Sprint Interval Analysis
A track coach records a runner's split times over six 200 m intervals during a 1,200 m session. The times in seconds are 31, 33, 36, 40, 45, and 52. The coach models fatigue with an exponential function of the form t(n) = a·b^(n−1), where t(n) is the time for interval n. A second runner's times are 34, 35, 36, 37, 38, and 39 seconds. The coach wants to compare how each runner's pace deteriorates and decide which runner to enter in a 1,000 m race, where the winner is decided by total time. Both runners cover 200 m per interval.
1. Fit the exponential model to the first runner's data using the first and last intervals, and evaluate how well it predicts the middle intervals.
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Fit. t(1) = a = 31. t(6) = 31·b⁵ = 52 → b⁵ = 52 ÷ 31 ≈ 1.677 → b ≈ 1.677^(0.2) ≈ 1.109.
Predictions. t(2) ≈ 31 × 1.109 ≈ 34.4; t(3) ≈ 38.1; t(4) ≈ 42.3; t(5) ≈ 46.9.
Comparison with actuals (33, 36, 40, 45). Errors: +1.4, +2.1, +2.3, +1.9 seconds. The exponential model overestimates the middle intervals by 1–2 seconds, meaning the runner's fatigue is closer to linear early and steepens late. The model is a reasonable summary but not exact; the residuals are all positive, which signals curvature mismatch rather than random noise.
2. Compute the total time each runner would take for a 1,000 m race (five intervals) using their actual split patterns, and identify who wins.
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Runner 1, first five intervals. 31 + 33 + 36 + 40 + 45 = 185 seconds.
Runner 2, first five intervals. 34 + 35 + 36 + 37 + 38 = 180 seconds.
Winner. Runner 2 wins by 5 seconds. Runner 1 starts faster (31 vs 34) but fades sharply, while runner 2 holds a nearly constant pace. This shows that a fast first interval does not guarantee a win over a longer distance; the cumulative sum matters.
3. The coach wants a single number summarizing pace decline. Compare the average rate of change over the six intervals for each runner and explain why this measure can mislead when choosing a 1,000 m entrant.
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Average rate of change. Runner 1: (52 − 31) ÷ 5 = 21 ÷ 5 = 4.2 s per interval. Runner 2: (39 − 34) ÷ 5 = 5 ÷ 5 = 1.0 s per interval.
Interpretation. Runner 1 deteriorates more than four times as fast per interval, consistent with the exponential fit. But this single number hides the starting level: runner 2 begins slower yet is far more consistent.
Why it can mislead. The average rate of change only uses the endpoints, ignoring the middle shape. A runner could have a slow first interval and a fast last, giving a misleadingly low rate. For a 1,000 m race, the sum of the first five intervals is the decisive quantity, so the coach should compare cumulative times, not just the slope. Runner 2 remains the better choice here, but the reasoning must rest on totals, not the slope alone.