Athletics Question: Establishing Relationship between Energy & Distance

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SUMMARY

The discussion focuses on establishing a relationship between energy expenditure and distance in athletics, specifically for middle and long-distance races. The energy expended during a run is calculated using the formula E = Distance * (Speed)^3. For a 5000m run, the energy is approximately 1,317,031 units, while for a 10000m run, it is about 2,309,070 units, indicating a ratio difference of approximately 1.753. The participants seek to derive a more sophisticated variable relationship between different race distances, such as 1500m and 3000m, rather than relying on fixed ratios.

PREREQUISITES
  • Understanding of energy expenditure formulas in athletics
  • Familiarity with cubic relationships in physics
  • Basic knowledge of middle and long-distance running times
  • Experience with mathematical tools like Maple or Mathematica
NEXT STEPS
  • Research the application of cubic relationships in energy expenditure for running
  • Explore the use of Maple or Mathematica for modeling athletic performance
  • Investigate empirical relationships between various race distances
  • Study the impact of air resistance on energy expenditure in running and cycling
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Athletes, coaches, sports scientists, and mathematicians interested in optimizing performance and understanding the physics of running energy expenditure.

eldrick
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Here is a problem to get your teeth around :

It is generally believed that a 5000m time of 13'00 is ~ equivalent to a 10000m time of 27'10 ( +/- 10s ). Now, it's pretty established that the energy expended in any run is given by :

E = Distance * ( Speed )^3

For 5000m, the Energy expended is : 5000 * ( 5000/780 )^3 = 1317031 u

For 10000m, the Energy expended is : 10000 * ( 10000/1630 )^3 = 2309070 u

( A ratio difference in this case of of ~ 1.753 )

I'd be grateful if posters could like at those energy values & "see" any empirical relationship to them ( to me it looked like 3^0.5, but we need something more sophisticated than by just doubling distance the energy expended only goes up by 3^0.5 . I'm looking for a variable relationship between different distances , e.g. between 1500m & 3000m rather than a fixed ratio which doesn't work as you increase distances ).

Thanks for any help.
 
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We are looking for "equivalent" performances for the main middle distance & distance track races : 800m, 1500m, 3000m.

For some guidance, it is generally reckoned that a 13'00 or a 27'10 is in ballpark of

800m : 1'43.5 +/- 1.0s

1500m : 3'31.0 +/- 2.0s

3000m : 7'30.0 +/- 5.0s
 
eldrick said:
Here is a problem to get your teeth around :

It is generally believed that a 5000m time of 13'00 is ~ equivalent to a 10000m time of 27'10 ( +/- 10s ). Now, it's pretty established that the energy expended in any run is given by :

E = Distance * ( Speed )^3

The energy expended when running is just proportional to distance, because the
resistance is approximately constant.
Even for an activity where nearly all of the power goes into overcoming air resistance,
such as cycling, the energy is only:

E = Distance * ( Speed )^2

or E = Time * ( Speed )^3
 
Believe me, the relationship is cubic :

http://www.atm.ox.ac.uk/rowing/physics/basics.html#section5

Use that.

Try & see if an empirical relationship using cubic relationship for energy &/or the suggested range of times above is obvious ( Maple or Mathematica ? Or just plain intuition/"eureka" moment )

Many Thanks
 

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