More Physics of Sprinting

Yesterday’s post on applying intro physics concepts to the question of how fast and how long football players might accelerate generated a bunch of comments, several of them claiming that the model I used didn’t match real data in the form of race clips and the like. One comment in particular linked to a PDF file including 10m “splits” for two Usain Bolt races, including a complicated model showing that he was still accelerating at 70m into the race. How does this affect my argument from yesterday?

Well, that document is really a guide to fancy fitting routines on some sort of graphing calculator or something. Which is fine as far as it goes, but I think it attributes too high a degree of reality to those unofficial split times, which are obtained from some unidentified web site. They proceed to fit a bunch of complicated functions to the data, but I think they’re overthinking it.

Let’s look at the actual data, graphed in more or less the way you would expect to see it in an intro physics class: as a plot of position vs. time:


The black circles represent the times from a race in 2008, the white circles times from a race in 2009. They’re practically right on top of each other, because in absolute terms, the difference in times is pretty tiny.

Their first step is to fit a straight line to the data, which works remarkably well, even though it can’t possibly be right. Looking at the graph, though, it does look awfully linear, particularly if you threw out the first point or two. That seems pretty consistent with the “accelerate to a maximum speed and stay there” model I assumed in the previous post, especially given that we don’t know anything, really, about how these numbers were obtained.

Of course, the real test is to look at the speed as a function of time:

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The Physics of Sprints and Kickoff Safety

Over at Grantland, Bill Barnwell offers some unorthodox suggestions for replacing the kickoff in NFL games, which has apparently been floated as a way to improve player safety. Appropriately enough, the suggestion apparently came from Giants owner John Mara, which makes perfect sense giving that the Giants haven’t had a decent kick returner since Dave Meggett twenty years ago, and their kick coverage team has lost them multiple games by giving up touchdowns to the other team.

Anyway, one of Barnwell’s suggestions invoked physics, in a way that struck me as puzzling:

Idea 3: The receiving team’s returner is handed the ball on his own goal line. His blockers must be positioned on the 20-yard line. The “kicking” team’s players are positioned on the opposition’s 40-yard line. Once the whistle blows, it’s a traditional kick return.

Advantages: Everyone on the field will still be colliding, but because the kick-coverage team will have been running for 20 yards as opposed to 45, there won’t be anywhere near as much momentum in those collisions. That should produce fewer injuries.

Now, if you know anything about introductory physics, you know that momentum is mass times velocity (for speeds much less than the speed of light). This doesn’t have any direct relationship to the distance somebody has run to get to that point, unless they’re accelerating the whole time. But it seems awfully unlikely to me that any of the whack jobs covering kicks in the NFL are actually speeding up appreciably between 20 and 45 yards– they probably hit their top speed well before that.

Ah, but is there a way to use our knowledge of introductory physics to test this idea? That is, can we estimate the distance over which an NFL player is likely to be accelerating? Well, I would hardly be posting this if I didn’t have a model for this sort of thing, so let’s have a run at it (heh).

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Are Books and Kindles Correlated?

I’m trying not to obsessively check and re-check the Dog Physics Sales Rank Tracker, with limited success. One thing that jumped out at me from the recent data, though, is the big gap between the book and Kindle rankings over the weekend. The book sales rank dropped (indicating increased sales, probably a result of the podcast interview), while the Kindle rank went up dramatically. This suggests that people who listen to that particular podcast are less likely to buy new books on the Kindle than new books on paper.

This got me wondering, though, whether this was an anomaly, or a general truth. That is, is there any correlation between the sales rank of the paper edition of a book and the sales rank of the Kindle edition of the same book? Happily, the sales rank tracker spits out all the hourly rankings in a nice table that I could copy into SigmaPlot and crank away on, producing the following:


This is a plot showing the Kindle sales rank of How to Teach Relativity to Your Dog (vertical axis) versus the sales rank of the paper edition (horizontal axis). I smoothed the hourly data a bit, averaging together five hours, because it’s really noisy, but that makes almost no difference.

What does this say? Well, that there’s a pretty weak correlation between them. The data points fall more or less in a wedge extending up and to the right, which tells you that when one is really high, the other tends not to be very low, and when one is low, the other also tends to be low, but the relationship between them is pretty weak. At a book rank of about 25,000, the Kindle rank ranges from about 14,000 to about 96,000.

This is for the recently released book, though. Maybe more data would make a clearer picture? In a word, no. In a thousand words (i.e., one picture):

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How Good Is BookScan Anyway?

One of the big stories in genre Internet news was Seanan McGuire’s post last week, about reactions to the early release of some copies of her book, and the hateful garbage thrown her way by people outraged that the ebook didn’t slip out early as well. And let me state right up front that the people who wrote her those things are lower than the slime that pond scum scrapes off its shoes. That’s absolutely unconscionable behavior, and has no place in civilized society.

That said, Andrew Wheeler picked up on something that also struck me as odd, namely the way McGuire was so upset about paper copies of the book being sold before the release date. Wheeler does a nice job, using numbers from Nielsen BookScan, of showing exactly why this might matter: McGuire’s past sales suggest that, if everything broke just right, rapid sales in the first week could put her book on the extended New York Times bestseller list, which is a Big Deal. That would require, however, that her book sell a lot of copies in the first week, which is hurt by having some copies slip out a week early. So there’s some reason why she and her publisher should be concerned about the early release.

Of course, this relies on BookScan, which is an imperfect measurement– Wheeler includes the usual explanation: “BookScan captures, by general consensus, somewhere from 2/3 to 3/4 of the book outlets in the USA.” But exactly how good a measure is it? which leads to this graph:


This shows the sales for the trade paperback of How to Teach Physics to Your Dog, normalized so as to obscure the proprietary values, for a period of several weeks. Black circles are numbers from BookScan (which I’ve used before in modeling sales), green triangles are point-of-sale values provided by Scribner, which capture all the books sold.

This shows more or less what you’d expect: the two track each other pretty well, with the BookScan numbers generally a bit lower (there’s one point that’s actually higher, which I think happened because I miscopied the number and included some hardcover sales). The big spike in the data is the week before Christmas, with sales almost five times the average of the other weeks.

This is, however, subject to some rather stringent limitations.

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How to Present Scientific Data

In the same basic vein as last week’s How to Read a Scientific Paper, here’s a kind of online draft of the class I’m going to give Friday on the appropriate ways to present scientific data. “Present” here meaning the more general “display in some form, be it a talk, a poster, a paper, or just a graph taped into a lab notebook,” not specifically standing up and doing a PowerPoint talk (which I’ve posted about before).

So, you’ve made some measurements of a natural phenomenon. Congratulations, you’ve done Science! Now, you need to tell the world all about it, in a compact form that allows the viewer to make a good assessment of your results. Here are some rough notes on the best ways to go about this, starting with:

STEP ZERO: Know what point you’re trying to make. If you’re trying to interpret brand-new data, in the privacy of your own lab, office, or coffee shop, you can just slap together any quick-and-dirty sort of graph that you like, so you can see what you’re dealing with. When you’re preparing to present data to somebody else, though, you need to have a specific purpose in mind. Are you just comparing two numbers? Looking at how some property changes over time? Trying to characterize a distribution of numbers? Different goals will be best served by different types of presentations, and it’s important to have a clear idea of what you want to accomplish, so you can choose the right sort of graph for the job.

STEP ONE: Know your options. There are a whole host of different options when making a graph, as even a casual glance at Excel will show you. Some of these are versatile and powerful, some are only useful for such a ridiculously narrow range of purposes that I’ve never seen one used effectively. And, of course, if you look into data visualization, you’ll find a whole community of people who are really hard core about this stuff, crafting wholly original graphics specifically designed for each new data set they work with.

If you’re at a point where you have need of my input, though, there are really only a handful of options that you need to be aware of. As you get a better feel for your subject, you can start to explore others, but these will get you started. The “starter set” of data presentation methods, with appropriate applications is:

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The Test(ing) of Time: The Surprisingly Good Hourglass

My class this term is a “Scholars Research Seminar” with the title “A Brief History of Timekeeping,” looking at the science and technology of timekeeping from prehistory through modern atomic clocks. This is nominally an introduction to “research methods,” though the class operates under a lot of constraints that fully justify the scare quotes, at least for scientists. As I am a scientist, though, I want the class to include at least one original measurement and the reporting thereof, so I’ve been thinking of really simple measurements that I can have them do independently and write up. I’ve previously blogged about some measurements using cheap timers and the NIST web site, that I did as a preliminary for this class.

In poking around a bit, I stumbled across this European Journal of Physics paper on the physics of a sand glass (thanks to a passing mention of it in The Physics Book), which is a really nice piece that we’ll be going over in detail in class. This suggested another possible cheap and easy measurement, though, so I bought a pack of cheap sand timers from Amazon, and intend to hand them out (along with digital timers from our teaching labs and the NIST time URL) as another possible measurement.

i-bfcad2807a36586071214f6f2a193fb5-hourglass.jpgBecause it would be the height of foolishness to assign students to make a measurement that I haven’t done myself, I pulled one out of the pack (shown at right), and tested it myself. The testing protocol was extremely simple: I fired up the stopwatch app on my Android phone, and measured the time required for the timer to empty, flipping it back and forth until I had 32 total measurements (these were spread over the course of a couple of days, in intervals when SteelyKid was watching tv or otherwise distracted). Averaging together all 32 measurements, I come up with an average emptying time of 177.5+/-0.6 s, where the uncertainty is the standard deviation of the mean of the 32 measurements (which were rounded to the nearest second).

That by itself is better than I expect for cheap plastic timers that sell for less than $1 each– the uncertainty in the time is about 0.3% of the time, which is pretty darn good. But it’s actually much more interesting than that, if you dig into the data a little. Here’s a histogram of the emptying time for the various measurements, with the height of the bars indicating the number of trials falling in each 1-s bin:


Pretty striking, isn’t it? This is a classic example of a bimodal distribution– the average of all the trials is 177s, but there were basically no runs that gave a time of 177s. Instead, there was a big clump at around 174s, and another big clump at 180s.

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Scientific Commuting: The Data

A few months back, I did a post about estimating the time required for the different routes I take to work, looking at the question of whether it’s better to take a shorter route with a small number of slow traffic lights, or a longer route with a bunch of stop signs. This was primarily conceived as a way to frame a kinematics problem, but I got a bunch of comments of the form “Aren’t you an experimentalist? Where’s the data?”

Well, here it is:


This is a histogram plot showing the number of times my morning commute fell into a given ten-second bin over the last couple of months. The blue bars are for the main-road route, with four traffic lights, the red bars are for the back-road route with nine stop signs, and the green bars are a “hybrid” route which takes the main road a bit more than half the way to campus, then cuts onto some side streets to skip the last two traffic lights, replacing them with two stop signs and shortening the distance by a fairly trivial amount.

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Scientific Commuting: When Does It Make Sense to Take Alternate Routes?

I am an inveterate driver of “back ways” to places. My preferred route to campus involves driving through a whole bunch of residential streets, rather than taking the “main” road leading from our neighborhood to campus. I do this because there are four traffic lights on the main-road route, and they’re not well timed, so it’s a rare day when I don’t get stuck at one or more of them. My preferred route has a lot of stop signs, but very little traffic, so they’re quick stops, and I spend more time in motion, which makes me feel like I’m getting there faster.

That’s the psychological reason, but does this make physical sense? That is, under what conditions is it actually faster to take the back route, rather than just going down the main road?

Some parameters: the main road route covers 1.7 miles and contains four traffic lights. The back way covers 2.2 miles and has nine stop signs. The speed limit on all of these streets is 30mph, but I usually drive more like 35mph, or 16 m/s to put it in round numbers. I don’t really gun my car after any of the stops, so the acceleration is around 2 m/s/s (I’m enough of a dork to have checked this with the accelerometer in my phone, as well as counting “one thousand one, one thousand two…” while accelerating up to speed).

Given that information, how can I estimate the conditions under which it makes practical sense, rather than just psychological sense, to take the longer route rather than the main roads?

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Economic Astronomy II: Gender Shares of Jobs

The other big gender-disparity graph making the rounds yesterday was this one showing the gender distribution in the general workforce and comparing that to science-related fields:


This comes from an Economics and Statistics Administration report which has one of the greatest mismatches between the tone of the headline of the press release and the tone of the report itself. Nice work, Commerce Department PR flacks.

There are a couple of oddities about this report, the most important being that they appear to have excluded academics from the sample, though that probably depends on how people reported their jobs (that is, if you’re university faculty, would you list yourself as a “professor” or as a “physicist”?). The main issue, though, is more or less the same as with the report discussed in the previous post: in some sense, they’re looking at a time average over the last forty years or so of changing conditions.

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Economic Astronomy: Gender Gaps in Lifetime Earnings

There are two recent studies of gender disparities in science and technology (referred to by the faintly awful acronym “STEM”) getting a lot of play over the last few days. As is often the case with social-science results, the data they have aren’t quite the data you would really like to have, and I think it’s worth poking at them a little, not to deny the validity of the results, but to point out the inherent limitations of the process.

The first is a study of lifetime earnings in various fields that includes this graph showing that women with a Ph.D. earn about the same amount as men with a BA:


That’s pretty damning. But also a little deceptive, because this is a plot of “lifetime earnings,” which means that they are necessarily doing a social-science analogue of astronomy to make this graph.

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