Athletics Question: Establishing Relationship between Energy & Distance

[eldrick]

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.

 

[eldrick]

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

 

[willem2]

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

 

[eldrick]

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

 

[pmacias]

I find this question very interesting and worthy of further investigation. The relationship between energy and distance in athletics is a complex one and is affected by various factors such as the individual's fitness level, running technique, and environmental conditions. However, let's take a closer look at the calculations provided and see if we can identify any patterns or relationships.

Firstly, it is important to note that the equation used to calculate energy expended (E = Distance * (Speed)^3) is based on the assumption that the runner is maintaining a constant speed throughout the entire distance. In reality, this is not the case as most runners tend to start off slower and gradually increase their speed as the race progresses. Therefore, the calculated energy values may not be entirely accurate but can still provide some insights.

From the calculations, we can see that the energy expended for a 5000m race is approximately 1.753 times less than that of a 10000m race. This suggests that there is a positive correlation between distance and energy expended, which makes sense intuitively. As the distance increases, the runner needs to maintain their energy output for a longer period of time, resulting in a higher overall energy expenditure.

However, the provided equation does not take into account the intensity of the run, which can greatly affect the energy expended. For example, a runner who maintains a steady pace throughout the entire 5000m race will expend less energy compared to a runner who sprints the first half and then slows down for the second half. In this case, the relationship between energy and distance may not be linear.

In order to establish a more accurate relationship between energy and distance, a more comprehensive study would need to be conducted, taking into account various factors such as intensity, terrain, and individual differences. Additionally, using a larger sample size and controlling for external factors would also provide more reliable results.

In conclusion, while the provided calculations do suggest a positive correlation between energy and distance in athletics, further research is needed to establish a more precise relationship. I would recommend conducting a more controlled study to investigate this topic further.