles, we whiteboarded some questions about circular motion. The biggest thing that seemed to trip us up was whether the force that pushes you in a circle is another force that should be drawn on the force diagram or a force we already know from our interaction diagram. We also had a great discussion about how much kinetic energy and potential energy you have to give a satellite to get it into orbit. (This is another reason I'm happy I postponed circular motion & projectiles until after energy & momentum.) We ended by talking about the forces on banked curves. We did it just qualitatively, but there were many, many questions about it.

Day 35: Every AP Physics Teacher Should Do This Lab

Have you seen the Veritasium video about what happens when you drop a Slinky? If you haven't, you must go there now. (And make a guess, watch the answer video, and make sure you watch the amazing slo-mo video that's linked to the answer video.) It's a great little piece of physics. 

You can do force diagrams of both the top of the Slinky and the bottom of the Slinky and the top of the Slinky accelerates at greater than 9.8 m/s². So we talked about that, and then I got out my foam apple. 

Where do I have to hold the foam apple so when I drop the Slinky and the foam apple at the same time, they hit the ground at the same time?

As soon as I asked the question, it was pandemonium. Kids started grabbing whiteboards. Heated discussions broke out everywhere. One student started counted the number of turns in the Slinky. There was much arguing. 

But what they all knew, what they all seemed to get, is that the center of mass fall at 9.8 m/s². (I think this lesson really helped.) Here were their whiteboards after 25 minutes of discussion:

They wanted to keep going. It was such an interesting lesson. Their answers weren't that bad; we only had to tweak their answers a little bit to get the great trial you see above. (Don't expect your students to get it perfect; it takes some pretty hard-core calculus to derive the theoretical value.)

Day 34: Break the Petri Dish!

Here are the fragments of a broken Petri dish and their calculations to break it.

Without much time spent on the mathematics of projectiles, we seem pretty comfortable using UAPM and CVPM to calculate the kinematics variables. We did lots of practice in TIPERs to make sure we understand the concepts behind projectiles. 

Tomorrow, we'll talk about the Slinky Drop Experience and start the next unit.

Day 33: Projectiles & PHeT

We started by going over some conceptual problems on projectiles, and students seemed comfortable with the shape of the velocity-time graphs. At the end of whiteboarding, after doing the last question about calculating where a horizontally-launched projectile would land, a student asked me how to calculate where a projectile would land if it were launched at an angle. He started mentioning components of the initial velocity vector, and I had to hold my tongue. I didn't want to get lost in the math of projectiles this year. We used PHeT instead to simulate the launching of many projectiles. We came up with some pretty powerful conclusions, most of which were suggested my more than one student before I summarized them on the board.

Day 32: Projectiles Using Our Four Models

Now that we've talked about the four major models of mechanics, we started projectile motion through video analysis. We immediately saw that the difference in the x-component of the motion and the y-component of the motion. Ut made sense using forces. Energy seemed like an easy way to figure out velocities. And momentum didn't seem useful at all.

We tried a problem about hang time. Two solutions are presented below. Almost every student liked the graphical way and didn't like the equation (which worked & which they learned in math class).

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.