Today we started the WB discussion of the ramp data. As explained yesterday, I assign the direction and initial position because it makes the post lab discussion more interesting and allow us to address a few things. Here is a screenshot of the two graphs we ended up with:
So far so good for the students. There were a few that had to think really hard about what the blue graph was telling us… the (magnitude of the) slope increases in the negative direction so the object is getting faster in the negative direction as it rolls down the ramp.
Next we went to how to make it linear… yes, I do linearization. I find it helpful to work on proportional reasoning , to work with variables only, and to identify what the slope and intercept mean. So, here is another screenshot of the linear graph:
Ok, we have established the square relationship between time and position. Now to identify the meaning of the slope and intercept:
We always focus on the units. The slope units are m/s^2. I explain that the meters are easy, they represent a displacement. But the square seconds…. we have no real conceptual basis for that. What is a square second? We understand a square meter, it represents an area. A ‘second’ represents a time, but what about a square second? By now the math kids are nearly foaming at the mouth. Finally one of the kids will say, “Yeah, but m/seconds squared is the acceleration, that’s what the slope is”. I simply reply, No, the slope is NOT ‘the acceleration’. And now some want to argue, some have their confidence shaken and some just want to be told what the hell the slope is (hell is my word, not theirs).
I explain that we are at a dead-end because of this, and when we are in a car, and we hit a dead-end, we go back to where we know something. or us, we go back to our original non-linear position-time graph. This is where I help the students develop the idea of instantaneous velocity. Another screen shot:
To help explain the idea of instantaneous velocity, I used the ‘selfie’ analogy (I added Snapchat) I read about on the modeling listserv a while back. It worked pretty well, especially since I snapped a self in each class that I will start with tomorrow as a review. I am careful to define instantaneous velocity as shown above and include the position piece. This is because of where we are headed tomorrow. Notice not a single mention of acceleration from me. It will be officially defined tomorrow. I LOVE THIS STUFF.
After we discussed the test they took yesterday, I shut the lights off and all of a sudden some laser dots floating around the room became visible. I sprayed some Fog in Can around the round and the beams burst into view. From a laser on my desk, to a plane mirror on the front board, to a plane mirror up on top of a cabinet, to our disco ball. GLORIOUS. So now we are on to mirrors. Following the same flow as we did with lenses, we start with looking at how the incidence angle compares to the reflected angle. We use our magnetic laser levels on the main whiteboard and a plane mirror to quickly establish the two angles are equal. From there we look at how the object distance compares to the image distance and how the object height compares to the image height. All of this in accomplished by using a CD case and a two identical legos. I read/heard about the AWESOME idea from Frrank Noschese here. This is the BEST way I have seen to build these concepts. GO ahead and watch it, I’ll wait.
See what I mean? It works so well. Tomorrow we tackle drawing a ray diagram to show how that image forms.