# Uneven Pace Running ?

#### eldrick

I hope someone can help me out with a question I've been trying to answer for quite some time with no success

A few years ago, Keninisa Bekele, the double Olympic Champ at 5 & 10km, won a World Championship 10km with incredible splits of 14:00 & 13:00 for each 5k for a final clocking of 27:00. The World Record at the time was 26:20 for 10km. Many observers felt this performance was superior to the World Record ( obviously run at even pace )

The question is :

Does anyone have a good method of working out theoretically what consecutive splits of 14:00 & 13:00 would imply for a race run at perfectly even pace from start to finish ?

I have tried various approaches using Kinetic Energies, but not really got anywhere.

Any help would be gratefully appreciated

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#### mgb_phys

Homework Helper
Not sure what you can really say in terms of physics, tt's more a question of physiology. Yes the second half was run at a pace which would have given a 26:00 record time - but he/she was presumably only able to run the second half at a 13:00 because they took the first half slowly, if they had run the first half at 13:00 they might only have been able to manage 15:00 for the second half.

#### eldrick

I agree physiology is a huge factor

However, consider the same problem by replacing the human with a point mass. Is there a way of working out the overall hypothetical even pace

#### mgb_phys

Homework Helper
If you knew how energy consumption (ie power output) varied with speed for a human then you could work out what 'average' speed would use the same amount of energy.

#### eldrick

I am trying to get a physics/mathematics perspective rather than a physiology one

Staff Emeritus
Not sure what you can really say in terms of physics, tt's more a question of physiology.
Exactly. The only "physics answer" (which is really a mathematics answer) is he can run a 10K in 27:00 or faster, and a 5K in 13:00 or faster.

#### Bob S

If air drag for a runner were similar to automobiles, the power expended goes as speed cubed. (Stokes drag power goes as speed squared). A runner has a much higher drag coefficient than an automobile, but a smaller cross section. So Berkele ran a 10k at an average speed of 0.370k per minute. Let's use this as a reference power level

So running at two splits of 13.5 minutes the relative energy expended (REE) would be

REE = 13.5m x (.370/.370)3 + 13.5m x (.370/.370)3 = 27.00

and running splits of 14m (at .357 kpm) and 13m (at .385 kpm) would be

REE = 14m x (.357/.370)3 + 13m x (.385/.370)3 = 27.22

So more energy is expended if Berkele had run two splits at different speeds. This would imply that Berkele expended about 27.22/27 = 0.81% more energy (not power) than if he had run 2 even splits.

#### rcgldr

Homework Helper
One of the reasons for uneven pace running is that the runner can use up "reserves" towards the end of a race, since there's no need for any "reserves" once the race is completed.

#### eldrick

If air drag for a runner...
I'd like to post some updates :

1) I think I have an answer for original question based on a very similar method to Bob's, from :

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

I tried to move on from there here (as ventolin^2 ) :

I would very much appreciate a critique of the various methods

2) I have tried to move on further to actually see if it's possible to use relative "energy equivalence" to find what a time over 1 distance implies for another :

The problem is, i've used sleight-of-hand with using exponent 3/4 for inverse of s^4/t^3 ( applicable to (s/t)^(4/3) but not s^4/t^3 )

Empirically, it's giving very good answers (!?!) but like I said, not strictly legit

It's a very important question & can be considered the Holy Grail of Athletics as there are numerous calculators based on physiology or statistical analysis of annual performance lists, all of which are very poor

Any help with getting to a correct physics based solution will answer the biggest question in the main Olympic Sport - there's a challenge !!!

( post replies either here &/or that athletics forum thread ( no registration required, so you can "freewheel" it there ) )

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