#### Wednesday, September 19th, 2012...12:26 pm

## Olympic Speed Limit

Just over a month ago, Usain Bolt electrified the Olympic track and field stadium in London as he won a second gold medal in the 100-meter dash. The New York Times published, in a stunning interactive graphic entitled *One Race, Every Medalist Ever* an analysis in which every Olympic medalist in the 100-meter sprint races against each other. How far ahead would Bolt have finished before Jesse Owens or Carl Lewis?

Below, we see a table of the times in the 100-meter dash that won the Olympic gold medal between 1896 and 2012. Warning – it’s a long table but this makes it easily cut-and-paste-able.

Year | Time |
---|---|

1896 | 12 |

19 | 11 |

1904 | 11 |

1906 | 11.20 |

1908 | 10.80 |

1912 | 10.80 |

1920 | 10.80 |

1924 | 10.60 |

1928 | 10.80 |

1932 | 10.30 |

1936 | 10.30 |

1948 | 10.30 |

1952 | 10.79 |

1956 | 10.62 |

1960 | 10.32 |

1964 | 10.06 |

1968 | 9.95 |

1972 | 10.14 |

1976 | 10.06 |

1980 | 10.25 |

1984 | 9.99 |

1988 | 9.92 |

1992 | 9.96 |

1996 | 9.84 |

2000 | 9.87 |

2004 | 9.85 |

2008 | 9.69 |

2012 | 9.63 |

Now, let’s plot these times. We see variability and also a certain trend in the overall improved times.

Suppose we model the decreasing times with a line and use least-squares to estimate this rate of improvement. As seen above, we find that the winning time *y* equals *y* = -0.0133*x* + 36.31.

Clearly, there is some limit at which a human can no longer run any faster. For instance, the 100-meter dash will never be completed in 2 seconds. However, where could the limit be between 2 and 9.63 seconds? John Brenkus, in his book *The Perfection Point*, considers such a question. Brenkus analyzes four distinct phases of the race: reacting to the gun, getting out of the blocks, accelerating to top speed and hanging on for dear life at the end. He lays out his analysis of why 8.99 seconds is the fastest 100-meter dash that can ever be run.

If we assume our least-squares line continues as the trend of improvements in speed in the 100-meter dash, then when would we reach the speed limit for this race? This involves solving:

8.99 = -0.0133*x* + 36.31,

which implies *x* = 2054.14. In fact, if you don’t round the slope and y-intercept of the least-squares line to two decimal places, you find *x* is approximately 2059.84. Therefore, we would reach the limit of speed in the 2060 Olympics! It’s just a model, so time will tell. But either way, we have exciting and memorable moments awaiting us.