Wasted Time is Very Tough to Make Up

I’m an amateur long distance runner and I’ve always found it surprising just how long it takes me to catch the slowpokes!

Here is an example: If I run twice as quickly as someone else and I give them a five minute head start, in my mind I expect to catch them in a few minutes. I am running twice as quickly as they are, after all, like a rabbit versus a tortoise! In fact, I will only catch them after running full speed for an additional five minutes.

What if I run 110% as quickly as someone else and I give them a five minute head start? Being 10% faster is very quick indeed. In a marathon I might finish 25 minutes earlier. However, to make up that small lead would take me an eternal 50 minutes. Even though I’m so much faster, I won’t pass my competitor for an hour!

How to calculate it:

Let v be the relative speed of the faster runner. So if the faster runner was 10% faster, v would be 0.1. d is the distance it takes for the two runners to be even, and t is the time for the same.

Then:

d/(1+v) = t and d/1 = t+5

d=t+vt and d = t+5

vt=5

t=5/v

In the case of a 10% faster runner, v=0.1, which makes t = 5/0.1 = 50. It would take me 50 minutes to catch him. If I was running at 10km/hr, it would take me over 8.3km to catch him, even though his lead was never more than about 750m: the distance he could run in five minutes.

In the real world this translates into a strong lesson. It means that wasted time is very tough to make up even if you are very talented.