Friday, May 2, 2008

Anders Södergren vs Gravity: Mordarbacken Power Output

February 23, 2008, Falun, Sweden. There are just over two kilometers remaining in the Men's 30k Double Pursuit, and Anders Södergren has a problem.

Södergren is the best distance skier in Sweden, and the course is lined with thousands of screaming Swedes, hoping that can win one for the home team. The pursuit has played out like most World Cup mass starts do -- there is a lead group of twenty or so skiers still in contention. Many of them are barely hanging on -- but if they can make it to the last 500m, anything can happen in the final sprint. And that's where Södergren's problem is. He's one of the worst sprinters on the World Cup circuit, and if he enters the stadium with any of his 19 traveling companions, he is likely to disappoint the screaming masses.

Södergren's only chance is to get away on field's last trip over the Mordarbacken (literally, "Murder Hill"), one of the steepest extended climbs on the circuit. He leads the field as they reach the bottom of the climb. He has no choice but to attack from the front, and ski it as hard as he dares after an hour of racing.

[ Photo by oskarlin ]

Södergren punishes the field mightily. Over the top of the climb he has a 5.1 second lead over the World Cup overall leader, Lukas Bauer, with the rest of the field is disarray behind him. On the descent, his five second lead means he has over 50 meters of free space over Bauer. To the casual observer, he has won the race.

But Södergren has skied too hard. Despite the brief recovery afforded by the descent, he's unable to keep his work rate high enough to hold his lead. He's caught first by Bauer, and then by Norway's Tore Asle Gjerdalen. He enters the stadium with these two men, and as the sprint starts it is clear Södergren has nothing left, his muscles saturated with lactic acid and his spirit broken after being caught. He coasts in well behind the other two for third place.

While Södergren ultimately failed to win the pursuit, his ascent of the Mordarbacken was truly impressive, as he had been leading the field for several kilometers before the climb and, despite this, managed to best all challengers by five seconds in under two minutes of climbing.

The video from the race can be seen (WCSN subscription required) here, around 70 minutes into the broadcast.

Mordarbacken rises 72m in about 500m of skiing, (course profile), giving it an average grade of 14.4%. Södergren scaled it in 1:48, an average speed of 16.6 kph (4.61 meters per second).

But, just how fast is this? If you started racing Södergren at the base of the climb, completely fresh, could you have made it to the top with him? After all, he had just skied for an hour, and had to leave enough in reserve to make it to the finish. So just how fast is World Cup Attack Pace?

The best tool for measuring this is power output, a subject that should be familiar to all cyclists. We shall determine how many watts Södergren averaged during this effort -- and then we have a single number to describe how hard he was skiing, that can be compared to any other physical activity.

Södergren's first opponent during the climb is gravity. 72 meters is roughly 22 stories, which means he was climbing a story every five seconds. Imagine running up a stairwell, hitting a new floor every five seconds, for 22 stories -- that's the effort he's putting out.

Since Södergren weighs 78 kilos according to his webpage. He's also wearing top of line skis, boots, and poles, and wearing a full race suit (and the corresponding clothing underneath). Assuming these accessories weigh about 2kg, Södergren moves 80 kg total from the top of the hill to the bottom.

We can use the potential energy equation to see how much work this is:
PE = mgh.

As mentioned before, m = 80kg, g = 9.8 m/s/s, and h = 72m. With these numbers, we find that moving 80 kg to the top of Mordarbacken is worth 56448 joules of potential energy.

However, an 80kg spectator can also walk up the hill in ten minutes, and generate the same potential energy, which is clearly no athletic accomplishment. We need to account for time to get a true picture. Thus we invoke the power equation:

Power = Energy/Time

With Energy = 56448 J and Time = 108 seconds, we get a power output of 522.7 watts.

Any cyclist who owns a power meter will immediately recognize that to be a very big power output.

Unfortunately for Södergren, gravity is not the only problem he must contend with -- snow is far from being a frictionless surface! He undoubtedly has the best wax money can buy on his skis, but it is a warm day (5 C), and no wax can completely remove friction. Thus he has energy sapped from him with every stride by the very medium that allows his sport to exist -- the snow!

To find out how much work he must do against friction, we can use the equation for kinetic friction:

Friction Force = (mu)N, where N is the normal force between a body and the surface friction is being calculated with, and mu is coefficient of friction.

While Södergren is on an incline, he is also digging his ski edges in to prevent slipping backward. We can assume that his normal force is directly onto the snow and thus is equal to mg, or 80 kg * 9.8 m/s/s. This gives a normal force of 784 N.

Next we must calculate the coefficient of friction for a ski waxed with pure flouro on those snow conditions. Unfortunately, there is no way to know what this number is. The most quoted number online for the coefficient of a waxed ski is 0.05 -- but this number comes from a 1976 study. Surely ski waxes have improved since then -- we will tentatively assume that modern pure flouros are twice as good as that, and give him mu = 0.025.

Plugging this in, we find a friction force of (784 * 0.025) 19.6 N. Multiplying this by his rate of speed (4.61 m/s) gives 90.3 W -- thus, he is losing 90 W to the snow he is gliding on.

As he is overcoming snow resistance and gravity simultaneously, we can say that he is putting out (90.3 + 522.7 W) 623 W as he climbs the hill.

But wait! Södergren is not exercising in a vacuum. In exchange for not dying of asphyxiation, he must also overcome air resistance as he skis. At high speeds, this can be a huge limiter, but Södergren is only skiing at a little over 16.6 kph.

To calculate his aerodynamic drag, we can use this equation:

Drag Force = 1/2 r Cd A V^2

Where r = viscosity of the substance being penetrated, Cd = coefficient of drag, A = frontal area, and V = speed.

For air at sea level and 0 C, r = 1.293. It's a bit warmer than that in Falun, and slightly above the ocean, but 1.293 is probably very close.

For the coefficient of drag, we'll rate Södergren at 0.34, which is comparable to a Ferrari F360 Modena (or a Ford Sierra!). In any case, the human body is probably not more aerodynamic than a vehicle specifically engineered against drag.

For his frontal area, we will estimate him at 1.5m high and 0.4m wide. 1.5m is obviously a bit short, but his head is considerably narrower than the rest of his body, so we'll model him as a 1.5m x 0.4m rectangle (arms at his sides) an assume the area of his head makes up for the fact that his legs aren't as wide and torso + arms. In any case, a frontal area of 0.6 square meters is reasonable.

Plugging all these numbers in gives us
(0.5)*(1.293)*(0.6)*(0.34)*(4.61^2) = 2.8 N of aerodynamic drag. Compared to everything else, air resistance is pretty negligible. Nevertheless, multiplying this force by his speed gives us an extra (2.8 * 4.61 =) 12.9 W of power that must be generated to cancel out wind resistance.

Adding all these terms together gives us a average power output of 635.9 W, which Södergren sustained for 108 seconds. This might only be 0.85 of a horsepower, but it's enough to power 12-25 laptops. And more importantly, it gives us the final number that we can compare with other athletic accomplishments.

The sport that has the most power data available for it is cycling, so it is what we shall compare against. Lance Armstrong is often quoted as putting out 500 watts for 20 minutes during climbs in the Tour de France, however, this seems ridiculously high. Bradley McGee's recent 4000m pursuit world record (3 minutes and 30 seconds) was quoted at being 530+ watts.

Both of these comparisons have lower wattage numbers, but are for longer durations. Another resource we can compare with is Dr. Andrew Coggan's watts/kg profile table (here), which shows the watts/kg that athletes of varying fitness levels can sustain for various durations. Södergren's watts/kg come out to be 8.15 -- the highest level on that chart shows a world class cyclist can theoretically put out 11.5 w/kg for 60 seconds and 7.6 w/kg for 5 minutes. Södergren's achievement of 8.15 w/kg when not rested and continuing after clearly put him in the "world-class" range of athlete.

But we already knew this. Looking at the chart linked, we can run down the "1 minute" column to find the type of person that can sustain 8.15 w/kg for 60 seconds. This person could ski with Södergren for a minute (starting fully rested, at the bottom of Mordarbacken) and would crack just over halfway up -- just before the steepest part. Sound like you? The chart rates this guy as a "Jersey Rider" -- that is, someone who is a local bike racer. I believe the analogous class of ski racer would be "citizen racer" -- so if you fancy yourself as better than a decent citizen racer, congratulations, you could hang at least a minute with Anders Södergren!

To make it to the top with him (starting fresh, and collapsing at the line), you'll have to be better still. We see that a mid level international rider can put out 9.7 W/kg for 60 seconds and 6 w/kg for 5 minutes. Interpolating and recognizing the non-linear nature of sustaining power, I'd wager this guy can probably hold 8.15 w/kg for almost 2 minutes. This person compares well with the average college skier -- an exceptional athlete, trains year round, but still a few rungs below world class. So a college skier can make it up Mordarbacken with Södergren, as long as he doesn't ski 28k first and doesn't have to make it to the finish line.

And who could actually make it 28k with the lead pack, then put out 8.15 w/kg for 2 minutes, and then make it to the finish? We know the answer to that -- Anders Södergren.

Of course, Södergren's charge ultimately left him short of the finish. I think it's safe to say that his effort was too hard, and the price he paid for such power output was a corresponding loss of power over the final kilometer.

But the man can't sprint. He had to try. It was good enough to drop all but two of the best skiers in the world -- and it would have killed you or me.


Cyrus said...

Wow! Now that was a cool exercise in high school physics!
Great blog!

mike said...

Good to see you got at least a B- in 112.

Why is sustaining power nonlinear?

Colin R said...

Why is sustaining power nonlinear?

Maybe I didn't phrase that very well.

What I'm saying is that, if I can sustain 10 w/kg for 1 minute and 6 w/kg for 5 minutes, you can't just assume that I can maintain 8 w/kg for 3 minutes by linear interpolation.

The curve is something like [Power that can be sustained = 5/x + 4], that is, it asymptotically approaches 4 (the power one can sustain almost indefinitely, if fueled properly)

mike said...

That's just because you're body is retarded.

So I asked my friend, who recently graduated with a degree in Engineering Physics from a prestigious university, and he said that normal forces are perpendicular to the inclined surface.

I think that means the normal force is something like:

Real N = Your N*cos(incline angle)

which, since you have a computer degree, we'll just call it Your N. If anything, always teach your kids this: For small angles, cos(angle) = 1!

For actual scientists, your power would go down three or four WATTS!. He should probably take off a jersey decal to make up for that.

You should see the power output of mother nature outside my house right now. I'm pretty sure my grill is about to get blown off the balcony and fall three stories.
That kind of power is rivaled only by my quads on a mountain climb.

Long ass comment...

Colin R said...

Real N = Your N*cos(incline angle)

Yeah, I'm down with that. But snow isn't actually a smooth incline, it's soft enough that when you ski up it your ski base is level (uphill edge digging in), so I think that cos(incline angle) = 1 directly under the ski.

Luke S said...

High school physics? I could never do that and I took high school physics...

I guess Colin is really just a huge nerd.

Howdy said...

"The curve is something like [Power that can be sustained = 5/x + 4], that is, it asymptotically approaches 4 (the power one can sustain almost indefinitely, if fueled properly)"

It's actually a natural log.
P(t)=Maximal aerobic speed+(Maximal anaerobic speed-Maximal aerobic speed)*e^((.013)(t))

(Doing my thesis on this)

Colin R said...

whoa, whoa, let's try to keep actual numbers out of this!

brisbin44 said...

Great idea for a blog. I am extremely interested in cross country racing and it's awesome seeing a blog dedicated to discussing world cup skiing.

I've got an idea for a post, it is a subject that I've always wondered about and I wanted to know what other people thought. The idea is comparing the different ski styles of racers. For example Jens Filbrich and Vincent Vitoz have two very different styles in both classic and skate but they are both successful, why and what makes each style work? There are plenty of other skiers with interesting styles (Johaug, Bjoergen, Kowalcyck, Hjelmeset) Also skiers from the same country tend to have similar styles. I've noticed Italian, Russian, German. Norwegian, and even North American styles. I know that nobody is in a position here to critize world cup technique but it would still be interesting to discuss.

Christopher Tassava said...

Brisbin44, thanks for the great comment. We're glad you liked the post. I'm not sure how Colin feels, but a post on technique would be well beyond my meager skills. We did, however, touch on this topic in the third segment of our podcast.You might like it.

Aaron D said...

In terms of converting raw muscle power output into speed wouldn't skate skiing be more or less efficient than biking(I would imagine less efficient)? That is, the bike's mechanism for transferring the power output from for legs to the rotation of the wheels might be more efficient than your biomechanics (V1 in this case) that cause your ski to glide across the snow while skating. If this were the case then wouldn't it make it hard to compare power outputs between biking and skiing based strictly upon the gravitational potential energy needed to overcome while ascending a hill?

Colin R said...

Right -- there's really two types of "power output," that being put out by your muscles vs that being put out by "the system." A man on skis is a system, as is a man on a bike. That's what we're comparing here.

Quantifying how much less efficient the skiing motion is than a cycling one is WAY beyond the scope of high school physics. it would be great if we could say that a skier with good technique is X% less efficient than a cyclist at converting glycogen to watts, but I have no idea what X would be there, which is why we ignored it.

Intuitively, I would think that skiing requires more balance and has fewer "gears" than cycling, thus is less efficient. A skier who can put out 800W, had he trained his whole life as a cyclist, would probably be able to exceed 800.

But, we are just doing the best we can with the numbers we have.

Aaron D said...

Not to go off topic but are you planning on attending any ski races soon?