## Wednesday, September 21, 2022

### Track: Correlating Running Times versus Distance

Dear Long-Distance-Runner Nephew, William:

Thinking about how to model expected running times (T, sec) versus distance (D, m),

I figured the relationship might be:

T sec = (10 sec) (D / 100 m)^b

Where b would be b=1.0 if you could run the entire distance at the same speed you ran the 100 m.

I expected fatigue would make b>1.0, but I did not know how much greater.

I consulted the internet:

there the exponent was found to be b = 1.1 from 100m to 10 km.

The best world-record coefficient (e.g., 10 s) might be Bolt’s 9.6 s for the 100 m, but

I was most interested in the exponent, the dependence of time on distance.

In high school, I ran the mile in about 5 minutes = 300 sec.

My pitiful 100-yard dash time was about 12 sec.

The ratio of these times was about (300/12) = 25.0.

The ratio of the distances was mile/dash = (5280/300) = 17.6

Which predicts a time ratio of (17.6)^1.1 = 23.4

not too different from 25.0,  assuming my approximate 12 s estimate.

For non-record-holders like myself, expect our times vs. our distances to go as

(T/To) = (D/Do)^1.1.

So, as you go from 6 k to 8 k, for example, you would expect a ratio of the times of about

(8/6)^1.1 = 1.37.

A 6k time of 21 minutes would be about 21.0 x 1.37 = 28.8 minutes, rather than simply (8/6)(21.0)=28.0. The extra 0.8 of a minute is 48 seconds.

Doubling the distance would tend to increase the time by the multiplier (2)^1.1=2.14, which is 7% longer than 2.0.

To some degree, goals like these target times can be helpful.

At other times, they might be unrealistic, too easy, or too hard.

They are interesting, though. (I was sort of a theoretical miler.)

I view this as a way to judge the likely impact of distance on your time,

rather than a way to compare oneself against world record holders.