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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:

enter image description here

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

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  • $\begingroup$ We would have to know the dimensions of the track. The total length is 400m, yes, but do we know the radius of the turn, for example? $\endgroup$
    – Matti P.
    Jun 24, 2020 at 11:30
  • $\begingroup$ You have 2 half circles (so one circle) and 2 straights. How long is each straight (it looks less than 100m)? Now work out the circumference of your inner circle. The next lanes circle will have a radius 1m wider. $\endgroup$
    – Paul
    Jun 24, 2020 at 11:30
  • $\begingroup$ for this example lets say the straights are exactly 100m each. Does that help? @MattiP. and @Paul? $\endgroup$
    – SOK
    Jun 24, 2020 at 11:32
  • $\begingroup$ 36.50m the radius of the semi circles $\endgroup$
    – SOK
    Jun 24, 2020 at 11:33

1 Answer 1

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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.

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  • $\begingroup$ This is great. Thanks very much. So for example if i had a much bigger track that had a circumference of 1000, a radius of 80 amd a straight of 248 . I would simply change the 80 to 81 in formula (1) and it should give me the result? $\endgroup$
    – SOK
    Jun 24, 2020 at 11:42
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    $\begingroup$ @SOK Exactly like that. Have a nice day! $\endgroup$
    – Matti P.
    Jun 24, 2020 at 11:46

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