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!

Goals for 2011

I made up my goals for 2010 as I went along, so I thought I should make my goals for 2011 in advance. Here they are.

Run 10 miles a day
For the past two years I have averaged 8 and 8.8 miles a day, respectively. I have found that if I run more, I get faster. So I want to run 10 miles a day, or 70 miles a week, on average.

Run every day
I don't see any pressing reason why I should skip running on any particular day. If something urgent comes up, such as debilitating illness, I will reconsider this goal.

Under 17:00 in the 5k, 10:00 in the 3k and 5:00 in the mile
In 2009 I broke 19:00 in the 5k for the first time, and in 2010 I broke 18:00 in the 5k. So in 2011 I would like to break 17:00. 10:00 in the 3k is a goal for winter track. 5:00 in the mile is a dream. You have to have dreams.

Run a NBB Tier 1 time
Since I ran under 10:15 in the 3k on January 2, I have the NBB Tier 2 standard for all of 2011. It would be awesome if I could run a Tier 1 time.

Win a race
Most likely, if I achieve any of my time goals, it will be in a race that I do not win. To run fast times, I must surround myself with people who are faster than me. However, I do like to win races and I would like to do that in 2011.

Win money in a race
What can I say, it's fun to have an envelope full of Benjamins in my sock drawer. However, the more often I race, the slower I will probably be in races, so I have to be choosy about the race schedule. Which brings us to:

Race fewer times than in the past two years
In 2009 I raced 35 times and set 19 PRs at 9 distances. In 2010 I raced 48 times and set 10 PRs at six distances. So maybe if I run fewer races, I will run more PRs. Mathematical analysis of those two (!) data points suggests that I can maximize my performance with about 27 races. That's about the number I was planning to do anyway. I don't plan to make the Red Rooster Ramble series a focus this year, so that will decrease my number of races by at least 10.

Run with lots of different people and have fun
Running with other people is one of the best parts of this sport. I ran with about 100 different people last year, and I think it really added to my enjoyment of running. My favorite part of running is running fast in workouts and races, but my second-favorite part is chatting with other people on a relaxed run.

Late addition: Run 1000 miles with Alan
I had to consult with Alan on this one, because it can't just be my goal; it has to be our goal. And we agree: another 1000 miles in 2011 is our goal. So far we have only run about 60 miles together this year, but I have high hopes that we can accomplish this together once again.

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A good start to 2011

On January 2, I competed at the BU mini meet, and ran PRs in both the 3000 and the 800. What a way to start the year!

Last year, my goal was to break 10:15 in the 3k so that I could join NBB. I ran 10:22.25 in my first race of the season, seven seconds from my goal. A few weeks later I tried again, ran a hard-fought 3k with much of the race in the second or third lane, kicked for all I was worth over the last two laps, and ran 10:21.95. Working so hard and taking off only 0.3 seconds was very disappointing.

However, it seems that I just needed to be patient. With another year of training, another 3000 miles on my legs, breaking 10:15 came naturally. Given the amount that I have improved in other distances over the past year, I thought I could run somewhere between 10:00 (40 sec/lap) and 10:15 (41 sec/lap), so when I visualized the race, I visualized running 40s and 41s.


Running the 3000

My seed time of 10:15 put me 4th in my section, so I was surprised when the gun went off and seemingly everyone passed me. I was content, though, as I went through the first lap in just under 40 seconds. I clicked off the first few laps in 40 seconds, then relaxed a bit and ran a few 41s. The race went by very quickly. Alan was at the far end of the track, and after 7.5 laps he told me that I was halfway done, and then I was shocked to realize, very soon after, that two laps had passed since Alan had made that remark, because usually each lap is very difficult. I was basically running alone; after the first lap, I passed one guy (actually lapped him) and one guy passed me. I went through 2k in about 6:48 (which is a new 2k PR) and I felt like I had the race very well in control. With two laps to go, I started kicking, and ran the last two laps in 39 and 38, for a final time of 10:07.56, a PR of about 15 seconds. I was thrilled with the race.

I cooled down, watched Alan and quite a few of my friends race the mile, and then after a few hours I warmed up for the 800. I added it up later and I had done over eight miles of running before I got to the line for the 800. I was also wearing brand-new spikes that I had just been given that day, after the 3k (thanks, NBB!). I had seeded myself at 2:30 because that is my PR. Both Alan and my coach said that I should have seeded myself much faster, and that my goal should be to win my section. I was in section 8 out of 9, the second-to-last race of the entire meet. I reminded myself that the 800 is a very short race, so you have to run hard for all four laps and not worry that it's too fast.


Running the 800

We started in lanes, a two-turn stagger, which was sort of weird. I went out at what I felt was the right pace, and when I cut in there were two men and two women ahead of me. I passed the two women on the second lap. On the third lap I intended to pass the two men, consistent with my goal of winning the section, but they seemed to speed up, so I was content to speed up and stay behind them. In the last lap I kicked past one guy and gave it everything I had to catch the last guy, but he finished ahead of me. Still, I ran 2:24.76, a 6-second PR. My lap splits were 35, 36, 38, 35, which is pretty much the canonical pattern for an immature racer (a mature racer runs even splits), but the 800 is not my specialty, so it's fine with me.

All in all, a great start to 2011. I am looking forward to racing the mile as my next event, and seeing what I can do.

2010 summary

Goals I set for 2010 and how they worked out:

Run under 10:15 for 3k and 18:00 for 5k:
3k didn't happen. I tried very hard to run under 10:15 for 3k all during winter track, but I wasn't able to do it. I ran 10:22 twice, running my heart out and coming up far short. I also tried to break my mile PR of 5:15.9, but was only able to manage 5:20 on the track in the winter and then 5:17 on the road in the summer. I also tried to break 2:30 in the 800, but I only ran 2:31, not even a PR. N.B.: I ran 10:08 for 3k and 2:25 for 800 yesterday. So, I just had to train more and be patient.

5k was better: All spring and summer, I ran times between 18:20 and 18:40. It didn't feel that hard to run 18:40, but I just hadn't found the right confluence of good conditions, good competition and hard effort. Finally, these all came together in September when I ran 17:40, which was a 41-second PR. That's a 3.8% PR! Big PRs like that usually only happen in when you haven't run that distance in a long time, not events like the 5k which I had run repeatedly. And yes, the course was USATF-certified! Running 17:40 changed my life (at least with respect to my running career). It qualified me to join NBB (which was my reason for setting my 3k goal at 10:15) and has put me at a higher level in my running.

Win every Red Rooster Ramble and win the points competition:
I did this. I drove down from NH, two hours each way, to continue running the races in the summer. I set the course record twice (it was broken in between) and got in lots of good tempo efforts. I really enjoyed the whole series, but I'm not planning to make this one of my goals for 2011.

Run under 19:00 for every 5k:
I did this. The only exception was a Halloween-themed 5k, which I ran for fun and still ended up running 19:04. The closest I came to not running under 19:00 was Mayor's Cup cross country race at Franklin Park, where I ran 18:50. That motivated me to make several changes to ensure that I wouldn't run so slow in the future. My fastest race was 17:40 and second-fastest was 18:01.3, so all but one of them were between 18:00 and 19:00.

Run every day:
I did this. I have now run every day for two years. My minimum mileage was 2 miles (when I was sick). I also count pool-running as the requisite 2 miles as long as I do it for more than 30 minutes. I pool-ran two days this year.

Run more miles than last year:
I did this. I ran 3211 miles in 2010 and 2907 miles in 2009. I averaged 8.8 miles per day. Here is a graph of my weekly mileage (click for larger) where blue is weekly mileage, red is cumulative average for the year and green is 8-week moving average:



I should point out the huge change that has taken place in my running over the past three years. The following graph shows my weekly mileage (and 8-week average) for all the weeks of 2008, 2009 and 2010:



Just for fun, I added a trendline to the weekly mileage data. I got y=0.28x+27, meaning that I have on average increased my weekly mileage by 0.28 miles per week (from about 30 to about 70 over the course of about 150 weeks).

Run 1000 miles with Alan:
We did this. In mid-November when we had only run 800 miles together, I didn't think we could possibly do it. However, we made a concerted effort in the last two weeks of the year, wherein we ran 53 and 61 miles together in two consecutive weeks, we managed to run a convincing 1043 miles together in 2010. (We ran 1178 miles together in 2009.)

Net over $1000 in prize money:
This is a goal that I set sometime in the fall, when I saw that I was close. As December came to a close, it became clear that I wasn't going to make it. However, if you include the profits I made when I sold things that I won, and if you consider gift certificates to be money (which I do, as long as I use them on things I would have bought anyway) then I netted $1001 this year. I spent $557 on entry fees and took home $1558. Here is a graphical view of my cumulative profits over the course of the year.



Of course, running is absolutely not about the money. I have enough money. I run to have fun and to run fast.

This year, I ran 49 races: Seven track races (5 indoor, 2 outdoor) and 42 road races, including 16 Red Rooster Rambles. I won 28 of them (including the 16 RRRs). Of the 26 road races that were not RRRs, I was 1st 12 times, 2nd twice, and 3rd three times.

I also achieved 10 PRs this year: three in the 5k (18:24, 18:21, 17:40), two in the 3k (10:22.25, 10:21.95), two in the 6k (22:24, 22:22), one in the half marathon (1:25:59), one in the 5-mile (30:09), and one in the 400 (65.0). These are percentage improvements of:

5k: 4.2%
3k 1.7%
6k 4.3%
half marathon 5.3%
5-mile 0.9%
400m 5.7%

I also set course records at the Red Rooster Ramble 5 mile (30:57 and 30:32), the Kingston Firemen's 5 mile (30:47), and the Rhode Races half marathon (1:25:59) (new course). The RRR is getting a new course next year; exciting!

People I ran with in 2010: Alan B, Melissa H, Jenna R, Haynes H, Sarah T, Jeff M, Kim C, Kim J, Bob W, Steph R, Jackie J, Karen B, Kristen P, Ian, Ben K, Michaela D, Katie T, Brian M, Mark B, Henry M, David L, Mike B, Edward T, Anna, Ben N, Greg, Erin H, Jamie J, Dan H, Oliver C, Joey G, Riadh, Jim M, Jen C, Tess A, Dan G, Scott D, Beth, Wendy, Nick S, Laura E, Michael C, Michael G, Bob J, Billy R, Russell D, Vicky J, Clay H, Matt H, Alex S, Scott C, Chris J, Michelle G, Cooper W, Stella W, Dave D, Mark C, Gwyn C, Eric M, Rob, Verity, David, Jimmy, Matt S, Paul L, Brett R, Erin L, Lauretta B, Mike O, Nate B, Sarah C, Pat M, Kyler E, Keith M, Carmen T, Emily D, Evan S, Jordan A, Erin D, Sasha V, Jen D, Emily M, Tom, Jenn D, Sarah C, Will V, Katie D, Karina J, Leah B, Erin B, Kathryn J, Sarah B, Tony W, Scott M, George R, Lauren, Vinit S, Laura K, Caitlin M, Chris P, Rich, Joe, Erin A, Zak K, Nikki. (This only counts people I talked to while running -- I did some running in large-ish groups and I don't count the people I didn't talk to. Also, just racing against someone doesn't count.) That's 105 people. Pretty awesome! Thanks, everyone!

2009 was my best running year ever, and now 2010 is my best running year ever once again. I ran lots of races and had lots of wonderful experiences. I ran lots of miles, got faster, got stronger, and ran with lots and lots of great people. That's pretty awesome, and it's what makes running so much fun. 2011 is off to a great start so far, and I hope it will be another wonderful year!