### Seeing who "hit the wall" in the New York City Marathon

A special congratulations to our very own John Lee, who ran the New York City Marathon this past weekend (his first full marathon)!

 Photo credit: Jake Park

John trained for months, and finished the race in a time of 3:21:11 (3 hours, 21 minutes, and 11 seconds). Given that every marathon is about 26.2 miles long, we can calculate that his average pace was 7:41 (7 minutes and 41 seconds) per mile.

Smart runners will pace themselves, and run at about the same speed the entire race. But runners who don't prepare as well will often "hit the wall" around mile 20, and will slow down toward the end of the race.

Let's look at graphs of how a few runners did. We'll specifically look at John ("The Real John Lee"), another person who happened to be named John Lee and who also ran the marathon ("Fake John Lee"), and the fastest woman to run the race, Kenya's Priscah Jeptoo. Here's how they did:

The dots in the graphs represent checkpoints along the race where the runners' times were precisely measured. We don't know exactly how fast the runners traveled between the dots, but let's assume they ran at a steady pace between consecutive dots.

Of the three runners, Priscah Jeptoo (in red) finished in the least amount of time, so she was the fastest (with a time of 2:25:07). You can also see that Priscah ran a smart race, without hitting the wall, because her graph is very close to a straight line. She ran at about the same speed the entire race.

The Real John Lee (in blue) also ran a smart race, keeping a steady pace the entire time. Fake John Lee (in green), however, started off on pace with Priscah Jeptoo, but slowed down more and more as the race went on. He went out of the gates too fast, and as a result, his graph is concave down, meaning it curves downwards.

We can see this even more clearly if look at the speeds of the three runners over the course of the race. Speed is equal to distance over time, so a runner's speed between two checkpoints is the slope of the line between those points. Here's the graph of the three runners' speeds:

At certain points along the race (like around mile 13), all three runners slow down. These are probably the uphill parts of the course! And it looks like there's a nice downhill stretch around mile 22. But you can also see that Priscah Jeptoo ran a smart race, with a pretty steady speed. The Real John Lee also maintained a steady speed throughout the race. But Fake John Lee kept slowing down throughout the marathon.

So by looking at the slopes at different locations of a runner's graph of distance vs. time, we can see how quickly that runner is moving. And in this case, we can see who ran a smart, steady race, and who hit the wall.