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My friends and I are having an argument.

I bet my friend that he could not run a mile in under 12:00 minutes, which he did, but it lead to a question. The mile he ran was on a field the size of half of a mile. He ran two laps. During his run, my friend accidentally skipped 5% of each lap. Since he had not run a mile (or anything) in over 5 years, he started out fast and decelerated over the course of the mile. In total, he finished 95% of the mile in 11:00.

My question is, if we were to project the total running time to 100% based on the information below, which of the two methods below (ONLY THE METHODS LISTED BELOW) would give the most accurate estimation?

  • 95% of a mile completed in 11:00.
  • Lap 1 completed in 3:40.
  • Lap 2 completed in 7:20.

Method 1:

Calculate the additional 5% from the total mile time (Laps 1 & 2 together) for a constant speed projection.

Method 2:

Calculate the additional 5% from the Lap 2 total time to take into account the deceleration over time.

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He'll only get worse so taking into account his declinde is the better choice. But maybe you can even make an two parameter ansatz for his velocity, as you have $d=\int_{0}^{t_1} v(\tau)\ \mathrm d \tau=\int_{t_1}^{t_1+t_2} v(\tau)\ \mathrm d \tau$. – NikolajK Sep 13 '13 at 8:24
^thanks for the comment :D – hiiiiiiiiiiiiiiiii Sep 14 '13 at 15:48

Nice question, although it's not really a mathematics question. It comes down to how fast you believe your friend could have run the final 0.05 miles.

His average pace was about 11m30s / mile for the course - so using method 1, and assuming that he completes the last 0.05 mile at his average pace, he would complete it in under 12 minutes.

However, his pace on the first 0.475 mile was 7m40s / mile, and his pace on the second 0.475 mile was 15m25s / mile. If he ran at his pace over the second half of the course for the remaining 0.05 mile, he would complete it in just over 46 seconds (still finishing in under 12 minutes).

It comes down to which of these you think is more realistic. I think that the most accurate method of the two you listed is to assume that he could run the final 0.05 mile at his lap 2 pace, i.e. 15m25s / mile.

However, you should consider that he probably ran the 0.9 - 0.95 section of the course much slower than the 0.5 - 0.55 section of the course, and so his average pace over the final 0.05 mile would be slower than the 15m25s average lap 2 pace.

Mitigating this, though, is the possibility of a sprint finish - when you know that you don't have to hold anything back you can often get an extra burst of speed (case in point - I recently ran a marathon, and although I slowed down slightly over the duration of the race, I was able to finish the final three miles at the same pace as the first three miles).

Overall, I suspect that if you really want to know whether your friend can run a mile in under 12 minutes, the only thing to do is to have him do it again.

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If he's the guy for a sprint finish, then he also did it on the shorter course which he already run. – NikolajK Sep 13 '13 at 8:31
That's a fair point! – Chris Taylor Sep 13 '13 at 8:32
Thanks for the comment – hiiiiiiiiiiiiiiiii Sep 13 '13 at 23:16

A lot of this is obviously moot. The number of variables in order to respond to this is too great. One can argue that one could take only the runner's average speed over the entire course and use that to calculate the remaining time for the remaining distance. One could also take the instantaneous speed of the runner at the end of the race (regardless of the potential sprint) and use that in conjunction with his overall negative acceleration throughout the entire run to calculate his final time should he have run the remainder. The other possibility is of the runner increasing his speed despite being "gassed". Oh, and don't forget, the runner's speed would have varied (most likely) throughout the race... if you were to plot his velocity over time on a graph, you'd see an OVERALL decrease in velocity, but his speed very well could have increased and decreased as time went on... again, noting the importance for using velocity.

Another point to note here is that the numbers that the OP has given are such that an "accurate" result cannot be obtained!

You cannot merely add 5% (or ANY percentage for that matter) of a given "time" to the observed time to come up with an overall total time. This completely omits the factor of "velocity" or "speed", both of which are functions of time. This is a flawed problem set-up, in my opinion. But in reality, the only way to determine how fast the runner would have ran the mile, he must do it again. Math and physics will only get you so far, and even when they do, you have to use accurate numbers and a valid methodology, which is not being used here.

Hope this helps!

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thx 4 the comment shoaib!! :D – hiiiiiiiiiiiiiiiii Sep 13 '13 at 23:01

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