# PTC Mathcad and the Marathon

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For those of you who don’t know, PTC’s headquarters are located in Needham, Massachusetts, just outside of Boston. And this week, on Monday, April 20, Patriots’ Day, around one million spectators gathered in and around Boston to watch over 30,000 participants take on the famed Boston Marathon. These last few days, the air in the city has been charged with a mixture of excited anticipation, as the Marathon is, according to the B.A.A. website, “New England’s most widely viewed sporting event,” and solemn remembrance, as the city honors those injured or killed in the 2013 Boston Marathon Bombing.

On the morning of Patriots’ Day, the day on which the Boston Marathon is held every year, my coworker, Tim Bond, and I were talking about how fast the race has been run, which led us to perform some quick PTC Mathcad calculations (because that’s what we do) to figure out average mile times and average speeds, thereby giving me the idea for this blog.

The discussion Tim and I were having started as we were wondering when the various roadblocks throughout the city would be opened up again. Tim, a Massachusetts native, mentioned that the winning time would be around two hours. I, a New England resident for little over a year, simply could not accept this. Two hours? To run 26.2 miles? I’m not even sure I can finish my laundry in two hours!

So Tim tells me to look up the fastest Boston Marathon time. The current record for the Boston Marathon was set in 2011 by Geoffrey Mutai from Kenya. His time was 2:03:01. My mind was, and continues to be, blown. Immediately thereafter, since that’s just the way our minds work, we pull up PTC Mathcad Prime 3.1 and do some calculations. Now, the calculations are very simple, so nothing too crazy here. However, we did make use of a unique type of function in PTC Mathcad that perhaps doesn’t get much attention—the affine functions.

Affine functions are automatically labeled by PTC Mathcad Prime as units, so they essentially double as both units and functions. In this case, we use the hhmmss function, which takes a string argument of the form “hh:mm:ss.sss” and converts it into the corresponding time. So, for example, Mutai’s record time can be used as the argument as shown below:

The same can also be done in a format that looks more like how PTC Mathcad typically handles units:

Now I can use this time to calculate Mutai’s average speed throughout the duration of the race:

To get an even better handle on what this is like, I calculate Mutai’s average mile time:

for which I can then change the units to display in hours-minutes-seconds format:

Let that soak in for just minute. Geoffrey Mutai averaged approximately 4 minutes and 42 seconds for each of the 26 miles he ran back in 2011! I can’t imagine being able to run one mile that quickly, much less over 26 of them!

Okay, so now we’ve seen the calculations for the current record-holder for the Boston Marathon. Let’s take a look at this year’s winning times and speeds.

For the wheelchairs:

For the handcycles:

And for the runners:

These times were taken from the B.A.A website.

Now, let’s take a quick look at the current all-time world record for fastest marathon time. This was set in 2014 at the Berlin Marathon by Dennis Kimetto of Kenya with an official time of 2:02:57.

Comparing that to the Boston Marathon record, we see that Kimetto ran each mile, on average, 0.153 second faster than Mutai.

To put that into perspective, this is approximately the same amount of time it takes for the brain to recognize what it is seeing, “from the moment light hits the retina to when the earliest recognition of basic object identity can occur.”

To wrap things up, what would it take to break the elusive 2 hour threshold? The calculation for this, as with the other calculations above, is very simple, but I’m going to use a Solve Block just to show off another bit of functionality in PTC Mathcad. Using the solve block here is a bit like using a chainsaw to cut a piece of paper—overkill but still pretty cool.

The target time is 2 hours. We will divide 26.2 miles by our guess value for the target speed (the average speed needed to break the 2 hour mark) and set it less than the target time as a constraint. Finishing it off with the find function, we see that an average speed of 13.1 mph and an average mile time of just less than 4 minutes and 35 seconds are needed to clock in at exactly 2 hours. This means that a runner would have to run each mile about 7 seconds faster, on average, than Kimetto did in 2014.

Do you think it’s possible?