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quantification of "performance equivalence curve" for speed and distance in running or swimming
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bornforthesummer
Aug 13, 2012, 11:45:41 AM8/13/12
to geoffr...@yahoo.com, rtar...@stanford.edu, lphop...@gmail.com, vic...@hotmail.com, ada...@yahoo.com, james...@hotmail.com
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?
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?
Androcles
Aug 13, 2012, 12:47:51 PM8/13/12
to
"bornforthesummer" wrote in message
news:d5e07103-50ba-4545...@sn4g2000pbc.googlegroups.com...
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.
Look at the winner of the Olympic Marathon, Kiprotich. He ran
the same distance as everybody else, and for 99% of the way
he ran at the same speed as everybody else. He won because
he put on a short burst that put him ahead, then he was able
to slow again, back to the same speed as everybody else.
Same in cycling, same in swimming, same in kayaking, same
in all races. Only in the 100 or 200 metres or a relay race does
anyone run at top speed ALL the way. It's the short burst of
extra speed that determines the winner, and it doesn't last
long. Nor does it matter when it occurs, the winner can lead
all the way or pass his opponent as he crosses the finish line.
The mistake you are making is to consider the speed as the
whole distance divided by the time as if the speed was constant,
but in reality it isn't. In sprint cycling, for example, the race
usually begins at a very slow speed.
bornforthesummer
Aug 13, 2012, 12:50:15 PM8/13/12
to
On Aug 13, 3:45 pm, bornforthesummer <bornforthesum...@gmail.com>
wrote:
clarification - really meant to ask if there was another formula with
ordering invariant to change in units period, not just distance
units. wouldn't do any good to just have the same problem with the
time variable instead.
wrote:
ordering invariant to change in units period, not just distance
units. wouldn't do any good to just have the same problem with the
time variable instead.
bornforthesummer
Aug 13, 2012, 1:22:23 PM8/13/12
to
miles, but that's my point, and the relationship I quantified. If
people could do a marathon at sprinting speed, they would, but they
can't. The best average speed for a marathon is still slower than the
best average speed for a 100m. So I guess I could have asked the
question differently: "if you're going to go farther, how much slower
do you have to go?" I wasn't concerned with whether or how people
pace themselves for any specific distance, or what the differences are
in those behaviors between different distances. Just in describing
the relationship generally.
Androcles
Aug 13, 2012, 1:40:03 PM8/13/12
to
"bornforthesummer" wrote in message
news:c76bdf06-078c-4fe7...@sn4g2000pbc.googlegroups.com...
A car goes 1 mile up a hill at 30 mph. How fast must it come down
again (same distance) for an average speed of 60 mph?
bornforthesummer
Aug 13, 2012, 2:04:04 PM8/13/12
to
2 miles @ 60 mph = 2 minutes, which is the time the car took to get up
the hill.
therefore it can only average 60 mph if no time lapses on the trip
down.
i hope you have a fast car.
Androcles
Aug 13, 2012, 2:40:49 PM8/13/12
to
"bornforthesummer" wrote in message
news:036c9a11-8e68-42d3...@vg10g2000pbc.googlegroups.com...
I hope you now know I have no idea what you mean by "best average speed".
bornforthesummer
Aug 13, 2012, 3:55:47 PM8/13/12
to
it in.
their average speed = that distance / that time.
for a specific distance, you calculate the average speed that way for
every time anyone has ever run it.
the best average speed for that distance, which will correspond to the
lowest time for that distance, is the highest of those numbers.
if you did a certain distance in a world record time, you also did it
with the best average speed ever. my understanding of the world
records is that split times are irrelevant.
Androcles
Aug 13, 2012, 4:27:46 PM8/13/12
to
"bornforthesummer" wrote in message
news:165cc508-b018-49db...@vz5g2000pbc.googlegroups.com...
A car travels up a hill 1 mile in two minutes, comes down the other side
1 mile at the speed of light, split times are irrelevant, the ordinary
average speed is 2 miles in 2 minutes, what is its best average speed
and why did the driver get a speeding ticket when the limit is 70 mph?
I hope you have a fast lawyer.
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