Middle-distance running

Worked example 1: the cost of running wide. On a bend, lane 1 is measured 0.30 m from the kerb (radius 36.80 m) and lane 2 is measured 0.20 m from its inner line (radius $36.50+1.22+0.20=37.92\mathrm{m}36.50\; +\; 1.22\; +\; 0.20\; =\; 37.92\backslash ,\backslash mathrm\{m\}$). How much further does an athlete run by staying in lane 2 for one whole bend, and how much time does it cost at $5.7\mathrm{m}{\mathrm{s}}^{-1}5.7\backslash ,\backslash mathrm\{m\backslash ,s^\{-1\}\}$ (a 2:20 pace for 800 m)?

1. A bend is half a circle, length $\pi r\backslash pi\; r$. Extra distance: $$\pi (37.92-36.80)=\pi \times 1.12=3.52\mathrm{m}\backslash pi\; (37.92\; -\; 36.80)\; =\; \backslash pi\; \backslash times\; 1.12\; =\; 3.52\backslash ,\backslash mathrm\{m\}$$
2. Extra time: $$\frac{3.52}{5.7}=0.62\mathrm{s}\backslash frac\{3.52\}\{5.7\}\; =\; 0.62\backslash ,\backslash mathrm\{s\}$$

A runner who stays on the outside of the pack round all four bends of an 800 m runs about 14 m further, more than 2 seconds. Overtake on the straights when you can, and get back to the inside before the bend.

Worked example 2: planning even splits. Uwase wants to run 800 m in 2:20 and 1500 m in 5:00.

1. 800 m in 140 s, two laps: $\frac{140}{2}=70\mathrm{s}\backslash frac\{140\}\{2\}\; =\; 70\backslash ,\backslash mathrm\{s\}$ per lap, 35 s per 200 m. Average speed $\frac{800}{140}=5.71\mathrm{m}{\mathrm{s}}^{-1}\backslash frac\{800\}\{140\}\; =\; 5.71\backslash ,\backslash mathrm\{m\backslash ,s^\{-1\}\}$.
2. 1500 m in 300 s: $\frac{300}{15}=20\mathrm{s}\backslash frac\{300\}\{15\}\; =\; 20\backslash ,\backslash mathrm\{s\}$ per 100 m. The race starts 100 m round the first bend, so she first runs 300 m to the finish line, then three full laps: $$300\mathrm{m}:3\times 20=60\mathrm{s}\text{each lap}:4\times 20=80\mathrm{s}300\backslash ,\backslash mathrm\{m\}:\; 3\; \backslash times\; 20\; =\; 60\backslash ,\backslash mathrm\{s\}\; \backslash qquad\; \backslash text\{each\; lap\}:\; 4\; \backslash times\; 20\; =\; 80\backslash ,\backslash mathrm\{s\}$$ Splits at the finish line: 1:00, 2:20, 3:40, 5:00.
3. She actually runs the 800 m in 65 s and 77 s. Total 142 s: 2 s slower than planned, after a first lap 5 s too fast. The fast start built up lactate early, and she lost 7 s against plan on the second lap.