It was 1900 at the Paris Olympics. The Dutch pair of François Brandt and Roelof Klein had qualified in the coxed pairs, but had lost a race to a French pair. They came to the conclusion Hermanus Brockmann, their 60 kg cox, was too heavy so they found a young boy in the crowd to do the duty. The boy weighed 33 kilograms and was light enough that five kilograms of lead had to be attached to the shell to make the rudder properly set in the water.

The flying Dutchmen leapt to an early lead but couldn't keep their initial pace. Halfway into the race the French started their move, but at the finish the Dutch were still ahead - but by less than a meter. A dramatic race by any measure, but more curious is the fact that the cox disappeared into the crowd never to be awarded his gold medal.^{1}

How important is that dead weight? In theory a cox aims the boat in the straightest possible line, gives pacing and coaching, and removes the effects of uneven thrust as much as possible. But he or she also adds to the load. Just how important is that?

Before diving in, I note that I'm going to give a sense of the sort of approximation that is common in the physical sciences during play. One of my advisors said that a 13 year old kid who starts thinking about what miles per gallon means is left with the conclusion the measure is terrible as it doesn't work well in comparing vehicles (you really want the inverse - gallons per distance), you have someone who may well become a good engineer. If, on the other hand, the 13 year old says "wow - I can express that as inverse acres!", the kid has physics potential.

In doing very quick calculations to test ideas dimensional analysis is frequently invoked. The kid destined to be a natural scientist would look at the units and note that gallons is a volume or a mass. Assuming you are using the same liquid in your comparisons - at least liquids of similar densities - it doesn't matter ... take your pick. Let's pick volume .. the dimension of volume is a length cubed - L^{3}. Miles are easy - just a length. So miles per gallon is dimensionally L/L^{3} or 1/L^{2} - an inverse area. It could be inverse square meters, inverse square feet, or ... hey .. inverse acres has a nice rural feel to it.^{.2}

I needed to do a quick estimate for what is described next and rushed through it in about two minutes at the blackboard using this technique. There have been complaints that some readers find my writing too technical. It is ok to skip the next section and I'll color the skippable text in blue to make it easy, but I thought it might be interesting for some to see how I play and this can form the basis of a good-enough model for understanding what is going on without extensive calculations and measurements.

So the question is how much does it cost for a racing skull to carry a cox? Rather than a boat with two rowers let's consider the more general case with N rowers. A four man boat has twice the power of a two man, but its weight is about twice as much. Does having twice the engine and almost twice the weight help or hinder?

The power the rowers deliver mostly is used to overcome the drag on the hull. It is

p = f_{d} * v

where p is power, f_{d} is the force of drag, and v is the boat's velocity relative to the water

The force of drag is reasonably complex and would require some careful measurements that depend on the shape of the hull and some other factors. The important thing to realize it is proportional to the wetted area - something that goes as a length squared - and the square of the velocity.^{3 } So

f_{d} ∝ L^{2} * v^{2}

the power needed is

p_{d} ∝ L^{2} v^{2} * v or L^{2}v^{3 }

The power to overcome the drag is supplied by N rowers (assuming all are about equally strong). A racing shell is pretty minimal, but its volume needs to increase to float with the heavier load. The volume goes as L^{3} and also is proportional to N, so L ∝ N^{1/3}. Getting tricky we substitute and now

p_{d} ∝ L^{2}v^{3} ∝ N^{2/3}v^{3}

getting to the end! .... We know power supplied is proportional to N so

N ∝ v^{3} N^{2/3}

solving for v we see

v ∝ N^{1/9}

The speed of the boat only increases slightly with an increasing number of rowers. For those of you who have rejoined us it only increases at the one ninth power of the rowers or N^{0.111}. The time is just to the inverse of the speed over similar distances so t ∝N^{-1/9}. If you look at world record times over similar distances this holds for one to four person coxless skulls reasonably well and is a good enough model for this purpose.

The original question was what is the impact of the cox - the little guy or gal with the rudder and the megaphone? I did a similar analysis increasing the size of the boat, and thus its drag, by going to N + 1 crew members, but assuming only N are contributing to the power. I won't go through the reasoning as it takes much more time to type than to think about it, but it is very similar. The predicted time with a cox, t_{cox} is:

t_{cox} ∝ (N + 1)^{2/9}/N^{1/3}

To get something that is easy to think about look at the ratio of cox and coxless times and apply a bit of rearranging

t_{cox}/t ∝ ((N + 1)/N)^{2/9}

(N + 1) is always bigger than N, so the cox always slows things down ... not by a lot and the effect decreases with increasing N. A two man boat should be about nine percent slower if the weight of the cox is similar to the rowers. But it isn't. The Dutch used a kid who was about a third the weight of each rower, so a reasonable guess would be about (2.33/2)^{2/9 }or something over two percent slower - probably several percent faster than an equal muscled crew with a cox that weighed as much as poor Hermanus Brockmann.

You can play with some numbers yourself - assuming the cox is just along for the ride the impact of having one decreases with crew number. Since races are tight you don't want to race coxed vs uncoxed boats and you don't want a heavy cox. Hopefully they add more value and perhaps you can get a sense of quantifying that value in real vs expected times.

A physics teacher usually doesn't do this sort of manipulation for non-major undergrads, but recommends checking answers by looking at the units - if you are calculating an energy it should have the dimensions of mass times the square of a velocity or ml^{2}/t^{2 - }if your answer has different dimensions, there is no way the calculation can be correct. From there you can see if you have picked the right units. It can be unfortunate to mix metric and Imperial units.^{4}

Most of us use abstractions - frequently symbolic abstractions like art - to think about our fields. They allow us to move quickly and play with the key concepts so we don't get lost in the forest of details. When necessary it is often possible to do something much more detailed - at least we know how to do it in principle.

You could ask me what is the predicted time of a eight man coxed shell and I could give some rough approximations, but building an accurate model would rely on careful measurements. But if you asked me for a ratio of times when the athletes and boat designs were roughly equivalent - then it is only a couple of minutes worth of calculating. You realize when all the tricky stuff drops out and the fundamental parts of the underlying physics are all readily available - in this case some analysis let us get down to just the number of power producing athletes.

You have probably seen Feynman diagrams. They are really visual short hand abstractions of some very complex physics. You can create and manipulate them with relative ease and make easy to calculate comparisons even though working through the underlying math can be extremely difficult. They enable play and play and curiosity are two of the big drivers in the field. Play greatly illuminates whatever you are working on and I encourage anyone to find the play component of their craft and hone it.

But in the experimental phase there has to be enormous caution and the analysis phase combines caution and self criticism. It is still fun, but calculation short cuts don't apply as much as they did early on in the game.

But back to the game at hand. What is the real value of a cox? I had originally intended write about reducing the wiggle waggle in the shell's motion that is produced by the fact that there are uneven torques produced by the arrangement of the rowers when you have one oar per person. Next time or the time after I'll write about suppressing the wiggle waggles.

And there is another really important issue in many sports. Larger people tend to have more muscle, but they are also heavier. Only a few sports adjust for this and the fact that many don't limits participation at elite levels in many sports to larger people. But smaller people tend to have higher strength to weight ratios ... and you can't have a Godzilla.

__________

^{1} I came across a brief reference to this reading about rowing history and tried to track down a bit more. There are numerous references and a starting point is here.

^{2} I don't mean to suggest natural scientists are better or worse than engineers. You can have amazingly brilliant people in either. The goals and nature of each of these sports attracts very different personalities and native inclinations. In the end engineers do much better financially than physicists, so that is in their favor, but physics is great fun if that is your inclination.

^{3} Force has the units of mass * acceleration or mL/t^{2}. Note that L^{2} * v^{2} is L^{2} * L^{2}/t^{2} or L^{4}/t^{2}, but m ∝ L^{3}, so we can express the ball of wax as mL/t^{2}. (I'm also using L capitalized as l is difficult to tell from the numeric value of one in this font.)

^{4} The Mars Climate Orbiter was lost at orbit insertion as the ground based computations assumed Imperial units for the retro-rocket thrusters ... specifically pound-seconds. The spacecraft was constructed and the numeric values were tabulated using metric units of newton-seconds. This destroyed the craft as it entered the atmosphere at a low altitude. Nearly $700 million and a lot of potential science quickly went down the drain.

A similar unit problem put a plane load of people at risk, but a pilot who knew a lot about sailplane flying was in the cockpit and the Gimli Glider became aviation legend.

__________

Recipe Corner

A really great snack from canned beans. All measurements are approximate and don't matter much - just play around!

**Roasted Garbanzo Beans**

**Ingredients**

° a can of garbanzo beans

° a tablespoon or two of olive oil

° non-iodized salt

° spice blends - use your imagination. I have a cajun seasoning that is very good.

**Technique**

° preheat oven to 400°F.

° Drain the beans in a strainer and rinse with water . Spread the beans on a paper towel and use another to absorb the excess water and roll the beans around to remove the skins.

° put the beans on a baking sheet and drizzle olive oil over them making sure they are somewhat evenly coated. Roast for 30 to 45 minutes until the beans are a deep golden brown and crunchy and don't let 'em burn

° season with salt and spice blend.

These are really addictive!

how's that for wimping out with a quick recipe?

Totally awesome Steve! I had wondered about the coxes but had no idea how you would solve it. This is so simple compared to what I was thinking. Thank you for showing me some dimensional thinking last year. It really helped me in a biology math class. I don't think naturally in it like you - yet:)

I did find a typo where you make use of N to the one third being proportional to L, you have pd ∝ L squared v cubed ∝ N to the two thirds v3. The v3 should be v cubed. You must have forgot the superscript:-)

I don't know how to make the sub and superscripts in this reply.

Posted by: Jheri | 10/01/2012 at 08:17 PM

Thanks Jheri! I have some feedback that this had too much math, but also your comment and an email that liked it. I suppose there are two very different audiences (at least)

And thanks for finding the typo - nothing like not proof reading. It is now correct.

Posted by: steve | 10/02/2012 at 07:54 AM