Virtual Nerd

If you haven’t guessed it already, we here at Virtual Nerd are not exactly the “athletic type”. That sad reality does not stop us from being big sports fans, and contests of speed are no exception. So we were quite excited to see Usain Bolt break the world record and run 100 m in 9.72 seconds. But I think it would be fun for all to practice some unit conversion and figure out just how fast he is.

So first off, lets get a quick estimate. Let’s assume that he was going at the same speed for the entire run. In that case the speed is simply found from the equation

Solving for speed we get

But what would that be in miles per hour? Well lets convert meters per second to miles per hour. We do that by converting seconds to minutes, and then minutes to hours. And then, meters to centimeters, centimeters to inches, inches to feet, and then feet to miles. Putting it all together it looks like this.

I’d recommend learning to do this on your own, even though you may be tempted to wait and just use the new iConvert application for your iPhone 3G. Anyway, once we calculate all this we get an average speed of about 23 miles per hour. Well, that’s pretty fast, but we know that Usain started at rest, and the 23 mph number assumes that he started going at 23 mph and kept going at 23 mph.

So to get a better sense of this we would actually make things more complicated, and assume that he is accelerating uniformly, getting faster and faster as he goes, until he reaches the 100 m distance. Then if you plotted his speed vs time you’d get a triangle, and the area of the triangle would be the distance, or 100 m. Since the area of a triangle is half the base times height, and the base in this case is 9.72 seconds, then we’d find that the height, or final velocity, would 20.56 m/s or twice the speed we found above. That means he would be going at 46 mph by the end of the run, and have an average speed of 23 mph as above.

If you think that sounds too fast, you’re right, and the problem is that a sprinter accelerates for only the first 30-40 m, and then tries to keep a constant speed for the rest of the race. So our graph of speed vs time now has an increasing line that turns into a constant horizontal line. If you look at the graph below, you’ll see that we have two equations and two unknowns, and it leads to an equation that says 9.72*s=130. Solving for the flat line speed we find that speed is actually 13.37 m/s.

Now if we convert that just as we did above, we get a maximum speed of about 30 mph. That seems more reasonable, and we can probably safely say that Bolt’s speed was somewhere between 23 and 30 mph.

[Update: Thanks to Andy M for this link to a great post at sportsscientists about Bolt's 100m performance at the Olympics]

So with this nice estimate of speed, you may wonder, how much energy does Bolt produce during his record breaking run? And how much power does he produce? This actually is a tougher question, and we will have to wait until a future post to figure that one out.