How Far is Mile Start Line from Finish Line at end of Lane 8

[eldrick]

I'm not proficient here, nor skilled, but it is it one of most important calculations in sport.

Every point on Mile start line means all run 1609.344m by hitting tangent at various points on 1st bend

Every guy has to hit a tangent at assumed different parts of bend

( 36.80m is shortest distance run in inner lane-1 to run 400m or 1609.344m, even though kerb is 36.50m

All runners obviously must run 1609.344m from staggered starting points on start line until gradually merging but never immediate coincide ! )

I have tried for weeks with my meagre brain & got nowhere but 5 simultaneous equations for this, involving Trigonometry

How far Mile start line at end of 8th lane from Finish line ?

 

[berkeman]

I'm not proficient here, nor skilled, but it is it one of most important calculations in sport.

Every point on Mile start line means all run 1609.344m by hitting tangent at various points on 1st bend

Every guy has to hit a tangent at assumed different parts of bend

( 36.80m is shortest distance run in inner lane-1 to run 400m or 1609.344m, even though kerb is 36.50m

All runners obviously must run 1609.344m from staggered starting points on start line until gradually merging but never immediate coincide ! )

I have tried for weeks with my meagre brain & got nowhere but 5 simultaneous equations for this, involving Trigonometry

How far Mile start line at end of 8th lane from Finish line ?

What have you found with your Google searches? I'd start with a Google Images search and some pertinent keywords...

 

[berkeman]

Also, why would a mile run use measurements in meters?

 

[berkeman]

This is close...

http://track.isport.com/Image.ashx?rs=800x600&dir=Images\\Guide&File=Img_Popup_16258101212011050649.jpg

 

[berkeman]

Actually that is a pretty helpful website...

http://track.isport.com/track-guides/track-field-dimensions

 

[NTL2009]

Maybe I am missing the question, or maybe I fully understand it?

This sounds similar to the question - how many grooves on one side of a 78 RPM record? One.
Isn't the answer to your question obvious: 1 mile? No math needed?

 

[berkeman]

Maybe I am missing the question, or maybe I fully understand it?

This sounds similar to the question - how many grooves on one side of a 78 RPM record? One.
Isn't the answer to your question obvious: 1 mile? No math needed?

No, I'm pretty sure s/he is asking about how to lay out the staggered start for a mile running race. They have to stay in their lanes for the first lap or so, I think. After that they all can go to the rail for the rest of the race.

 

[NTL2009]

OK, but then I guess I don't understand why it isn't some very basic math and a bit of trig. The straight lanes are all equal, the curved tracks are just a circumference calculation ( 3 turns being 3/4 of the total circumference), then calculate a ratio of the last turn to make the length equal to the inner track calculation.

Oh, OK, sorry - I wasn't following that they go out of their lane. Never was a sports fan! Hah!

 

[eldrick]

I would not ask this question if it was simple & I could do it myself !
It works out with 5 horrible equations with 5 unknowns :
Here is more detail posted elsewhere :

"this is a relevant track problem
it's about the curved start for races not 100 or 200 or 400
it's about the curved-line start :

[link]http://3.bp.blogspot.com/-63-cvJfRh8I/UTpWShIStFI/AAAAAAAAaMI/202g4mxFhd4/s1600/IAAF_400m_Track.jpg[/link]

or you can find plenty others with better image specs

the question i want answered is :
how far horizontally from finish line is curved mile start line at it's end at end of 8th lane ?

it's 9.344m for guy in lane-1 who is assumed 30cm out from kerb which is 36.5m from origin of curve
but we are interested in end of curved-line of mile start where it hits end-of-lane-8
some useful info :

- each lane is 1.22m ( historically 4' )

- the guy theoretically in end-of-lane 8 of start line is expected to run tangent to 1st bend to hit it at 36.8m from origin

- he therefore, theoretically will coincide with guy in pole in lane-1, who has run 9.344m straight, then continues at somewhere along 36.8m curve from origin until they collide, former having run straight line/tangent, latter having run 9.344 + some part of 36.8 radius, but both have run same distance at collision point !

- the distance from origin to vertical at end of lane-8 is 36.5 + ( 1.22 * 8 ) = 46.26

i have ended up with 5 equations with 5 variables, so it's not easy !
if someone can solve it, i will be very, very impressed !

( in comparison, the horizontal distance from the 2k or 10k start on outer edge to vertical start line is trivial calculation at :
5.11m

check on diagram at max size & use ruler :
it shoud be ~ 5.0 - 5.2m if you can measure closely enough )

the mile start line distance on end of lane-8 to start line is immensely difficult
i have reduced it to 5 equations, with
z
being answer

i've looked for it for days & best i can reduce it to is to solve for
z

the equations are :
y^2 = x^2 + 36.8^2
y^2 = z^2 + 46.26^2
x = 9.344 + ( 36.8*pi * ( a / 180 ) )
x = 36.8 * tan ( a + b )
y = 46.26 / cos ( b )

the required is
z
obviously, solving for any variable will get the rest
however, it is 5 equations with 5 variables & it's ugly with trig functions involved"
I hope this helps !

 

[eldrick]

Anyone have time to input those 5 equations into Maple or Mathematica as I don't think there is any easy way of solving those 5 equations otherwise.

Looking at scale diagram, the answer should be around 5m mark