### Quantitative description of the tradoff between running faster vs. farther: performance = ln ( 1 + distance ) * ( distance / time )

Michael Redman - 2012 08 13

My original post to Usenet:

```2012 August 13 1545 UTC

i sought out to quantify the idea that there is an "equivalence"
between either running the same distance a certain amount faster, or
else running a certain amount farther at the same speed.

same with swimming.

by the same idea all the world record performances in, say,
backstroke, across all distances, ought to be "equivalent" in the
sense they all represent the best people can do.

after some investigation i found this formula

performance = ln(1 + distance*scale) * (distance*scale / time)

i was able to find scale parameters that tightly focused my samples of
scores for some current world records.

for running, with scale=3.05, which physically is like using distance
units of 1/3.05 m, or about a foot (anatomical coincidence?), the
average was about 19x the standard deviation for the 11 current world
record runs i sampled from 100m-100km (approximately doubling in
distance each time), and about 29x for the middle 9.

for swimming, with scale=116, the scores focused to average about 39x
the standard deviation.

i got my world records from wikipedia

men's running

meters	seconds	score, scale=3.05
100	9.58	182.2226989661
200	19.19	203.9191842155
400	43.18	200.811928634
800	101.01	188.4209275493
1500	206	187.188135625
3000	440.67	189.3997974528
5000	757.35	193.9580186086
10000	1604	196.33927671
20000	3321	202.3900565264
42195	7418	204.1143559653
100000	22413	171.8449762171

men's freestyle

meters	seconds	score, scale=116
50	20.91	2403.7090604354
100	46.91	2314.2745753825
200	102	2286.3260325101
400	220.07	2265.5134268295
800	452.12	2347.7531143433
1500	871.02	2410.5371180953

the physical interpretation of the log term, is that at a given speed
you have to go an amount of extra distance proportional to what you
are already doing, in order to be equivalent to improving your average
speed on the same distance by "1 unit".  if you improve your speed for
the race by "1 unit", since it is an average speed that is something
you sort of did over the whole race.  so it is not equivalent just to
run 1 extra unit of distance at the end of the race.  the log makes
incremental distance proportional, just like exp models how the money
in your bank account grows in proportion to what you have.

so the basic form of the formula is really

performance = ln(distance) * speed

unfortunately the way that formula orders a given sample of races of
different distances, is not invariant to your choice of units for
distance.  moreover with long distance units like kilometers, some
races would be for less than 1 unit of distance, so in the basic
formula the log term would turn negative and the formula would be
physically meaningless

so the 1+ term in the full formula reflects the physical reality that
all races are for at least some positive distance.

and the "scale" term is  your choice of distance units.  the log term
is close to linear near 0 so when the distance units are long (fewer
units of distance traveled) the formula approximates distance^2/time,
which gives an edge to longer races.  and the log term gets flatter
towards infinity so when the distance units are short and more are
traveled the formula approximates distance/time, which gives an edge
to the sprints.

however as the data above demonstrates, with a suitable choice of
distance units the formula behaves well across the 3 or so decimal
places of distance scale that the different races cover.

does any one know if there is (or even can be?) another formula, which
does the same thing, except that its ordering of performances is
naturally invariant to a change of distance units?
```

Copyright 2015 Michael Redman.

IN GOD WE TRVST.