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Dan Meyer is an artist. He makes some very compelling educational videos. Here is his latest.
Height v. Time from Dan Meyer on Vimeo.
I can't stop from getting sucked into his math-world. So, how about a quick video analysis of this motion using Tracker Video? It is a fairly good movie to analyze. Really, there are only two problems with it:
- There is no obvious scale in the video. But this is ok - I can figure it out anyway.
- There is some parallax issues. Notice when Dan is at the bottom of the swinging motion, you only see one of the swing chains. At the ends, you can see both. You could fix this if you had the video camera farther away. But I don't think this will be too big of a problem.
Scaling the video
This is a great trick. If you have something with a known acceleration, you can use that acceleration to find the scale. In this case, Dan jumps off the swing. After he leaves, his vertical acceleration should be around -9.8 m/s2 since he is in free fall. So, I picked something to measure the length. In this case, I picked the chain on the swing. Next, I fit a quadratic function to Dan's position after he left the swing. This is what that looks like.
This is after I adjusted the scaled length of the chain. Here you see the following fit equation:
I left off the units because computers don't need no stinkin units (or better - computers don't understand units). Anyway, compare this to the kinematic equation for a free falling object:
The (1/2)g term should be the same as the -4.79 in front of the t2 term. This means that g = 9.58 (m/2) - close enough to 9.8 for me.
Ok. Now the video is properly scaled. We can do some useful stuff. Just how tall is Dan? Skipping forward, I have this.
I think Dan's head is tilted down a little bit - so I extended the measuring tape a little. This gives a height of 1.94 meters or about 6 ft 4 inches. The guy has a wingspan like a condor.
What about Dan's graph?
Here is my data on top of his data. The vertical axis for mine is still in meters - you know, more sensible.
There are some difference. Dan's change in height goes from about 7 feet to 2 feet. If you convert my change in height to feet, you get 4.2 feet. So, that is a little off. Also, Dan's data seems to be a sort of a sine function. In this case, the vertical motion as a function of time would not be a sine or cosine function. Oh, pendulums = simple harmonic motion and all. Well, not true. Actually, pendulums can be approximated by simple harmonic motion if the amplitude is small. This is probably way more details than you wanted regarding pendulums. However, it does show how you use the small angle approximation to get simple harmonic motion.
Another problem with simple harmonic motion approach is that this is not even a simple pendulum. I guess you could model this as three masses (head, torso, legs) that all change positions relative to each other. Here is a teaser graph showing the motions of the head and the waist.
Notice how the head is out of phase with the torso? This is one of the keys to "pumping" the swing - although I haven't fully studied this.
Ok, so the graph doesn't completely match. So what. I attribute it to "artistic license". Dan can still use his prettyfied graph to do some cool math. For me, I am more interested in how you move to make the swing go higher.
Check out GraphingStories.com
So, apparently Dan has decided to make some more graphing stories like the one above. Here is your chance to submit some ideas. He wants this to be huge, so make it huge. HUGE. Hopefully, there will be a ton of new videos that teachers of all types can use to make learning interesting.
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