Over the years, I’ve struggled a bit with teaching Special and General relativity to my second year physics students. And, if we’re really being honest, I’ve struggled myself to understand some of its finer points. It’s very easy to find yourself confusing reference frames and losing track of an observer or two. And, to top it all off, it’s difficult to use experimental results as a basis for understanding (despite my continual requests to build a particle accelerator behind our football field, my district has yet to agree to the proposal… something about budget cuts and zoning laws). Because of this, I think that it’s vitally important to get students thinking critically about reference frames and observers for more common everyday occurrences so that when we make the jump to relativistic speeds, the transition is a little less jarring. I do admit that despite this belief, I’m not sure I’ve ever really done a fantastic job at it.
My typical introduction to the Galilean version of relative motion has usually involved some teacher led discussions regarding reference frames, usually using examples of “freeway” relativity with cars passing each other on the highway at various relative speeds, followed by a teacher led derivation of the Galilean equations for relative motion and the ever present S and S’ observers. Ho hum. Where was the student thinking here?
This year, I was determined to do it differently. As I thought about alternate approaches I recalled a Dan Meyer posting where he had created a video of himself walking up and down stairs and escalators. I think his intention was to use the video to get students to work with systems of equations, but I thought it could be used to analyze relative motion. Particularly, I thought it could be a good way to practice “ninja physics.” The students would be using relativity concepts without really knowing they were doing relativity.
I showed the video clip of Dan walking up and down stationary stairs and walking up an escalator that was also moving upwards. The clip ends with Dan about to walk up the down escalator. Afterwards, I had the students write down three questions they had after watching the video. I immediately saw where some of them were going with this and decided that in order to reign in their, um, “creativity”, they were allowed one question that probably couldn’t be answered by using physics, and two questions that could. After we laughed a bit about questions like:
- Where did he film the video?
- Was the person in the video a clone/quadruplet? (four videos of Dan are shown simultaneously)
- Why did he choose that particular music?
- Where did he get the cool shirt? (actually, one of the students realized he possessed the exact same shirt and wore it the following day just to verify).
On the physics front, there were questions regarding the individual velocities of both Dan and the escalators, as well as the $64,000 question, “how much time will it take Dan to travel up the down escalator?” With this, we were off discussing things they felt they needed in order to answer the question. In the end, the students decided that they would need the time information provided by the video, as well as the height and depth dimensions of each step on the stairs and escalators (conveniently available on the additional images Dan provides).
The students then went into their lab groups and started to work things out on whiteboards. There were some questions about whether or not they could assume that the stair distance and the elevator distance were equal, some groups decided they could, while others decided to treat them separately. A couple of the groups did stumble at first, trying to quantify everything they could about the situation before they realized that in terms of answering the question, most of the information they were attempting to find was not necessary. In the end, after some great student to student discussions and some mental exertion, the groups each arrived at an answer (values ranged from as low as 19 seconds to as high as 45 seconds). I had students present and compare their thinking processes to each other and made sure that they were explicit about any assumptions they had made. I was actually a bit surprised at the diversity of solutions the students came up with. Most groups determined the velocities of Dan and the escalators in inches/sec, while one group used a rate of steps/sec. Another group determined the distance traveled per step in inches/step and then used a step rate of seconds/step to determine the velocities. Some of their board work is shown below.
As they presented and discussed their work, I surreptitiously recorded their findings on a whiteboard of my own:
I asked the students what they noticed about their different approaches and if there was anything the approaches had in common. In very little time they noticed that, without even explicitly realizing they had done it, they had used vector addition of velocities because it just “seemed” right to them. In general, they determined that they had used the following relations:
velocity on steps + escalator velocity = velocity of Dan going up the up escalator
velocity on steps – escalator velocity = velocity of Dan going up the down escalator
Eureka! They had just used and determined the Galilean equation for the superposition of velocities!
I then decided to take this a step farther and asked them if they could use their information to determine an equation that relates Dan’s position on the escalator at any time with the velocity of the escalator if they knew how far Dan had traveled relative to where he started on the escalator? Admittedly, I’m not too happy with how I asked this question. I think I may have guided them to a place where they might not have naturally gone by limiting them to the specific terms that I wanted them to have in the equation. Next time, I think I’ll try to phrase the question more generally. Perhaps have them come up with several different expressions that all describe his position on the escalator. With multiple options, we can then discuss the advantages/limitations of each one and see if we can come to a group consensus on the most useful expression.
In any case, the students went back into their lab groups and after some group discussion about the necessity to include/not include Dan’s velocity relative to the escalator, within about 5-10 minutes they had all determined some form of the following relation:
position relative to ground = position relative to escalator + (escalator velocity)t
Eureka again! The Galilean expression for relative position!
By this time, the students were dying to see how close their results were to reality, so in addition to checking the answer video Dan provides for the actual time, we also used it to check their position equations. Obviously, the groups that had arrived at a time that was closer to the actual time (~ 22 secs) had better luck in verifying their position equations, but overall, the students felt they had done a reasonable job at describing the escalator/stair scenario.
Following this, we formalized their relationships into the traditionally accepted forms of the Galilean relativity equations. We also discussed the S and S’ reference frames and how the students had already used these two different frames (stairs and escalator) to analyze the situation. We also tested the waters a bit and tried to switch reference frames to describe what someone traveling on the escalator would see if they were describing the motion of the camera or of someone walking on the stairs.
So, how did this different approach to introducing relative motion impact student learning? Primarily, it took me off the center stage, which is always a good thing. It provided the students with an opportunity to develop an intuitive sense for relative motion first, before we officially formalized the concept. They also exercised their abilities to approach new questions, determine a possible path to a solution, and try to take that path to a sensible conclusion (with some back tracks and stumbles along the way). Particularly, they had to critically evaluate the assumptions they could or could not make as well as determine how to best use the video to obtain the information that they needed in order to answer the question. Additionally, I think it helped that we focused on some key concepts before getting bogged down in discussions of S and S’ or x and x’. Many of the students were surprised to find out that they had done “relativity” on their own without even realizing it. Interestingly, a few astute students did mention that it would seem like somebody would have already developed equations to deal with these types of scenarios. To which I smiled and replied that it was possible, but I doubt they would have used such a cool video to do it!
It’s still early in the relativity unit and I’ll be continually checking to see if the intuitive sense they developed regarding reference frames and observers as a result of working with the video carries over into their work on other everyday situations. From there, we’ll see if it helps them to deal with the non-intuitive cases of relative motion at close to the speed of light. I’m still trying to come up with ways to approach the ideas of Special relativity in a more student-centered way, so feel free to offer any advice/suggestions you may have in the comments below.