first time poster and definitely no maths expert.
I am trying to solve a basic problem using an athletics track. The total distance around a standard athletics track is 400m:
If you run in the first lane you run 400m, I am trying to work out the formula to estimate the distance run in the 2nd, 3rd, 4th lane etc if they all start at the same point (no staggers)
If I assume that the distance between lane 1 and two is 1m how would I go about calculating?
Any help would be much appreciated!
Thanks
Best Answer
Let $L$ be the distance of the straight part, and $R$ be the radius of the turns. Now, the total length (when you're running on lane 1) is $400~\text{m}$, or $$\tag{1} 2L +2 \pi R = 400 $$ We also assume that the straights are exactly $100~\text{m}$. Then we see that on lane 1, the radius of curvature is $R = \frac{400-2\cdot 100}{2\pi}\approx 31.83~\text{m}$.
When moving from the first lane to the next one, the radius $R$ increases by 1 meter, and the straight parts remain the same. Therefore, as we can see from Equation (1), the total length increases by $2\pi$ meters per every meter that the radius increases.