# What is the most accurate way to project a total running time based on a 95% complete run?

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 $d=\int_{0}^{t_1} v(\tau)\ \mathrm d \tau=\int_{t_1}^{t_1+t_2} v(\tau)\ \mathrm d \tau$. –  NiftyKitty95 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. –  NiftyKitty95 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