I’ve been wondering occasionally how we should describe group actions. What determines what the group can accomplish vs the individuals acting independently?
One particular example of this is a relay race vs individual runners. We could think of the task as requiring getting a token across a certain distance, the shorter the time, the better. What’s the fastest an individual could get something a certain distance vs a team? What is the team’s total time as a function of the speed of its individuals over each segment?
Usain Bolt set the world record for the 100 meter dash in 2009: 9.58 seconds.
He was also part of the team that set the world record for the 4 x 100 relay in 2012: 36.84 seconds.
One curious thing: I had expected the 4 x 100 record time to be greater than 4 times the individual record time for 100 meters. But it’s actually less. 4 * 9.58 = 38.32, 1.48 seconds longer than the actual relay record.
To really get at these questions, it’d be better to look at the individual 100 meters times for the participants, their 400 meters times, then look at the times for relay teams composed of the same individuals. But the world records data are easily available so I’m starting there.
How about an individual running 400 meters? The world record time was set by Wayde van Niekerk in 2016: 43.03 seconds. So, 6.19 seconds longer than the 4 x 100 relay record; this at least seems less paradoxical.
Usain Bolt also holds the record for 200 meters: 19.19 seconds, also from 2009. (Doubling his 100 meter time gives: 2 * 9.58 = 19.16, off by just 0.03 seconds from his actual time for the 200 meter dash … so his pace for the 200 meter dash was very similar to that for the 100 meter dash.)
The world record for the 4 x 200 meter is 1:18.63 = 78.63 seconds, set in 2014; this is 1.87 seconds greater than we get multiplying the record time for one segment: 4 * 19.19 = 76.76 seconds.
David Rudisha holds the world record for 800 meters: 1:40.91 = 100.91 seconds, set in 2012. So, 100.91 – 78.63 = 22.28 seconds longer than the relay for the same distance.
I’m looking at https://en.wikipedia.org/wiki/List_of_world_records_in_athletics for most of the above. It has some nice plots of time vs distance and average speed vs distance, also including links to download the raw data.
Here is record time vs distance:
library(ggplot2)
m <- read.csv("~/Downloads/Running records by race distance and time.tab.csv", stringsAsFactors=F)
m$distance <- m$x
m$time <- m$y1
ggplot(m, aes(x=distance, y=time)) + geom_point() + geom_line() + scale_x_log10()And record average speed vs distance:
s <- read.csv("~/Downloads/Running records by race distance and mean speed.tab.csv", stringsAsFactors=F)
s$distance <- s$x
s$speed <- s$y1
ggplot(s, aes(x=distance, y=speed)) + geom_point() + geom_line() + scale_x_log10()So, as expected based on human frailty, times go up, speeds go down as distances increase.
Going back to the original theme, looking at the table of record times, I see that in 2023 Aleksandr Sorokin ran 100 km = 100,000 m in 6:05:35 = 21935 seconds. If we had a team of 1000 Usain Bolts arrayed along the route, maybe they could run the same total distance in 1000 * 9.58 = 2:39:40. Or perhaps it’s better just to compare their average speeds: around 4.6 m / s for Sorokin across 100 km, around 10.4 m / s for Bolt over 100 m, roughly 2.3 times faster.
(I’m also noticing that the time provided in the data underlying the Wikipedia plot doesn’t seem to be for Sorokin’s 2023 run. For 100 km, the data table has a time of 22154 seconds… maybe an older world record?)
One big flaw in this analysis of team vs individuals is that it ignores the cost of positioning the team along the route.
What area of study do these questions belong to? Economics? Social psychology?
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