Monday, July 9, 2012

Power Up a Tower: Part 2

In Power up a Tower: Part 1, I showed the relationship between work (J) and power (watts). I also introduced the power to mass ratio metric (watts/kg) which can be used to measure the performance of different climbers climbing the same building. Finally, I showed an example of how the power to mass ratio of climbers depends on the height of a climb. In this post, I will attempt to predict how an athlete's average power to mass ratio changes with respect to the height of a climb.
Before I discuss climbing, I’m discuss to show a related topic; running. Admittedly, I’m neither a serious runner nor a scientist, but I conjecture that running speed is proportional to power output. At the very least, here is a paper that supports this assumption. Although this paper ignores important terms like drag and acceleration, at low speeds these terms are relatively small.
Since a runner’s sustained power output will drop as race length increases, it would be interesting to see how much it drops depending on the total distance run. I’m particularly interested in races lasting between two and twenty minutes in duration, nearly all stair climbing races fall into this range. I couldn’t find data for individual runners across this range, so instead I plotted data for world class athletes. In most cases, I took the average pace for the top 10 races done at a given distance. See table below:
Average Running Pace per Race Distance
Here is a plot that shows the pace as percentage of the 100m dash pace, a pace very close to a runner’s top speed.

Figure 1
Here are a few notes and observations about this data:
  • The shape of the curve looks like a decreasing exponential.
  • This data represents a theoretical maximum for humans. Even for world-class athletes, it would be nearly impossible to replicate these speeds across the whole range of distances. For example, Ryan Hall would handily beat Usain Bolt in a marathon, but Usain would dominate in a sprint. With training, these athletes might be able to excel in other races, but they probably would still have different (yet similarly shaped) speed vs. distance curves.
  • Max speed (and hence power output) looks like it can only be achieved for about 20 seconds. You will note that the men’s 200m has a greater top speed then the 100m since the initial acceleration is less of a factor in the longer race. However, for the women’s race (which lasts over 20 seconds) the average pace for the 200m race is slightly slower than the average pace for the 100m race. For longer races, there is a significant drop off in average speed.
  • There is an odd drop off for the women’s 300m race. This is probably because very few races are done at this distance; hence the world record pace is somewhat “slower” than what would be expected.
  • For races shorter than a mile (about 4 minutes in duration) average speed (and hence power output) drops significantly as race length is increased. However, for longer races, average speed drops very little as race distance is increased (about 60% - 70% max speed). In fact, if you extended the curve out to include the marathon distance, you would see that the marathons are done at about 55% max speed.
  • There is a distinct gap between the curves for males and females. Part of the gap can be explained because top women are a bit slower than men. For reference, the world record for 800 meters is 1:41 and 1:53 for males and females respectively. Since it takes females about 12 seconds longer to cover the same distance, it would be expected that a female’s % maximum power during the race would be slightly lower at this distance.
To make this information useful to stair climbers and eliminate some of the discrepancies highlighted in the last bullet above, I’ve transformed the X axis to show “Time” instead of running “Distance”. I’ve also relabeled the Y axis to show “% of Max Power” rather than “% of Max Speed”.
Figure 2
A couple observations:
·        The data is virtually identical for males and females up to about 2 minutes (about 800 meter mark) but then diverges by about 2% of Max Power.
·        I made the assumption that Power & Running Speed are directly proportional.
This is a very useful tool for stair climbers since it can be used to predict race times for any building assuming the climber knows the vertical height of the race course and has results from at least one other climb.
Here is an illustrative example: A climber wants to race up a building which is 300 meters tall. Prior to this race, the climber raced up a 100 meter tall building in 4 minutes. What is the predicted time up the 300 meter building?
Step 1: Find where the climber’s previous race lies on Figure 2. Example: For a four 4:00 minute race, the climber was working at about 69% of maximum power (taking the average between male & female curves).
Step 2: Estimate how long it will take to do a long race. Example: For a 300 meter race, it is going to take a bit longer than 3X as long as the 100 meter race. Therefore, let’s guess it will take about 15 minutes.
Step 3: Find where the climber’s estimate lies on Figure 2. Example: For a 15 minute race, the climber would be working at about 62.5% of maximum power (taking the average between male & female curves).
Step 4: Calculate the climber’s pace & predicted finish time using % maximum power found in step 3. Example: The climber should be about 10% slower when climbing the 300 meter race compared with the 100 meter race (i.e. 69%/62.5% = 1.104). At that pace, the climber would be able to climb 100 meters in 4.42 minutes (i.e. 1.104 x 4 minutes). This translates into 13.248 minutes (i.e. 3 x 4.416 minutes) in the 300 meter race.
Step 5: Repeat steps 2 through 4 using the predicted finish time from step 4 as your new guess. Stop once your guess matches your results from step 4. Example: Guessing that the finish time is 13.248 minutes (@ step 2) then the climber would be working at 63% of maximum power (@ step 3). This value predicts that the climber should be able to do 300 meter in 13.14 minutes (= 69%/63% x 3 minutes x 300m/100m) which matches closely with the initial estimate of 13.248 minutes.
A few final notes:
·    As mentioned before, the curve is derived from a “theoretical best”. Some climbers will do better in longer races and others will do better in shorter races. In my case, I tend to do better in shorter climbs, so using results from a shorter race to predict how I’ll fare at the Sears tower won’t be very accurate (i.e. my race time will be slower than what is predicted).
·    This method only takes into account vertical height of the race. It won’t be very accurate for races like the ESBRU which includes a fair bit of running.
·    This method assumes that races will be done at a constant pace (i.e. constant power) throughout the entire race. From experience, I know if I go out too fast in the beginning, I’ll pay for it dearly in the 2nd half of the race.
·    Other variables like number of turns, hand rail placement, etc. are also not factored into this method. When using this this curve to predict how you’ll do in a race, hopefully your trial time was done in a similarly configured building. From experience, I know that turning around corners eats up time!


  1. This is great, Alex! Even with all the caveats, it's great to have some metric with which to set a pace for ourselves!

  2. Hi Alex! I discovered your blog last week after also discovering StairLife. Thanks for sharing all of this information chronicling your climb to the top of the ranks. I found this post the most interesting because I've kept track of my 38 stairwell PRs, & noticed the similarity between my curve & yours. When I scale the 100m-5k data in your "Average Running Pace per Race Distance" to mine, the curves overlap amazingly well. When I initially included the 10k, the middle of my curve was way above that of the runners.