Friday, May 11, 2007

Updated Pacing

[Finally pulled from the drafts folder ...
... update with graph added on Saturday, 19 May.]

My usual practice when teaching is to preview the next major topic as a way to warm up the kids brains and point to key topics in the next day's readings. On the first day of the gen-ed class I am teaching, I always make the point that we are going to work at connecting the principles of physical science to events from daily life. As a preview, I asked how many of my students had seen the Kentucky Derby, or news coverage of it. The answer was none.

None !! So much for being digitally connected to news and current events. (I didn't ask if they knew the Queen of England had visited the U.S., or why.)

Heads up to newbies (and oldies): Assume nothing! I learned that I had to change my plans for Thursday to include showing them the real-world context of the example.

Fortunately, the massive copyright violations available on YouTube made it easy to show them the entire two-minute race:

The eventual winner, Street Sense, is the horse that disappears from the screen at the 0:22 mark, reappears at the 0:28 mark (he is the last horse on the rail through the 0:38 mark). He vanishes from the coverage until the 0:54 mark (again on the rail, in 19th place). The last two horses seen at the 1:09 mark end up finishing 1st and 4th, from 19th and 20th.

The "come from behind" starts at the 1:22 mark with Street Sense at the very bottom of the frame, still on the rail. But Street Sense does not speed up; the other horses slow down. He stays on the rail until the 1:41 mark when he has to move outside to take second, and then first.

[Graph added Saturday, 19 May ...]

The distance vs time graph that I used in class (cleaned up a lot for public viewing) provides a mathematical version of what you see in the video. [A functional representation of it, if Jennifer O is looking in.] The blog-display version is just barely legible; you need to click on it to see some of the detail, such as the green line showing a constant pace. The graph clearly shows that Street Sense ran close to a constant pace, and that Hard Spun fell back after a much too-fast start.

The second place horse, Hard Spun, shown with a blue line, started out fast. His highest speed (marked with an asterisk) is for the first quarter mile. He (along with the other horses in the lead group) is slowing down the rest of the way. A slight increase in speed at the end enabled him to hold on to 2nd place after leading from the start. His slowest speed was when running the final curve, not unlike a runner hitting the "wall" at an uphill finish.

The red line shows the performance of Street Sense, the winner. He starts slowly, slower than any of his other segment times. As slow, in fact, as the second place horse is going at the finish! His fastest segment is the 2nd quarter, but even his highest speed is less than what Hard Spun did over the first half mile. He is saving energy for the rest of the race. That increase, sustained over the 3rd quarter, was just enough to get him close to the ideal pace (54.0 ft/s, shown with the green line) for the race. He ran the last half mile right on that constant pace.

Technical detail for any physicists who might read this and want to use in when introducing motion diagrams and graphs: You can barely see the negative curvature of the graph for Hard Spun. I used a yellow shading between the constant speed line in green and the blue curve to highlight this, but it would have to be pointed out that the numbers are a better measure of the acceleration than your eye is.

The procedures used to get the "data" plotted here are described
in the previous post. The minor flaws in that analysis don't affect the qualitative conclusions discussed here. Anyone want to hand-time the video?

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