Both Simpler And More Complex

From this week’s The Week That Was:

In a recent headline regarding Usain Bolt, we pointed out how his 200m world record of 19.19 is almost exactly twice his 100 meter world record and wondered how that could be. Well, the people over at speedendurance.com have written a nice piece explaining how it all works out.

You’d be surprised to find out how often people go for the complex explanation and miss the simple one. The linked article talks about acceleration and maximum speed and all that stuff and uses all kinds of physics terms.

But it makes the mistake of taking times and distances at face value. Sports measurement is not the same as physics measurement. Track & field is generally closer than any other sport, but it’s still not quite the same.

In a track & field sense, Usain Bolt ran 100 meters in 9.58 seconds and ran 200 meters in 19.19 seconds, showing no slowing over the longer distance. In a physics sense, Bolt ran those in 9.43 and 19.06 seconds, which does show some slowing. The difference occurs because the clock started at the report of the pistol, which was before Bolt began running. In track, we call this “reaction time”. It’s part of the sport, but it’s not part of how we measure speed in a physics class.

Then there’s the distance. We all understand that a football player credited with a 90-yard touchdown run may actually have moved more than 90 yards, because the field is measured in a straight line but the athlete probably didn’t move that way. In track, the issue is similar, but subtler and more complex.

The 100m is run in a straight line, and Euclidean geometry tells us that’s the shortest distance between two points. Because of this, we can’t run any less than 100 meters in a 100m race, but a sloppy runner can run more. We can confidently say Bolt ran 100 meters in 9.43 seconds for a speed of 10.60 meters per second.

But the 200 is run partly around a curve, and that makes distances less than clear. Athletes must stay in their lanes, but what does that mean? Track & field rules say an athlete’s feet must stay in his lane, but from a physics perspective the important thing is the center of gravity, and an athlete’s center of gravity can go into the next lane. Look at a photo of a sprinter hugging the lane line and leaning aggressively, and his COG is clearly in the next lane.

Then there’s the issue of how the lanes are measured. If you take a measuring wheel and roll it from a 200m start line to the finish line going along the left-hand lane line, it will give you a distance of somewhat less than 200 meters. This is because the true distance on tracks isn’t on the lane lines, but 30 cm (1 foot) outside of them. The shortage from 200 meters varies depending on the lane; while they all spend the same distance on the turn, outside lanes go through a smaller central angle. (If you want this part explained to you, seek out a pre-calculus teacher.)

Figuring out exactly how much distance a sprinter saves in a 200m race isn’t easy, and if they’re in the middle or outside of their lane they’ll have run more than 200 meters. But at this point I’ll just guess that Bolt probably ran about 199 meters.

So Bolt’s official times, from a track & field perspective, are 9.58 and 19.19, showing only an 0.16% slowdown. But his estimated speeds, from a physics perspective, are about 10.60 m/s and 10.46 m/s, showing a 1.49% slowdown.

As defined by sports, Bolt doesn’t slow down. As defined by physics, he does.