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:
Here, the points have the same meaning as before, in terms of which race is which. In this case, though, we're plotting the average speed as determined by the very crude method of taking the difference between two consecutive split times and using the fact that each split was for a 10m distance. If you look closely at the two graphs, you'll see that the points aren't at the same times, because these crude velocity measurements are really giving the average velocity over the 10m interval. Accordingly, I've assigned a time to each one that corresponds to the midpoint of that split interval. So, for example, the first 10m took around 1.8s to run, giving an average velocity of 5.5m/s, which I put at 0.9s, the middle of the time interval. The second 10m took about 1s to cover, for a speed of 10 m/s, at a time of 1.8+0.5 = 2.3s. And so on.
Again, while you can construct a complicated fit function that reproduces all the little wiggles in the data, what's important here is the general form, which involves a rapid acceleration up to a relatively flat plateau. Just as you would expect for the toy model I proposed yesterday.
And, in fact, the toy model from yesterday, with a top speed of 1 m/s and an acceleration of 6.4 m/s/s, is plotted on there as the solid line. It's not a great fit-- the initial acceleration is too fast, and the final speed is too low, but that's spectacularly good agreement for such a spherical-cow model. If this were astronomy, people would gush about how well the theory matches the data.
Interestingly, the agreement gets even better if you repeat the analysis from yesterday, but throw out the data from the 200m dash. The uncertainties get a little bigger, but the maximum speed increases a bit (to just over 12m/s) and the acceleration decreases (to about 4.5 m/s/s), fitting the actual race data even better.
This does push the distance over which the runner accelerates out a little farther, a bit over the 20 yard threshold discussed yesterday. Most of the acceleration takes place well before that, though-- at 20m, the speed is around 10 m/s, and it increases to 12 m/s by 40m. Which isn't nothing-- that's a 44% increase in the kinetic energy of the eventual collision, if this is a football play-- but it's not a revolutionary improvement in player safety, either.
So, all in all, I think this toy model worked surprisingly well when confronted with actual data (albeit of uncertain quality). It's always nice when introductory physics turns out to actually work to describe the real world...
A 44% change in KE isn't significant? Tell you what - we'll assign Ray Lewis to cover you, and the cheerleader can cover me.
What would be most interesting is a comparison with the numbers of a lesser sprinter - i.e. the data represent not just "maybe bad measurement" but "maybe measurement of a bad example".
If this were astronomy, people would gush about how well the theory matches the data.
Nope, astronomers never use linear fits for anything...
Your toy model assumes that the force that the sprinter can apply to the ground is constant (so that acceleration is constant). In practice the force that can be applied at a given power grows lower at higher speeds. At low speeds, the sprinter can compensate by generating more power to keep acceleration constant, but once the sprinter reaches the maximum power she can generate, then force (and acceleration) will decline as speed increases, until she's reached her maximum speed.
There's been a good deal of study of this effect, not for sprinters, but for railroad locomotives! There's a nice web page, written for train enthusiasts, which tries to explain the physics in words of one syllable at http://www.twoof.freeserve.co.uk/motion1.htm. Figure 3 looks like the general shape of your white dots.
For biological sprinters (including those of a 4-legged persuasion), the practical limit comes from maximum muscle contraction rates, not muscle strength. Given fixed gearing (i.e. leg and stride length), increased speed demands a higher stride rate.
So, I'm not sure the power-limited (also traction limited?) behaviour of a locomotive is a good match. A locomotive like a biological sprinter would have astoundingly high power-to-weight, but a strict RPM limiter on the wheels. The locomotive would surge up to the RPM limit and sit there.
So are any biological sprinters power, rather than stride-rate limited? Whales?.