After reading my friend Stan's excellent blog for some time, I've decided
to do a bit of number crunching, too. My end goal is to be able to predict performance
and benchmark other athletes depending on the height of a climb. I won’t be
able to do it all in one post, so I expect this will be one of many.

Like Stan, I believe an important measurement is power. To get a good
understanding of power, here is a brief explanation on the difference between work
and its close relative, power.

Work, is defined as force (e.g. weight) multiplied by distance. The
standard SI unit of measurement of work is called a joule (J). It takes 9.8
joules of work to lift 1 kilogram (kg) of mass one meter (m) straight up in
earth’s gravity field [written as 9.8 J = (1 kg x m

^{2})/s^{2}]. The 9.8 factor comes from earth’s gravity field acting on a mass (i.e force = mass x acceleration). Basically, a mass will drop (accelerate) at a rate of 9.8 meters per second each second (9.8 m/s^{2}).
Power is closely related to work. It is defined as the amount of work done per
a given time period. The standard SI unit of measurement of power is called a
watt. For example, it would take 9.8 watts of power to lift 1 kilogram (kg) of
mass one meter (m) straight up in exactly one second [written as 9.8 watts = (1
kg x m

^{2})/s^{3}]. Likewise it would take 4.9 watts of power to lift up the same 1 kg mass if the work is spread out across two seconds. In essence, you’ve done the same amount of work at half the power output, but it has taken twice as long to do so.
Power and work are very import in stair climbing because you are primarily
acting against gravity when you climb up a stair case. This is unlike other
sports like running or cycling that primarily act against friction and drag (unless
you are climbing uphill).

Here is an illustrative example. If a 100 kg climber (about 220 lbs) races
up a 100 meter stair case (roughly 30 stories) in 300 seconds (5 minutes) then
that climber has done:

100 kg x 9.8 m/s

^{2}x 100m = 98,000 joules of work
98,000 joules / 300s = 326.67 watts (on average during those 5 minutes)

So how would your climbing ability compare with this person? It is hard to
tell just by looking at a person’s power output. After all, a lighter climber
(say someone who only weighs 110 lbs (about 50 kg) only needs to do 49,000
joules of work to climb this stair case. Similarly, the lighter climber would
only need to exert a paltry 164 watts to climb up this staircase!

A good measure of relative performance would be to compare the power to
mass ratio of these two climbers. In the case above, each climber has an equal
3.27 watts/kg power to mass ratio.

Interestingly enough, to calculate this ratio you don’t even need to know
how much someone weighs! Since mass is used to calculate power, you effectively
eliminate mass from the equation since you divide by mass when you calculate
the power to mass ratio. This simplifies things greatly:

Power to mass ratio (watts/kg) = [Height of climb (meters) x 9.8 m/s

^{2}] / Climbing time (seconds)
This ratio gives a good performance indicator when climbing up given
height. But how do you compare performance between climbs of different heights?
The answer is somewhat tricky. After all, athletes can only maintain a
difficult pace (i.e. a high power output) for a relatively short amount of
time. The length of the race really determines what pace you are able to
maintain. Here is an illustrative example: Runners can’t sprint an entire
marathon. Conversely, runners can’t competitively jog a 100 meter dash.

To better make this point, here are the power outputs and power to mass
ratios of the last two races I participated in. I also included data for
Sproule Love and a couple other fellow Tower Masters so you can see some relative
data points. Weights are approximate except for my weight and Sproule’s. I specifically
asked him how much he weighs after our race up One Penn Plaza :-)

*note* I think One Penn Plaza climb
is actually about 10 meters taller than my estimate above. Does anybody have a
tape measure I could borrow?

As you can see, since the Trump Parc is only 100 meters tall, the climbers
were able to put out around 20% more power than in One Penn Plaza which about
twice as tall.

Also note that although I had a larger power output than Sproule, he
climbed much faster because he weighs a good 25 lbs less than I do. He has a
superior Power to Mass ratio.

Finally, can you tell who use to be an elite marathon runner? As you can
see from the two data points above, Steve has a very good power to mass ratio
on tall climbs. I have a feeling he’ll kick my butt at the ESBRU, which is
nearly 100 meters taller than One Penn Plaza (^_^).

So now that I’ve written about work, power, and the power to mass ratio, I
hope to show how power output will vary depending on the height of a climb. It
will probably be a while before my next post since it will take a bit of time
to collect the data I’m going to need. Until then, I’m going to be busy
climbing stairs. I hope to improve my power to mass ratio in time for the
ESBRU!

Thanks for putting these power equations up, Alex. I've been training with HR and would like to move to training with power. These will allow me to easily do the calculations I need.

ReplyDeleteI hope to see you at Strat this weekend! -Bob Toews