Running around a track II

Task

An Olympic $400$ meter track is made up of two straight sides, each measuring $84.39$ meters in length, and two semi-circular curves with a radius of $36.5$ meters as pictured below:

In a $400$ meter race, runners are staggered with those in the outermost lanes starting the furthest ahead on the track: this way they can all complete the race at a finishing line perpendicular to the track in the straightaway where they begin the race. The width of each lane is $1.22$ meters. Also important for this problem is the fact, as per Olympic guidelines, that the $400$ meter distance for lane $1$ is measured $30$ centimeters from the inside of the track, and $20$ centimeters from the inside of each other lane . This is pictured below:

1. How does the perimeter of the track $20$ centimeters from the inside of lane $2$ compare to the perimeter $30$ centimeters from the inside of lane $1$? How far ahead should the runner in lane $2$ start, compared to the runner in lane $1$, if they are both to complete $400$ meters at the finishing line on the straightaway section?
2. How does the perimeter of the track $20$ centimeters from the inside of lane $3$ compare to the perimeter $20$ centimeters from the inside of lane $2$? How far ahead should the runner in lane $3$ start, compared to the runner in lane $2$, if they are both to complete $400$ meters at the finishing line on the straightaway section?
3. In a longer distance race where the runners are all toward the inside of the track, why is it more efficient for a runner wishing to pass others to do so in the straightaway section of the track instead of through the curves?

IM Commentary

The goal of this task is to model a familiar object, an Olympic track, using geometric shapes. Calculations of perimeters of these shapes explain the staggered start of runners in a $400$ meter race. The specifications for an Olympic track indicating that the distance around for lanes $2$ and greater is measured from $20$ centimeters of the inside of the given lane is found on pages $35$ and $36$ of the following document:

• Olympic Track Specifications

For lane $1$ this measurement is taken $30$ centimeters from the inside of the lane while for the other lanes it is taken $20$ centimeters from the inside of the lane. The teacher may wish to explain this to the students so that the values in the set-up of the problem make sense.

In order to get the values in the solution below, an approximation of $3.1416$ for $\pi$ is sufficient. If students are using a scientific calculator to evaluate the expressions they can use the $\pi$ button but if not teachers may wish to share this value with students.

This task addresses the ''staggered start'' of racers who stay in their lanes for one lap. Because the runners on the outside lanes have further to go through the curves, they start an appropriate distance ahead of runners inside of them on the track. The staggered starts can be calculated explicitly as is done here.

This task is primarily intended for instruction as it builds on ''Running around a track I'' and some time is necessary to explain the different numbers occurring in the task (namely the distances of $30$ and $20$ centimeters where perimeters are measured.

Solution

1. We are given that the perimeter of the track $30$ centimeters from the inside of lane 1 is $400$ meters. For the perimeter $20$ centimeters from the inside of lane 2, we can calculate as follows. The straightaway sections are still each $84.39$ meters long. However, the curved sections form a circle whose radius is now $$36.5 + 1.22 + 0.2 = 37.92$$ meters. The diameter of the circle will be $2 \times 37.92 = 75.84$ meters. So the perimeter of the track $20$ centimeters from the inside of lane 1 is $$2 \times 84.39 + \pi \times 75.84 \approx 407.04.$$ So the perimeter of the track $20$ centimeters from the inside of lane 2 is approximately $7.04$ meters larger than the perimeter $30$ centimeters from the inside of lane 1.

2. For the perimeter $20$ centimeters from the inside of lane 3, we can calculate as follows. The straightaway sections are still each $84.39$ meters long. However, the curved sections form a circle whose radius is now

   $$36.5 + 2 \times 1.22 + 0.2 = 39.14$$ meters. The diameter of the circle will be $2 \times 39.14 = 78.28$ meters. So the perimeter $20$ centimeters from the inside of lane 2 is $$2 \times 84.39 + \pi \times 78.28 \approx 414.70$$ meters. So the perimeter of the track $20$ centimeters from the inside of lane 3 is approximately $7.66$ meters larger than the perimeter $30$ centimeters from the inside of lane 2.

   Note that to compute the difference in length between the perimeters in different lanes, the straightaway sections do not need to be considered because they are the same for all lanes. So we are only interested in comparing the circumferences of two circles with different diameters $d_1$ and $d_2$. This will be given by the formula

   $$\pi \times (d_2 - d_1)$$ assuming $d_2$ is the larger of the diameters. So as we continue to move out the track, comparing lane 3 to lane 4 and so on, these values will all be the same since the $d_2 - d_1$ term will always be $2 \times 1.22$ meters.

3. If a runner wishes to pass in the straightaway, the only extra distance to be travelled is the distance to move outside of the other runner when passing. If a runner passes on the curved section of the track, however, then she is running around the circumference of a bigger circle and consequently is running further, in addition to the distance to move outside of the other runner and then back again.