Problem Statement:
At the Boston Marathon, the runners’ finish times are adjusted based on a formula that accounts for their position on the starting line. At our own Auburn Engineering Cupola Short Circuit 5k, finishing times are currently unadjusted for time delays and path diversion at the start line. Using engineering judgment, how could race time accuracy be improved?

Given:
Race Distance (d) = 5 kilometers
Longest Finish Time = ~45 minutes
Number of People (n) = ~150

Assumptions:
Temperature and humidity change between first and last racer’s start time is not significant in short race period

Using tracking chips is prohibitively expensive

The average runner is running at a constant rate of 6 mph or 8.8 ft/s. (v)

The average runner has a personal space radius of 2 feet (r_p)

Solution Process:
Solution 1:
The runners will be divided into groups of 15 people based on ascending order of their race number. Each group will be delayed by one minute. The large delay time would account for path diversion while passing, because the groups would have sufficient time to disperse. The race time would be adjusted based on starting position: t_f’=t_f-t_0. Where t_f’ is the adjusted time; t_f is the raw time; and t_0 is the start time based on group.

Solution 2:
The runners will be divided into groups of 15 people based on ascending order of their race number. Each group will be delayed by 20 seconds. Because the runners are closer together, their time will be adjusted for diverting their path to pass in the first quarter mile. One passing motion generally results in passing two to three people. Each passing motion will credit the runners’ time in the first quarter mile.

Passing Time = (πr_p)/v

Conclusion:
While Solution 2 may be more accurate, Solution 1 would be easier to implement.