How Do You Ensure Track Runners Go an Equal Distance? Math!

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In every Olympic event, officials try to keep things is as fair as possible. In track, this means making sure runners cover the same distance. Short distances make this easy–simply paint a perpendicular line across the track to denote the starting and finishing points. This works quite nicely for the 100 meter race.

But what if the distance is longer than 100 meters? Of course, officials could use a longer track, creating, say, a 400 meter track for the 400 meter hurdles. But with longer distances, it starts getting difficult for spectators to see all the action. Instead, the answer is a curved track. Modern track and field tracks typically are an oval.

This presents a problem. If you run around the entire track, an inner lane is a shorter distance than an outer lane. The solution to this is to make the athletes start at different points on the track. But how far apart should they start?

Let’s look at an arc length of a circle:

Spring 2016 Sketches key

Above you can see two different track lanes. Both lanes have a circular shape. They share the same center, but have different radii. If two runners started and finished next to each other (on a super-short race) they would have the same angular displacement, which I am labeling ?. But they would have different linear distances. It’s fairly easy to calculate both of these arc lengths (which I am calling s):

La te xi t 1

Warning: You must have the angle in units of radians or this won’t work.

Now let’s build a course for the 200 meter sprint. Let me assume the track consists of two 88 meter straight sections joined by circular ends with a radius of 35.75 meters (for the inner lane radius). I obtained these values by measuring the distances as they appear for Hayward Field on Google Maps, so my values are approximations. Maybe I should call these values L for the length of the straight part and r0 for the inner radius.

Clearly there is not enough room to have the entire 200 m race on straightaway. Instead, I’ll start on the curved portion and place the finish line on the end of the straight. This means the finish line will be perpendicular to the track but that the runners will have to start at different positions. Each path will have a straight length of L plus some amount of curved track. I can write that as:

La te xi t 1

The innermost lane must have a larger angular size to equal the same length as outer lanes. If I know the width of a lane, I can find the incremental decrease in angular size as the lane get farther from the center. Wikipedia cites a lane width of 1.22 meters. I will call this value ?r. I can write the first two lane distances as:

La te xi t 1

Since they are both 200 meters, I can set the two right sides equal to each other and simplify a bit to get the following:

La te xi t 1

It probably is safe to assume the product of ?r and ?? is small–so I will drop that term. Now I can solve for ??:

La te xi t 1

Now I can use my value for r0 and a distance of 200 meters to find the angular position of the inside lane with a value of 3.13 radians (right about at 180 degrees). So, this inner lane starting position will be at the end of the circular part of the track.

The next lane will have a larger radius and thus a smaller starting angle. With these values, the angular adjustment will be 0.107 radians (or 6.13 degrees). Each successive lane will start with a lower angular position by approximately the same amount.

But what about the 400 meter starting position? These will be spread out even more. Since the 400 meter race includes a greater section of curved track (almost 180 degrees worth) the angular position for the inside lane has a larger value and gives the change in angular position a greater value as well.

Some race events do not have lane restrictions and runners can move to the innermost lane. For these events, the starting line is curved such that everyone starts about the same distance from the first inside curve.

Although all the lanes in a race cover the same distance, some runners prefer particular lanes. The innermost lane has the disadvantage of not being able to see the other runners and also has the smallest radius of curvature (so you must turn harder). I would guess that this would be the least desirable lane, but I’m not a runner.