Does a hill bring you closer?

As I have mentioned before, sometimes things come up in running that I wonder about mathematically, like time intervals around a track or what you can count as a PR. Both of those questions occurred to me within the past few years, but here is something that I wondered about way back in high school:

I was in a cross country race, running about 50 feet behind a girl from the other team. During a straight, flat section of the course, we had been running the same speed, so this 50-foot gap had remained constant. I knew that a hill was coming up, so I wondered: If we both slow down the same amount on the hill, will the distance between us stay the same, or will it get bigger, or smaller, while we are climbing the hill?

It's a good question. At the time, I didn't know how to figure out the answer (and I didn't ask anyone). But after a few years of high school math and physics, where I learned that (for example) you can easily toss a ball up and down while you are on a train as long as the train isn't accelerating or decelerating, and I learned about the "twin problems" in relativity, I figured out how to think about the problem, and I figured out the answer.

The answer is that the time gap between my opponent and me stayed the same throughout the race, but the distance between us decreased while we were on the hill.

You can think of it like this: Imagine that we were identical twins who run exactly the same speeds, but she started 5 seconds before me. Then the gap between us will be 5 seconds for the whole race. If we both run at a speed of (say) 10 feet/sec on a flat course, then we'll be running 50 feet apart for the flat sections. But if we slow down to 8 feet/sec on the uphills, we'll still be 5 seconds apart, but only 40 feet apart, so the distance between us decreases on the uphill.

If we both speed up to 12 feet/sec on the downhills, the distance will increase to 60 feet on the downhill -- even greater than it was on the flat. But when the course flattens out again, we'll still be 5 seconds apart, so the gap will be back to 50 feet whenever we are both running on a flat section. (If she is on a hill and I haven't gotten there yet, the distance between us will be somewhere between 40 and 50 feet, and the same with the other transitions.)

I even have a way of visualizing this happening. Way back in middle school, our coach had us all running around the indoor track, half a lap sprinting and half a lap jogging. There were 100 kids of widely varying abilities, so we were spread out all around the track, with people crossing the starting and finishing lines at all different times. But the key was that when you got to the line on the track, your speed suddenly went from jogging to sprinting, and then vice-versa. I imagined a screen where the left half was blue and the right half was red, and there were red blood cells flowing across the screen, and as soon as they hit that imaginary line, they changed from red to blue, and sped up. So as soon as they turned blue and sped up, they had to get farther apart. This is what happens when my cross country opponent and I reached the top of the hill and started running down the other side: suddenly, we sped up and got farther apart!

Three for three (GBTC mile)

(Results) Yesterday I ran the mile at Harvard, and ran 5:12.24, which improved on my previous best of 5:15.90 set way back in December 2009 at the Alden Invitational. It wasn't as fast as I wanted to run -- I wanted to run five-oh-something -- but it was still a PR.

I used to think that my problem was lack of mental toughness. I would finish a race like the one I ran yesterday, and I would think, "if only I had more mental toughness, then I could have stayed with the girl ahead of me. I could have run faster if only I had pushed myself harder." I shared this outlook with my friend GC one day last summer, after we had completed a morning track workout together. She pointed out that I had just done [such-and-such impressive workout] under [such-and-such difficult conditions], so clearly I was not lacking in mental toughness. I trust GC's judgment, so from that day forward I no longer beat myself up about inadequate mental toughness.

Now I just say, "Didn't run fast enough? All right. Better train harder."

I had wanted to run all of my laps in 38 seconds, for a time of 5:04 (Harvard has a 220-yard track). I had expected that, with the excitement of the race, I would fly through the first three or four laps in 38 without it being too hard. However, that didn't happen; I ran closer to 39s, and when I tried to speed up in the second half of the race and run negative splits and catch the girl ahead of me, I nonetheless slowed down and she got further away. So, I just have to train harder, so that my fitness allows me to run eight 38-second laps in a row.


Me in one of the last laps of the race. Photo thanks to Yvonne.

Before the race, I was a little annoyed because I wanted to be in the invitational section -- "the fast heat" -- but I wasn't chosen. After the race, I saw that my time would have been the slowest time in the section, so it's completely justified that I wasn't in that section. However, the way that race went, it would have been perfect for me: It went out reasonably slowly for the first six laps, so I could have hung on the back end of the strung-out pack, at a perfect pace, for most of the race. In the last two laps, there would have been people right ahead of me, who I could have tried to catch (in my section, the first girl was 9 seconds ahead of me, too far ahead to be a motivator). So it would have been good for me, and I probably would have run a little faster than 5:12.24 -- but I probably would have been last. The moral is: If you want to be chosen for the fast section, you have to run faster.

So far, I have run three races in 2011 and have set PRs in all three. In fact, in every race since Mayor's Cup where I have worn the blue and white NBB uniform, I have run a PR:

Mayor's Cup: 18:50 (5k xc PR)
USATF-NE XC: 22:24 (6k xc PR)
Pie Run: 30:09 (5 mile PR)
Club nats: 22:22 (6k xc PR)
BU open meet: 10:07.56 (3k PR)
BU open meet: 2:24.74 (800m PR)
GBTC invite: 5:12.24 (mile PR)

This can't possibly last, but it's nice to improve. It's nice to keep running the fastest I have ever run. I will just have to keep training hard and see how it goes.

Mixing speeds

Sometimes, when I'm running, I come up with a mathematical question, such as the question about PRs and average speeds. Here is another one.

I was doing 200m intervals on the indoor track, one lap fast and one lap slow, and I wondered: is it the same time interval between when my coach sees me, and when the people on the exercise bikes see me?

More concretely: Suppose that a runner is alternating fast laps (which take 1 minute) with slow laps (which take 2 minutes). She passes her coach, who is standing at the start/finish line, after 1 minute, then 2 minutes later, then 1 minute later, then 2 minutes later, etc. How about the people on the exercise bikes 1/4 of the way around the track? Do they also see her every 1, 2, 1, 2, minutes? Is there a place you could stand so that you could see the runner with an equal time interval between meetings?

The answer is that depending on where you are standing, the interval between meetings with the runner is different. If you stood halfway around the track (diametrically opposite the start/finish line), then it would always be 1:30 between meetings with the runner: If she passed you on a slow lap, she would then run half of a slow lap (1:00) followed by half of a fast lap (0:30) before seeing you again. Similarly, if she passed you on a fast lap, she would then run half of a fast lap (0:30) followed by half of a slow lap (1:00) before seeing you again.

If she passed the people on the exercise bike on a slow lap, then it would be 3/4 of a slow lap (1:30) and 1/4 of a fast lap (0:15) before she passed them again, so that's 1:45. When she passed them on a fast lap, it would be 3/4 of a fast lap (0:45) and 1/4 of a slow lap (0:30) before she passed them again, so that's 1:15. So the people on the exercise bikes see her every 1:45, then 1:15, then 1:45, then 1:15, etc.

You can get any time intervals you want between (1:00/2:00) and (1:30/1:30) by standing in various locations around the track.

This is basically a mixing problem: Mixing various proportions of fast and slow laps. It is similar to the question: Suppose you have one liter of apple juice, which costs $1, and one liter of grape juice, which costs $2. You have to make two fruit punches, each 1 liter. What are the possible costs of the two drinks?

In this case if you keep the juices separate, the punches are $1 and $2 (like standing at the start/finish line). If you mix the juices 50%/50%, both punches are $1.50 (like standing diametrically opposite the start/finish line). If you put 25%/75% in each punch, the costs will be $1.25 and $1.75. You can get anything between those by varying the concentrations (like you can get any time intervals between the given ones by standing in different places around the track.)

N.B. My fast and slow intervals were not 1 minute and 2 minutes. That was just for purposes of illustration, to make the numbers easier.

Records start at the gun

Back in September, in my post The Intermediate PR Theorem, I discussed whether, if my 2-mile split in a 5k was 11:09, I can multiply this time by 1.864/2 to convert it to a 3k time of 10:24. So if my 3k PR had been 10:30, could I now claim a new 3k PR of 10:24? A legitimate question.

In discussing this with one of my professors, I discovered that the answer is no. Consider this:

The current world record for the 100 meters is 9.58 seconds, which Usain Bolt ran in Berlin. Bolt has only run 9.58 once in the 100-meter dash. But in fact, lots of people have run 9.58 for 100 meters before. How? In the last 100 meters of the 200-meter dash!

When you compare 100m and 200m times, it's interesting to note that individuals' 200m times are usually faster -- less than twice their 100m times. For instance, Michael Johnson's 100m PR was 10.09 and his 200m PR was 19.32. In fact, he went through the first 100m of his 200m in 10.12, and then proceeded to run 9.20 for the second 200m. Could he therefore say that his 100m PR was 9.20? No, because he had what's called a "flying start" -- he didn't have to accelerate from motionlessness in the blocks; he started at full speed.

This effect is much smaller in longer events; 200m times are much faster than 400m times. However, the fact remains that in order to be a record for a particular distance, the performance has to start at the gun. For instance, Haile Gebrselassie set the world record for 20k en route to his 1-hour world record, and this record was the first 20k of the race. Even if he ran faster between 1k and 21k than between the start and 20k, it wouldn't count as a record because he had a flying start.

So if I want to run a PR for 1500m in a mile, it had better be the first 1500m!