Mathematics & Running

I am a I mathematician by training. I spent many hours as a college student studying Laplace transforms, Fourier series analysis, differential equations, etc.  I can't say I have ever come remotely close as a professional to using any of the above mathematics.

To date, I have not played my personal math geek card.  It's not a popular card.  Cutting to the chase, the graph above is a representation of my last training run where I tried to run at a faster pace at the conclusion of each mile. I began the first mile at a pace of 534 seconds per mile (8:54) and finished mile six with a pace of 470 seconds per mile (7:50).

The horizontal purple line within the graph above represents my average pace for the entire run, eight minutes and twenty seconds. This equates to a five hundred seconds per mile pace.  The green line represents my actual pace for the entire run.

My pace slowed two seconds per mile within mile four due to a large hill. That's why the green line between miles three and four does not decrease.

The red triangular area represents the distance where I was running slower than my average pace while the green triangle represents the distance where i was running faster than my average pace.

For the Beach to Beacon 10K Race I want my average pace for the race to be seven minutes and thirty two seconds (452 seconds) in order to finish with a time less than forty-seven minutes

The 'takeaway' from this graph is the larger the red triangle (distance slower than average pace), the larger the green triangle (distance faster than the average pace)  has to be to compensate for the initial pace deficit.

Why don't I just simply run at a 7:32 minute pace for each mile during the race ?  I don't believe many runners exhibit a flat-line pace graph for any of their races. Less experienced and less disciplined runners such as myself have paces which slow over the course of a race. More experienced runners have paces which quicken over the course of a race.  They place a priority on a strong finish.

Their will be 6,500 people at the starting line for the B2B race. It may take me five minutes just to get to the starting line from my vantage point within the starting line gathering. It may also be a mile into the race before I will not be enveloped by a corral of runners. The silver lining in this cloud is I may be forced to start this race at a modest pace by virtue of the congestion at the onset of the race.

Math Geek addendum: To get a picture of what I need to accomplish to finish the Beach to Beacon 10K (6.2 miles) at an average pace of 452 seconds I would shift the entire graph down 48 seconds, from the current 500 second location. All that is required then is to subtract 48 seconds from each mile pace time above to obtain goal paces throughout the race.

For example, my first mile pace would then be 486 seconds (8:06) while the sixth mile pace would translate to 422 seconds (7:02).  The problem with this scenario is I am not going to finish the last mile at this pace unless the last mile is entirely downhill and I have a ten mile per hour wind at my back.(i.e. it's too fast of a pace)

I need to construct a B2B pace chart with triangles which are more flat. In other words I need to quicken my initial pace to allow for a more reasonable finishing pace.  The chart above contains a pace variance of  one minute and four seconds (8:54 - 7:50).  I would choose a pace variance closer to  twenty seconds.

And you thought I just run during a race ?