PREFACE

*In this installment of The Science of Paddling we take a brief diversion from paddling, and explore what “science” means, and how that can inform one’s reading of these articles. So kick back, grab your favorite adult beverage, relax and float downstream.*

INTRODUCTION

When I was an undergraduate – you know; back when dinosaurs roamed the Earth – I presented my senior thesis before members of our department faculty. The presentation was titled, “An Acoustical Measurement of Fundamental Parameters in Porous Sound Absorbers.” Sounds like a real page-turner, eh? Without getting into the details, the thesis combined a mathematical model of how sound travels through porous materials (like, say, fiberglass) with the design, construction, and operation of an experiment, the data from which could be used to determine the physical constants in the mathematical model.

The presentation went great; I had fun giving it. Then came the Q&A part. The first question went to Prof. Earl Dowell who asked, in essence, “That’s all fine and dandy. But *where’s the science?*”

Now aside from being an excellent and insightful teacher, noted researcher, and a generous and patient person, Prof. Dowell is also the founder of the field known as aeroelasticity and a member of the National Academy of Engineering. It wasn’t quite like having Isaac Newton stand up and ask a pointed question, but in my world it was pretty close. In response I babbled on about my approach, the model, the results found, and the implications thereof. I pointed out one unexpected result from the experiments (an acoustical anisotropy), and that seemed to suffice. Afterwards, my thesis advisor pulled me aside and noted, “That was an *excellent* question from Prof. Dowell. You handled it pretty well, but…” I won a couple of awards for my research. It was only later that I realized a Zen master had confronted me with a koan, and instead of achieving satori I remained mired in samsara.

So why the stroll down memory lane? Does this have something to do with paddling? Well, no; this doesn’t have anything to do with paddling. It does, however, have something to do with The Science of Paddling (see how meta this is getting?). In particular, where is the science in my articles? Is there any? (Hint: Yes.) And is there something we can draw from science and the scientific method itself to learn how to most profitably read these articles? (Another hint: Yes. Question everything!)

This article was prompted by some of the emails I’ve received about TSOP, where the emailers were trying to see more in these articles than is really there. Or in some cases, less. So if you feel like pairing a little philosophy with your paddling, read on! We’ll start… with science.

SCIENCE!

So what is science? “Science… is the organized, systematic enterprise that gathers knowledge about the world and condenses the knowledge into testable laws and principles.” [E.O. Wilson, *Consilience*, pp. 58.] Features of science include repeatability of results, the ability to abstract results into theories, and perhaps most important, *falsifiability*, e.g., a scientific theory can be proven true or false using universally-accepted methods. Further, a scientific result can be true within a certain context (anti-CD28 monoclonal antibody TGN1412 successfully stimulates T cells in monkeys), and false outside of it (anti-CD28 monoclonal antibody TGN1412 triggers a potentially fatal immune response in humans). Consequently, science must also bound the range of applicability for its results. These bounds generally arise from underlying assumptions or limitations regarding a theory, test result, or test method.

Now most Science of Paddling articles are seasoned with a bit of mathematics, not just science. In contrast to science, mathematics is not concerned with empirical validation through observations made in the physical world. Mathematics is a collection of concepts that are tested – in other words, proven true or false – using the tools of logic. Some of you may recall doing “proofs” in a high school geometry course. You can thank the ancient Greek geometer Euclid for that! A mathematical proposition – which can also be thought of as a theory – can be proven true or false in light of certain agreed-upon “facts,” called axioms, or in light of previous mathematics that has withstood similar rigorous analysis. In that way mathematics is falsifiable, too.

*Applied* mathematics is used to build and analyze models related to observable reality. For example, around 600 BC Thales of Miletus developed a geometric tool called “shadow reckoning” to calculate the height of buildings and towers, as well as the distance of ships from a harbor, using a method we now refer to as similar triangles. It’s fairly easy to show that Thales’ work – and later, Euclid’s more encompassing geometry – has a correspondence to objects in the real world. Empirical models based on geometry can be constructed with rulers, dividers, and lengths of string. You may have done this yourself, bisecting a line segment with another perpendicular line drawn through intersecting arcs having their origin at the segment’s endpoints; dividing a circle’s circumference into sixths using a divider set to the circle’s radius; and so forth. These models allow us to demonstrate that many geometric theories are true via construction: you draw a picture, you make a measurement, and can see with your own eyes that shadow reckoning is true. Geometric propositions are falsifiable.

Similarly, we can understand that arithmetic works based on empirical observation: If I place two apples into a basket that already contains three apples, then the resulting numbers of apples in the basket is five. And while some of you may shudder when you remember algebra, it’s also easy to show how algebraic statements like, “If a train leaves New York traveling west at 50 mph, and a train leaves Los Angeles traveling east at 60 mph, which train will reach Chicago first?” directly correspond to empirically-testable results.

But things got weird when Newton came along and developed the foundations of classical physics. Isaac Newton and Gottfried Leibniz – at the same time, and independently – developed calculus, the branch of mathematics that among other things is used to model *changes* in physical quantities like position, speed, and acceleration. Calculus is built on the foundations of algebra and geometry (the slopes of lines, tangents, areas, and the like), plus a couple of revolutionary new ways of viewing the world (limits and differentials). Newton employed the new calculus to develop the branch of physics called mechanics, which described the relationship among force, mass, momentum, acceleration, and velocity. And yet… how obvious is it that gravity is an acceleration, and not a force? (You can thank Newton for that one.) Despite these less-intuitive mathematical underpinnings, classical mechanics derives its power from the ability to model and predict things in the physical world: The trajectory of a bullet, how to launch a spacecraft into orbit, or the fastest way to paddle across a river in current.

A proliferation of other disciplines followed including fluid mechanics, acoustics, thermodynamics, heat transfer, and (thanks to Faraday’s experiments and Maxwell’s analysis) electromagnetics, all of which have formalisms that rely on further branches of mathematics know as differential equations. Differential equations are built upon the foundation of calculus. In each of these fields sophisticated mathematical models are developed based upon simpler models that are generally accepted as true. As an undergraduate I recall deriving a series of coupled partial differential equations called the Navier-Stokes equations that are based, when all is said and done, on Newton’s 2^{nd} law of motion (force is proportional to the time rate of change of momentum, aka mass times acceleration for fixed-mass systems). The Navier-Stokes equations allow us to do fun things like design jet airplanes, sewer systems, and automotive fuel injectors. And it lets me write Science of Paddling articles.

What is the link between sophisticated mathematical models and the real world? Philosophers have puzzled over this for centuries, and I’m not qualified to add to their corpus. I’ll let noted scientist E.O. Wilson step in:

For reasons that remain elusive to scientists and philosophers alike, the correspondence of mathematical theory and experimental data in physics in particular is uncannily close. It is so close as to compel the belief that mathematics is in some deep sense the natural language of science. “The enormous usefulness of mathematics in the natural sciences,” [mathematician Eugene] Wigner wrote, “is something bordering on the mysterious and there is no rational explanation for it. It is not at all natural that ‘laws of nature’ exist, much less that man is able to discover them. The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.” [E.O. Wilson, *Consilience*, pp. 53]

What I’ve personally experienced is that the predictions of applied mathematics, *when properly qualified*, match phenomena I’ve seen and empirically measured. For those of you wondering when I’d get to it, that’s the (long-winded) takeaway from this article.

We’ll revisit what “when properly qualified” means in a moment. But first, consider a simple example: an electrical circuit composed of resistors, capacitors, and inductors. If I apply a particular voltage signal to this circuit, I can predict what the voltage and current signals will be anywhere in the circuit via a mathematical model readily derived using college sophomore-level linear systems theory. Consequently, one can demonstrate this direct correspondence between the mathematical model and the physical circuit. Ditto for the feedback control of a flexible mirror’s shape based upon a set of mathematical functions called Zernike polynomials, and the design of a sonar sensor to “automatically” resolve the angle of an underwater sound source. I’ve personally modeled and built each of the systems listed above; in the last case I relied on the differentiation property of Fourier transforms – you know, math – to design, develop, and patent an entirely new class of sensor… and it worked in the field as predicted. In each case I was left with a sense of both delight (akin to, “Nailed it!”), and wonder: How can mathematical equations “tell” me what to do, and why does the thing I build subsequently work? And as important, when does it *not* work?

It’s easy to be pragmatic about these things, accept that it works, and get on with your life, *especially as an outside observer*. This perspective has limits, though, especially in reading technical articles. It can foster complacency, as embodied in the reaction, “Well, that’s sure some sophisticated mathematics – it all must be true.” Or it can cause one to latch onto something familiar, and base their reaction on the habitual / familiar rather than thoughtfully considering the work as a whole. I caution you, gentle reader, not to fall into these traps.

I’ll let you in on a little secret; give you the secret handshake of science and engineering. Applying calculus and other sophisticated mathematics to objects in the physical world relies upon *assumptions*. For example, Newton’s physics relies on reference frames that are either static or move with a fixed speed. If a reference frame is accelerating, or you start approaching the speed of light, classical physics falls apart and you have to jump on board with Einstein – and things get *really* weird. Underlying my favorite fluid mechanics expression, the Navier-Stokes equations, is the continuum hypothesis. The continuum hypothesis requires that you limit your analysis to dimensions a whole lot larger than the distances between atoms in the fluid. This means you can say “the Charles River is flowing at 900 cfs” rather than having to specify the velocity of each and every water molecule in the river. Seems obvious, but there are entire branches of fluid mechanics (e.g., non-equilibrium statistical mechanics) that treat fluids statistically. Or that nifty sonar sensor I invented: It works great resolving a sound source along a two-dimensional plane, but once you start looking at sound sources in three-dimensional space (aka, “off axis”) performance begins to suffer.

If you are mindful of the *underlying assumptions* you can bound when a system will perform correctly, and when performance will degrade. It is equally important to understand the underlying assumptions in *any* physical model or analysis, *for the exact same reasons*. If you are mindful of the *underlying assumptions* you can bound when the results presented in a technical article work, and when performance will degrade (or where the model has less fidelity, or is not applicable). This is part of the scientific method through and through, harkening back to Socrates when he noted, “The beginning of wisdom is the definition of terms.” And that’s the lens you can and I hope will employ when you read any Science of Paddling article, or any article for that matter. In that way, as the reader you too are a scientist.

TELL ME ABOUT PADDLING ALREADY

So where does this leave The Science of Paddling? Where’s the science?

Well, certainly the article *The Deflection Point* has elements of science: the relationship between heart rate and effort during exercise will experience a break point at the anaerobic threshold owing to our physiology, and I saw this in data I took myself. That’s an example of first-person science: test a theory via experiment, then see what the data shows. If others conduct the same experiment and obtain the same result, that’s even better; if enough confirmations are found to make the result statistically significant, Bingo! All I contributed was a framework for making this break point easier to see when plotted based on a model of how hulls experience resistive forces when underway. That’s applying a mathematical concept, rooted in a simplified physics model (drag force in racing hulls is proportional to the square of velocity above a certain speed), to experimental data. Is that one tweak by itself… science? I’m not convinced that it is, but it sure is useful.

“Linearizing the Field” was all about exploring underlying assumptions. The NECKRA Points Series utilizes a straight-forward process to weight finishing times of various canoes and kayaks, paddled by various combinations of male, female, youth, and senior / veteran paddlers, in order to rank race results. The underlying assumptions of this approach were likely never stated before this article was published, perhaps because they were implicit and common-sensical. But as pointed out in the article, this approach breaks down when more than one weighting factor is applied to a given finishing time, and grows worse as more are introduced. True result; possibly useful; likely of interest to a tiny fraction of the readership. But this article shone a light on the role of underlying assumptions in something many of us in New England took for granted. Is this science? Yes, in that it built upon a previous model (the Points Series) and developed new conclusions, verified by testable mathematics, that reveald inherent assumptions in the NECKRA model. The article gets at method, e.g., question everything.

Finally, how about the predictive model employed in “About the Bend”? For those of you who haven’t memorized all of my articles (which is all of you, I hope!) I applied someone else’s empirically-derived sprint canoe dynamical model to assess the role of the bend angle in bentshaft paddles for the marathon stroke. I wasn’t attempting to make any *quantitative* performance prediction per se; sprint and marathon canoe stroke mechanics are different, but at least there was some data to work with. All I was looking for was insight into whether the analysis suggests that the bend angle makes any difference (it does), and whether the model suggests there could be an optimal bend angle (it does). That’s it. There was definitely no science involved, just some math twiddling where I extrapolated data taken from a sprint canoe paddler, who had employed a sprint canoe stroke, and applied it somewhat tenuously to marathon paddling.

Yet despite my efforts to couch the analysis in qualifying statements, “About the Bend” generated more email traffic than any other I have written. At the risk of being rude – which I don’t intend to be in the least – most of the questions could be answered by, “Please re-read what I wrote about the *underlying assumptions*,” and, “No, the article doesn’t quantitatively predict X,” for whatever conclusion ‘X’ was. All predictions were qualitative.

In essence, this is what Professor Dowell was asking about, all those years ago. Back then I had taken a mathematical model, constructed an experiment that embodied certain features of the model, took data, and used the model to interpret the experimental results. In a way I was fitting data to a model without fully appreciating (or at the time, understanding) the assumptions implicit in the model itself. Kind of like “build your own tautology,” where I was asking the model to be consistent with itself. In retrospect, some of the assumptions I employed at the time were obvious and reasonable since the acoustic wave equation is derived from the Navier Stokes equations (which are based upon Newton’s Second Law of Motion, which we accept as correct because it works) and the equation of fluid mechanical continuity (which is based on the notion that fluid mass is conserved, and not lost; since the experiment didn’t “leak” air this was a valid assumption). However, the thermodynamic equation of state – which relates acoustic pressure and density – I used to derive the model had an implicit assumption that I could only articulate a year later, and it was the crux of the matter. I later understood that the experiment was actually looking for when and how a certain property embodied in this equation of state changed with frequency; it was *testing an assumption of the model*. As an undergraduate I was looking at it from two levels of abstraction higher. And I was enamored with all the nifty equations.

We’re all busy; we all like what we like; I for one am fascinated by, well, everything. In my periodic science and engineering news trolling across the Interwebs I stumble upon headlines touting remarkable achievements in medicine, engineering, astrophysics, and the like. Yet when I read an interesting article a little voice starts whispering in my ear, “Where is the source article that this is based on?” And when I finally locate the source article I first search for disclosures about the underlying assumptions: test population size, measurement limitations, whether experiments were performed on humans or fruit flies, etc., in order to separate the wheat from the chaff. In understanding the assumptions, you can quickly bound the range of applicability and thus the utility of any result in science, engineering, mathematics, medicine, etc.

One might argue that this process of examination and discernment sucks all of the fun out of life – or that I don’t have a life if this is what I do for fun! For me, searching for and understanding underlying assumptions and limitations lets me know when to discount a headline or claimed result, and when to really get excited about it. That way I can put my energies into following science and engineering that will have, in the words of my friend and thesis committee member Dr. Dan Hegg, “Depth, breadth, and permanence.” That’s where long-term satisfaction lies. It also helps me decide when I can put a particular result into practice, hopefully to make my life easier, more fun, or best of all, provide more insight.

© 2019, Shawn Burke. All rights reserved.

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]]>*by Shawn Burke, Ph.D.*

INTRODUCTION: PAIN WITH NO GAIN

I’m sure some of you have had the following experience at a race: You go out hard at the start, really “putting the coal to it” to hang with the leaders. Then, maybe thirty to forty-five minutes into the race your body says, “No more!” Crash and burn. Coming in to the race you felt well trained; rested; nourished and hydrated; calm and focused. But as you cruise – slowly – into the finish you wonder what the heck just happened?

While there are any number of reasons for this kind of performance – and I’ve experienced ‘em all – you may have been pushing a bit too hard, at an anaerobic pace, and your body just wasn’t able to keep up with its production of blood lactate, hydrogen ions, and the other by-products of muscle metabolism. Remember what we covered in Part 6, What Fuels You?^{[1]} Well your body really does all that stuff. Not only did you crash and burn, but you experienced science!

As we saw in Part 6, the various energy systems – aerobic, anaerobic, and phosphate – are called upon in proportion to the duration and intensity of exercise. The body knows perfectly well when it is performing aerobically or anaerobically. How do *we *know whether our workouts are aerobic, anaerobic, or somewhere in between? And what can we take from this to help understand how hard to go during a race?

In this installment of The Science of Paddling we’ll explore a way to measure paddling intensity and determine whether your body is performing aerobically, anaerobically, or “somewhere in between.” In particular we’ll focus on a way to determine, using a paddle ergometer and a heart rate monitor – in the comfort of your own home gym^{[2]} – the transition point called the “lactate threshold” above which your body is unable to keep up with the production of lactate. Think of it as the red line on a car’s tachometer. Once you exceed the threshold and stay above the red line, the clock starts ticking until you flame out.

Much of what follows may be well known to runners and cyclists. We’ll assume that you’re familiar with the use of heart rate as a measure of exercise intensity; the harder you work, the faster your heart beats. For sports such as running the relationship between heart rate and intensity is pretty much linear up until the lactate threshold; if you increase your speed by a certain amount your heart rate will rise in direct (aka, linear) proportion. But as we learned in Part 1: Tandem vs. Solo, at racing speeds the drag force on a hull is proportional to the square of speed. It’s not a linear function. As a result, the work you expend paddling a hull increases nonlinearly with speed, too; it not only gets harder, but it gets harder fast. As we’ll see, this complicates things a bit if you want to determine your lactate threshold, since tests that don’t rely on taking blood samples to directly measure blood lactate concentration generally assume that this relationship is linear. Which is where the physics of paddling comes to the rescue.

Now we could dive right in to a prescription of how to measure your lactate threshold for paddling, and if that’s what you want please feel free to skip ahead to the section titled “The Deflection Point and Lactate Threshold.” But since many readers may be less familiar with the minutia of heart rate data and blood lactate concentration you’ll find a review of these topics in the next two sections. Then, we’ll use these tools to understand heart rate data obtained in a so-called “step test.” You get to be a paddling scientist in the comfort of your own abode.

HEART RATE

The body’s circulatory system transports oxygen and glucose to muscles during exercise and removes the waste products of exercise such as muscle-generated heat, carbon dioxide, water, hydrogen ions, and adenosine. These metabolic byproducts cause the capillaries to expand, enabling the heart to deliver more oxygenated blood to the muscles^{[3]}. At rest the heart pumps about 5 liters of blood per minute. As you begin to exercise the autonomic nervous system tells the heart to beat faster. At maximum output the heart of a highly trained athlete can pump as much as 40 liters per minute.

As noted above, there is a relationship between heart rate and exercise intensity. Consequently, heart rate is commonly used as a measure of exercise intensity, as well as for planning a targeted training program. The maximum and resting heart rates are of particular interest, as these are used to prescribe training zones, track the progress of your exercise program, and monitor recovery from workouts.

Everyone has a *maximum heart rate *(HR_{max}) that does not change appreciably with endurance training. The maximum heart rate does, however, decrease with age. It is difficult to determine HR_{max} without directly measuring it. However, HR_{max} can be *approximated *via

HR_{max} ~ 205 – (age ÷ 2) for men,

HR_{max} ~ 210 – (age ÷ 2) for women.

According to this formula, a 44-year-old woman has a maximum heart rate of 210 – (44÷2) = 188 beats per minute (bpm). Note that an athlete’s true maximum heart rate can differ significantly from what is predicted by this simple formula, so it is best to actually measure HR_{max} if at all possible. We’ll see how to determine HR_{max} more precisely using a heart rate monitor during a threshold workout or race.

A chest strap heart rate monitor is a most convenient tool for measuring your pulse^{[4]}. These devices consist of a sensor, a transmitter, a receiver, a little computer, and a display. Sensing electrodes are located on the chest strap. These electrodes measure electrical activity in the body when nerves fire to make the heart beat. A transmitter in the chest strap sends this information to the receiver using a low-power radio signal. The receiver can be built into in a plastic watch housing and worn on the wrist, or mounted on a thwart, bulkhead, or the like. The wrist unit displays heart rate; more advanced models can also measure speed and distance using other sensors such as GPS or accelerometers.

You can measure HR_{max} accurately in the field using a heart rate monitor. Here are two methods:

*Threshold Test*: First, you must be fully recovered from any earlier anaerobic workouts or races, and well rested. The test begins with a warmup of 15 to 20 minutes, with gradually increasing intensity. This is followed by an all-out effort of 20 minutes. The level of effort during the last 4 to 5 minutes should be maximum. The final 30-60 seconds should be an all-out sprint, and you should feel that there is nothing left in the tank at the end. The heart rate reached over this final 30 seconds will be very close to HR_{max}, usually within 2 bpm or less.*Race Test*: At the end of a 5-kilometer to 10-kilometer race, finish the event with a kick of 4 to 5 minutes. The final 30-60 seconds should be an all-out sprint, and you should feel that there is nothing left in the tank at the finish line. The heart rate reached over this final 30 seconds will be very close to HR_{max}.

An important caveat about HR_{max} is that it is *sport specific*. HR_{max} for running generally will be higher than HR_{max} for paddling, cycling, or swimming; differences of 10bpm to 15bpm are common. This variation in HR_{max} can be attributed to the differing mix of muscles engaged for running, paddling, swimming, cycling, etc. For example, it is estimated that a paddler uses 85% of the muscle engagement of a runner. This is because a paddler’s body is supported against gravity by the canoe (a “weight supported sport”), and the paddler’s legs are only performing isometric contractions during the leg drive. Since HR_{max} is used to plan training, if you are a multi-sport athlete it should be measured for each sport using the methods outline above.

So why do we care about HR_{max}? Because every paddler’s HR_{max} will be *different*, for starters. I’m always amused at races when I hear someone say, “My heart rate got up to 175!” Well, OK; is that good or bad? Does it mean they were working really hard, or not? Unless we know what that person’s maximum heart rate is the number is meaningless. Further, since most training prescriptions based on heart rate are expressed as a percentage of HR_{max}, knowing HR_{max} sounds like a good thing.

Figures 1 and 2 are sample heart rate records plotted over time. The first shows heart rate as a percentage of HR_{max} over a 55-minute race. Nearly steady-state heart rate is reached in about 3 minutes, reflecting how long it takes for the aerobic system to fully engage. There was a buoy turn on the course at the 19-minute mark; note the increase in heart rate as the paddler makes the turn and briefly accelerates. Otherwise heart rate is fairly uniform, reflecting a constant effort throughout the event. Heart rate rises during the finishing kick, reaching 98% HR_{max} at the finish. It turns out that the steady-state heart rate during this race, 89% HR_{max}, was likely very close to this paddler’s lactate threshold. And as we’ll learn below that’s a very good thing for such a race length.

*Fig. 1:* *Heart rate plot for 55-minute race.*

Figure 2 plots heart rate as a percentage of HR_{max} of a canoeist paddling at a constant level of intensity (e.g., pace). Note how the heart rate increases over time during the work interval. This phenomenon is called *heart rate drift.* Heart rate drift in fit athletes can be attributed to increases in core temperature, or on long races to dehydration – with less blood volume, the heart has to pump harder to supply the muscles with oxygen and remove waste products. In really long races heart rate can even drift downwards at a constant level of effort.

*Fig. 2: Plot showing heart rate drift.*

Trained endurance athletes have a lower *resting heart rate *(HR_{rest}) than untrained individuals. The average person’s resting heart rate is between 70bpm and 80bpm. As fitness increases HR_{rest} gradually decreases. A well-trained athlete can have a resting heart rate as low as 40bpm to 50bpm. This is due in part to conditioning-based changes in the heart. Endurance training increases the heart’s stroke volume, hence it can pump more blood per beat. Also, the muscles’ metabolic systems become more efficient with training, requiring less oxygen at rest.

You should measure resting heart rate in the morning, right after you wake up. Studies have shown that HR_{rest} is generally lower in the morning than in the evening^{[5]}. You can go so far as to use your heart rate monitor to measure HR_{rest}. However, manually measuring the radial pulse will suffice. An obscure but true fact is that your heart rate will be elevated by around 2-3 bpm if you need to go to the bathroom. So if you need to pee when you wake up do so, then climb back into bed. Your heart rate will settle back down in a few minutes and you can get a more accurate reading of HR_{rest}.

Some people count their pulse over ten seconds and multiply the result by 6 to get their heart rate in beats per minute. Taking your pulse over a full minute will increase accuracy; if you miscount your pulse over 10 seconds by one beat, your estimate of HRrest will be off by 6 bpm. Are you really in such a hurry that you need the extra 50 seconds in your day? As you take your heart rate you will notice that your pulse increases when you breathe in, then decreases when you breathe out. This phenomenon, called *heart rate variability*, is perfectly fine normal reflects a healthy heart. Just breathe normally and take your pulse over a full minute.

It is a widespread myth that HR_{rest} is itself a measure of an athlete’s fitness. Rather, gradual *systematic decreases *in HR_{rest} over months and years reflect improved conditioning. More importantly, HR_{rest} is an important indicator of recovery from the previous day’s workout or race. If HR_{rest} is elevated, say by 5bpm compared to the usual baseline established over time, then recovery may be incomplete; you should consider substituting a recovery workout for that day’s training. The morning pulse should recover to baseline by the next day. Increases in HR_{rest} that do not go away can be an early indication of overtraining, stress, or looming viral infections (e.g. colds, the flu).

An example of how the resting heart rate can change over time is shown in Figure 3, a plot of my morning rest heart rate over one week. Back when I took this data my resting heart rate was 54 bpm. Note how HR_{rest} rose by 8 bpm when I was fighting a cold on Tuesday and Wednesday, then returned to normal levels after the cold had passed. The elevated resting heart rate on Sunday was the result of a hard 2-hour overdistance workout on Saturday.

*Figure 3: HR*_{rest}* over time.*

LACTATE

As you exercise your glycolytic system produces lactic acid. Before a training session most paddlers have a blood lactate concentration of approximately 1.5 millimoles per liter (mmol/l). Once you start paddling aerobically the concentration drops below 1.0 mmol/l, and as you continue to paddle it will rise depending on the level of effort.

The relationship between blood lactate production and exercise intensity has both linear and non-linear components, as shown conceptually in Figure 4. This relationship is fairly linear below a lactate concentration of 1.5 mmol/l, then starts to bend sharply upward as the concentration approaches 4.0 mmol/l. The 4.0 mmol/l blood lactate concentration point – often called the *lactate threshold *– is of great interest here, since it occurs at roughly the transition point between aerobic and anaerobic metabolism. A rule of thumb is that you can race continuously at the lactate threshold for about an hour before flaming out; less if you race harder; longer if you periodically drop down below the lactate threshold to recover (like when doing intervals, or dropping in behind a boat to stern wake).

*Figure 4: Blood lactate concentration as a function of exercise intensity.*

The blood lactate vs. exercise intensity curve will shift to the right as you become more fit. This is because with increasing fitness your aerobic system will be able to clear the metabolic byproducts of exercise more efficiently. The glycolytic system and the Cori cycle will become more efficient as well, especially after anaerobic training such as tempo paddles, hard intervals, and racing. With this kind of training you’ll be able to paddle harder for longer periods of time because your lactate threshold will shift to increasingly higher exercise intensities.

This *training effect *makes the determination of metabolic training zones and zone boundaries problematic, since they shift with your changing fitness level. The only truly precise solution is to periodically measure blood lactate production as a function of workout intensity. While not practical for most of us, they may work for a paddling club or team.

Blood lactate can be measured by taking a drop of blood from a fingertip or ear lobe much the same way diabetics monitor the level of blood sugar. Samples are taken during exercise at specified intensities and plugged into the analyzer, whereupon your blood lactate concentration corresponding to that intensity is measured and displayed. Portable analyzers like the Lactate Pro^{TM}, shown in Figure 5, the Lactate Scout, etc. enable paddlers and coaches to accurately determine blood lactate levels in the field to precisely determine individualized metabolic training zones. They are the gold standard.

*Figure 5: Lactate Pro*^{TM}* portable blood lactate analyzer.*

Other products have been introduced over the past few years that use near infrared spectroscopy to determine blood lactate concentration, such as the BSX Insight (which was recently discontinued). However, this product only provided a blood lactate measurement using post-processing at the company website, and it was designed for use in running and cycling. Consequently, the rest of us are left with indirect methods for determining the lactate threshold.

THE DEFLECTION POINT AND LACTATE THRESHOLD

The relationship between blood lactate production and exercise intensity is non-linear, especially at higher levels of effort. We saw that the lactate concentration vs. exercise intensity curve, Figure 4, has a “knee.” And we also know that for paddlers there is a nonlinear relationship between heart rate and exercise intensity. (We engineers just love this kind of stuff…) With a little tinkering, you can take advantage of these facts to determine non-invasively an important blood lactate concentration level and its associated heart rate.

The *heart rate deflection point* (HR_{defl}) is defined as the heart rate above which the accumulation of lactate exceeds its recycling by the liver. This corresponds to the *lactate* *threshold* of 4.0 mmole/l blood lactate concentration. A fit paddler can perform at the intensity level corresponding to HR_{defl} for a long period of time, as there is equilibrium between lactate production and removal; a rule of thumb is that a fit endurance athlete can perform at HR_{defl} for about an hour. There is a good correlation between the anaerobic threshold and HR_{defl}. Once you know your lactate threshold heart rate you can determine your *lactate threshold pace* by paddling at HR_{defl} and noting your pace on a GPS sports watch; easy peasy. And most importantly, this pace corresponding to HR_{defl} can be increased through training.

Italian physiology professor Francesco Conconi developed a means to non-invasively measure the lactate deflection point; otherwise, to date blood samples have to be taken to determine the heart rate and exercise intensity at which blood lactate concentration reaches lactate threshold. The so-called *Conconi Test* maps heart rate vs. exercise intensity at a number of intensity levels, e.g. at a number of pace/speed set points. For runners, the plot of heart rate vs. pace will initially have a generally linear trend, as shown in Figure 6. At the deflection point the heart rate versus intensity plot curves downward – what mathematicians call an “inflection point.” The aerobic system – which is based upon the heart’s ability to deliver oxygen and remove lactate – can no longer “keep up” with higher intensities, and the energy mechanism transitions from aerobic to anaerobic metabolism. This knee in the curve corresponds fairly well to the 4.0 mmol/l blood lactate threshold.

*Figure 6: Idealized Conconi Test plot*.

The Conconi test can be conducted on a paddling ergometer equipped with a performance meter like that shown in Figure 7. You’ll also need a heart rate monitor, a sports watch that can be programmed to create custom workouts, and software that can average heart rate data, plus software like Excel that can plot and perform basic mathematical computations. You must also be fully recovered from any race, interval workout, long paddling session, etc. before your start. Check your resting heart rate, for example, to be sure.

The process is pretty straight forward, but expect your first test to be a learning experience as you experiment with the ergometer’s damper setting, starting pace for the first step, and the number of steps. Once you have all of these parameters sorted you can repeat the test every 4-6 weeks to assess the impact of your training program on (potential) race pace. Just note that for the step test to be a valid indicator of training progress, it must be conducted under identical conditions each time.

*Figure 7: Paddle ergometer with performance monitor.*

If you are using a Concept 2 rower, set the damper to 2 or 3; this will take a bit of experimentation. An extensive warm-up of 15 to 20 minutes is followed by a “step test.” During this step test the pace is decreased at regular intervals, approximately 10 times, as defined in Figure 8. Program your sports watch to do an interval workout^{[6]} comprising: (1) a 20-minute warmup of gradually increasing (but still easy) intensity, followed by (2) ten to twelve 2-minute segments with no target pace or heart rate (and no recovery phase), then (3) a 5-minute cool down. The first step is done so that you finish at a pace corresponding to 55-60% HRmax; for me this was 3:00/500m; your pace may be different. After reaching each step’s target pace keep your pace constant for 2 minutes. This will take a fair bit of focus and concentration. Record your heart rate over the entire test with a heart rate monitor.

*Figure 8: Conconi Test logic.*

The pulse over the last minute of the step is the steady-state heart rate for that pace; the heart rate needs a period of time to adapt to the new speed over a step as seen in Figure 9. In this Figure you see that the heart rate spends most of the first minute rising, then is fairly level through the rest of the two-minute segment. For analysis you can either average the data over the last minute, or over the entire segment; just be consistent. If you choose the latter your lactate threshold heart rate estimate will be a couple of bpm low.

*Figure 9: Heart rate (red trace) drift over a 2-minute segment.*

Pace is incrementally decreased by 5sec/500m for each segment, and data points recorded until points are logged at and above the anaerobic threshold, typically somewhere around 90-94% HRmax.

When I plotted average heart rate data vs. step from my own step tests I noted something odd. You’ll see the data in Figure 10. The heart rate data doesn’t fall on a straight line before it (apparently) curves off to the right. Instead, it exhibits a bit of a curve. For comparison, I plotted a red dashed line where I tried to find a linear behavior among some of the upper points, and a green dashed line where I looked for that kind of behavior among the lower points. While you might argue that the curve “deflects” at the 8^{th} segment, you’d be hard pressed to justify that choice. Why not the 7^{th}? Or the 9^{th}? Also, why is it bowed along the lower 8 segments?

*Figure 10: Heart Rate vs. Segment (blue trace).*

Well; duh. Physics! As we learned in Part 1, the resistive force on a hull goes as the square of the velocity at race paces. This means the resulting intensity should show that kind of quadratic behavior as well.

This is where high school algebra helps us. As you may recall there was this function called a logarithm. Logarithms were invented by a fellow called Napier back in the 1800s to aid in computing the products of large numbers, among other things. He developed a mechanical means to perform these computations, the forerunner of slide rules.^{[7]} Since then several other interesting properties of logarithms have been deduced. Namely, taking the logarithm of any function that is expressed as the power of a number reduces to a simple linear form:

That is, something that showed a quadratic (or any other power) form when plotted would now take a *linear* form if you plotted its logarithm instead.^{[8]}

Armed with this insight, I used Excel to calculate the natural logarithm (“ln”, or “log* _{e}*”) of my heart rate data, and re-plotted the data. The result is shown in Figure 11.

*Figure 11: Log _{e}(Heart Rate) vs. Segment.*

I added a straight line fit to the lower data points, which now looks to be a good fit. And the deflection point is very clear, occurring at or just above segment 8’s heart rate. Segment 8 had an average heart rate of 159 bpm, which for me is 88% HR_{max}. As you can see, leveraging our knowledge of physics made analysis of the heart rate data much easier.^{[9]}

While these test results don’t map to performance on the water based on *pace* since they are done on an ergometer in the comfort of your home or gym, they do map well based on *heart rate* since the ergometer does a good job mimicking the type of resistance and paddling motions encountered in the canoe. Consequently, you’ll have a hard time using erg pace data to predict race times. But once you know your in-boat pace at HR_{defl}, you can predict potential race performance fairly well. Plus you’ll know where the “red line” is on your internal tachometer, and can modulate your level of effort accordingly.

You can also conduct a Conconi test on the water if you keep your wits about you re. environmental factors, and if you have a data recording GPS/heart rate monitor. You can program this unit to prompt you to paddle at target paces in successive steps. As you paddle each step it the will record distance traveled, your pace, and your heart rate. Use averaged data, and perform the analysis as outlined above. Note that you should paddle each step on a long deep-water course, with no wind and no current. Otherwise shallow water effects will corrupt your data, as we learned in Part 4: Shallow Water, as will the speed boost from wind or current.

The primary limitation of the Conconi test is that it doesn’t work for every paddler. Some paddlers will perform a Conconi test and the resulting data won’t show an inflection point. While this may reflect an insufficient warm up or step lengths that were too short, some paddlers will have an indistinct deflection point at best. If this happens to you, try the test again in a couple of weeks, paying careful attention to pre-test recovery, proper warm up, and step length. If the resulting test data is still ambiguous, consider using a blood lactate test to determine your anaerobic threshold. Or, use time trial results to estimate HR_{defl}. Recall that you can paddle at your anaerobic threshold heart rate for about an hour, so find a suitable race or course offering that duration, paddle while holding a consistent heart rate and level of effort, and listen to your body – and write down that heart rate!

APPENDIX: A NOTE ABOUT OXYGEN

Some of you may be thinking, lactate threshold sounds interesting, but isn’t VO2_{max} of greater importance? Important; yes. But *more* important? For determining race pace, not really.

Aerobic metabolism uses oxygen to fuel the electron transport system. Oxygen binds to an iron- rich protein in the blood called hemoglobin, and is transported from the lungs to the muscles. Oxygen unbinds from the hemoglobin to feed the various muscle oxidative sites, whereupon the hemoglobin binds to carbon dioxide produced by muscle metabolism. This carbon dioxide is carried to the lungs and expelled when you exhale. The volume rate of oxygen you can process metabolically is often written as VO2 (“Volume of O_{2}”), and the *maximum volume rate of oxygen *that you can process is called VO2_{max}.

VO2_{max} is the most common measure of *maximal *aerobic capacity. It depends on oxygen diffusion in the lungs, hemoglobin binding efficiency, blood transport, and mitochondrial density and efficiency. It also depends pretty strongly on genetics. Athletes reach VO2_{ max} when their oxygen consumption stops increasing while physical exertion still increases. To reach VO2_{max} the body needs to produce approximately 10 mmol/l of blood lactate, and a respiratory quotient – the ratio of carbon dioxide output to oxygen consumed, denoted as “RQ” – in excess of 1.1.

VO2_{max} can increase with training by only about 20 percent at most. Once a certain aerobic capacity is established, VO2_{ max} will only increase marginally with additional aerobic training. Further, aerobically fit athletes show little fluctuation in VO2_{max} over a season.

Studies have shown that you will see the greatest improvements VO2_{ max} by exercising at intensities corresponding to 90 and 100 percent VO2_{ max}. A 90-100% VO2_{max} effort roughly corresponds to exercise intensities of around 90-100% HR_{max}. This range corresponds to sprint interval and high-intensity continuous training, an intensity perhaps most appropriate for dragon boat paddlers and sprint canoeists and kayakers. In contract, marathon canoeists need not train at this level of intensity for more than 5% of their total yearly workload.

Since VO2_{max} occurs at approximately 10 mmol/l blood lactate concentration, you will only be able to perform at VO2_{max} intensity for a few minutes at most. For longer races, your actual performance potential depends far more on the submaximal percentage of VO2_{ max} that you can sustain.

Think of VO2_{max} the as an indicator of an athlete’s aerobic *potential*; anaerobic threshold is an indicator of the athlete’s *utilizable aerobic capacity*. Anaerobic threshold is more directly affected by the volume and intensity of training than VO2_{max}. For example, a well trained athlete’s oxygen consumption at the anaerobic threshold is almost 90% of VO2_{max}; in untrained individuals the anaerobic threshold occurs closer to 60% of VO2_{max}. Race times correlate very well with blood lactate production. This reinforces the importance of aerobic utilization; you can have all the VO2_{max} you want, but you won’t race fast if you can’t use it.

REFERENCES

Burke, Edmund, *Precision Heart Rate Training*, Human Kinetics, Champaign, IL 1998.

Janssen, Peter, *Lactate Threshold Training*, Human Kinetics, Champaign, IL, 2001.

Nolte, Volker, *Rowing Faster*, Human Kinetics, Champaign, IL, 2005.

Sleamaker, Rob, and Browning, Ray, *SERIOUS Training for Endurance Athletes*, 2nd Edition, Human Kinetics, Champaign, IL, 1996.

P. Droghetti, C. Borsetto, I. Casoni, M. Cellini, M. Ferrari, A.R. Paolini, P.G. Ziglio, and F. Conconi, “Noninvasive determination of the anaerogic threshold in canoeing, cross-country skiing, cycling, roller and iceskating, rowing, and walking,” *European Journal of Applied Physiology* (1985) 53:299-303.

© 2018, Shawn Burke. All rights reserved.

v1.2

- If not, you might want to give it a quick read. ↑
- And, possibly, on the water. With a lot of qualifiers and requirements. ↑
- At the same time the brainstem is constricting the flow of blood to the internal organs, enabling more blood to flow to the muscles. ↑
- For those of you who rely on wrist-mounted optical / IR heart rate sensors, note that these can produce inconsistent data depending on how they are worn, and even where on the wrist or forearm they are mounted. Your experience may differ, but consider using a chest strap if you can. ↑
- If your resting heart rate is consistently lower in the evening, however, then by all means measure it then. ↑
- Sports watches that have an “auto lap” feature based on time are ideal since they will break each step into its own lap, and you can extract and average the heart rate for each of these segments separately. ↑
- His proto slide rules were nicknamed “Napier’s bones.” ↑
- For those of you keeping score at home, this property is true irrespective of the base of the logarithm. For the analysis here we’ll use the natural logarithms, with base
*e*. ↑ - I was stunned at how good the data came out. So I repeated the test a week later, and it was just as good as long as I plotted the logarithm of the average heart rate data. ↑

*by Shawn Burke, Ph.D.*

In 1971, famed Minnesota paddler and canoe racer Gene Jensen noticed that by bending a canoe paddle at the neck, where the shaft meets the blade, a paddler could move a hull faster than with an equivalent straight-shaft paddle. Jensen noted that paddlers using straight shafts, “had to dig these enormous holes, and their paddles would really cavitate. They didn’t seem to go as fast as they should.” The rest, as they say, is history. You can scarcely go to a downriver or flatwater race and not see carbon fiber bent shaft paddles everywhere. And because of their efficiency they’ve become rightfully popular with recreational paddlers as well.

So naturally, I had to ask “Why?” Why is it that bent shaft paddles are more efficient at moving a hull? Is there an optimum bend angle, and if so, what does it mean for it to be “optimal”?

To answer these questions I used a simplified physics model of how a paddle generates force to quantify “efficiency” and “optimal.” Fortunately, I also stumbled upon a nice article by Nicholas Caplan which provided valuable biomechanical data that I used as input to the model. It turns out that you maximize peak padding force and impulse (the time integral of force) if the blade is perpendicular to the water at the instant that the relative velocity between the paddle and the hull is maximum. And using Caplan’s data, based on this preliminary analysis, the optimum bend angle is 12 degrees. Subject to a few caveats.

So now that you know the bottom line feel free to move on, go train, etc. Or if you’re curious like me about the *why*, read on.

EIN BISSCHEN PHYSIK^{[1]}

While Isaac Newton wasn’t a paddler, perhaps he was inspired by the punters at Cambridge when he developed his laws of motion. His Second Law states that forces acting on a body will accelerate it in inverse proportion to its mass^{[2]}. For a hull, paddler, and paddle with a combined mass *M*, the Second Law may be stated as

where *d*/*dt* denotes the time rate of change of the hull’s velocity^{[3]} *v _{hull}*,

We’ve discussed the contributors to the drag force *D* in “Part 3: The Rough Stuff.” Ultimately the components of drag – friction, form, and wave – are properties of Nature and the hull design. What we’re left to consider is how to maximize the propulsive force *P* to create the greatest hull speed.

The propulsive force from the paddle may be expressed in terms of the *relative velocity* between the hull and the paddle – in the direction of travel – as

where * *(“rho”) is the density of water, *A* is the area of the blade, and *v _{paddle}* is the velocity of the paddle blade’s center in an inertial reference frame (e.g., relative to the Earth). The matter of reference frame is important since it is the relative velocity that contributes to the propulsive force. The hull could be moving swiftly downstream, and the paddle along with it (for instance, if you are not paddling); in that case you’re exerting no propulsive force but are just along for the ride, yet the velocity of the paddle by itself could be large.

The term *C _{D}* is the drag coefficient, which is a dimensionless quantity that characterizes the blade’s resistance to motion through water. It is a property of the blade design, and in the equation above is distinct from the blade area

*Figure 1: Coordinate definitions.*

The propulsive force *P* has a direction that is perpendicular to the blade’s power face; force is a vector. And as paddlers all know the angle of the power face changes over the course of the stroke. Since we’re interested in the propulsive force in the direction of travel, and assuming that the center of pressure on the blade’s power face during a stroke does not move up or down^{[5]}, the angular dependence of the drag coefficient can be approximated by

where (“theta”) is defined as the shaft angle shown in Figure 1, *C _{D}*

Nicholas Caplan employed high speed video cameras to measure the shaft angle over time, and the relative hull/paddle velocity, of an elite outrigger paddler using a straight shaft paddle^{[6]}. Data! Now it would be natural to just dive in and use his data without thinking about it. But where’s the fun in that? Plus this provides us with an object lesson in interpreting the results derived below, or more broadly, in reading any reporting on science or engineering: read the results in light of the underlying assumptions. By using Caplan’s data *we are assuming that the stroke mechanics employed by one outrigger paddler will be the same for all of the rest of us, using either straight shaft or bent shaft paddles*. That’s a big assumption, so please keep it in mind. But it gives us a place to start.

While I did not obtain a copy of Caplan’s raw data, I was able to approximate it rather well using simple mathematical functions. The relative velocity he measured can be expressed by

This approximation is valid from the time of the stroke’s catch at *t* = 0 seconds until the exit at *t* = 0.4 seconds, and is plotted in Figure 2 below. The relative velocity is negative because the magnitude of the paddle velocity is greater than the magnitude of the hull velocity during the power phase, as we discussed in “Part 5: What Moves You.” It also means that the paddle velocity is directed toward the stern. (Remember our friend momentum?) The relative velocity clearly varies over time, and its magnitude achieves its maximum value just a bit after mid-stroke (as defined by the time interval between the catch and exit). Hmm… That’s very interesting. Might be worth keeping that fact in our back pocket.

*Figure 2: Relative velocity from catch to exit (after Caplan).*

Caplan discovered that the shaft angle for his outrigger paddler varied linearly over time from a minimum of minus 30 degrees at the catch to a maximum of 50 degrees at exit. Consequently, the shaft angle he measured can be represented by

This angular dependence is plotted in Figure 3 below. Since it is linear, the shaft angle doesn’t show any inflection point with maximum magnitude, just minimum and maximum angles at the catch and exit.

*Figure 3: Shaft angle from catch to exit (after Caplan).*

Given all that, it’s time to compute some paddle forces!

PADDLE FORCE AND BEND ANGLE

As shown above, the propulsive force from the paddle can be expressed in terms of the time-dependent shaft angle and relative velocity. Most generally,

In this equation we’ve introduced an additional parameter (“phi”) which represents an offset to the shaft angle. In other words, phi represents the *bend angle* for a bent shaft paddle. Now the density of water, blade area, and baseline drag coefficients are all constants. So what do you do to maximize the peak paddle force? You look for variable terms in the equation, then determine how (or when) to maximize their product.

We know that the direction cosine and the relative velocity change over time, so it’s reasonable to conclude that we want them to reach their maximum magnitudes at the same time. Engineers call this “in phase.” You will recall from high school trigonometry that the cosine function varies in magnitude from 0 to 1. The cosine reaches its maximum when its argument is either zero or 180 degrees. Here, the zero degrees corresponds to “downward vertical,” and the 180 degrees corresponds to “upward vertical,” e.g. when doing a paddle salute.

In Figure 4 you’ll find a plot of the relative velocity’s magnitude vs time, as well as the direction cosine in the equation above based on Caplan’s data. In order to fit things nicely on the same plot, and for clarity of exposition, the direction cosines have been multiplied by 4 in this figure. The direction cosine is plotted for two values of the bend angle * *: 0 degrees (no bend), and 12 degrees. Since from Caplan we know that the shaft angle varies linearly over time, the figure shows that the straight shaft paddle reaches its maximum direction cosine value when the shaft (and hence the blade) are perpendicular to the water. Unfortunately, this occurs *before* the relative velocity reaches its maximum value; engineers would conclude that the direction cosine “leads” the relative velocity for this paddle. By contrast, the paddle with a 12-degree bend maximizes its direction cosine at the moment that the relative velocity hits its maximum. The bent shaft’s blade angle is *in phase* with the relative velocity. *The blade is perpendicular to the water’s surface when the relative velocity is maximum*. This is a good thing.

*Figure 4: Relative velocity magnitude and direction cosine from catch to exit.*

In Figure 5 you’ll find a plot of the resultant propulsive forces for the straight shaft and bent shaft paddles, computed using the equation above. Since we’re interested in comparing data, and not in the force’s absolute magnitude, for simplicity I set the values of the water density, blade area, and nominal drag coefficient to one. The paddle force for both cases starts at zero since at the catch you haven’t started to pull. The bent shaft’s force profile – the variation in force over time – lags the straight paddle; the bend introduces a phase lag in the force^{[7]}. As you can see from the plot the bent shaft paddle hits a higher peak force than the straight shaft paddle, again due to this “phase alignment.” Further, it hits this peak a bit later than for the straight shaft case. This is because the relative velocity had maximum magnitude a bit later than the power phase’s midpoint, and the bent shaft’s blade is perpendicular to the water at that moment.

I had expected the peak force for the bent shaft paddle to be a bit higher in comparison to the straight. But again, we’re taking data from an outrigger paddler using a straight shaft, and extending it using mathematics to the bent shaft case, assuming all along that the mechanics are the same. Still, even in light of those assumptions, the underlying physics is true. Phase alignment is good.

*Figure 5: Normalized paddle force vs. Time.*

WHAT DOES IT ALL MEAN?

So your paddle blade should be perpendicular to the water at the same time that the relative velocity between the blade and the hull is greatest. Great. Why not just cut off the stroke at that point? Wouldn’t that make the hull move fastest?

Short answer: No^{[8]}. But great question! In order to arrive at this conclusion I took the time integral of these normalized paddling forces in order to compute what physicists and engineers call *impulse*. An integral is merely the area under the force curve, here from catch to exit. It captures the entirety of the power phase’s propulsive contribution, not just the peak force. Impulse produces a change in linear momentum, which as we learned in Part 5 determines cruising speed. Using Caplan’s data the bent shaft produced 2.4% more impulse than the straight shaft; in other words, you’ll go faster with the bent shaft. A bit less than I would expect, likely due to our underlying assumptions, but an increase nonetheless.

All of the above computations were done in MATLAB. I studied other bend angles than 0 and 12 degrees. But the optimum bend angle *for the given relative velocity vs. time, and shaft angle vs. time*, was 12 degrees. More fundamentally one would need to measure these quantities *for each paddler*, and then investigate how to optimize bend angle for them individually. But for a given paddle it is best if the blade is perpendicular to the water when the relative velocity is maximized. Phase alignment is good.

And for this first pass, we’ve seen that Gene Jensen was right. Bent shaft paddles are more efficient at generating greater propulsive force.

V. 1.02

Copyright (c) 2018, Shawn Burke. All rights reserved.

- Well if Mozart got “Eine Kleine Nachtmusik,” why can’t physics? ↑
- Actually, the applied force changes the object’s momentum, but if the mass is constant then the equation simplifies to this familiar form. ↑
- This time rate of change of velocity is the acceleration, and d/dt is a derivative. Just dropping a little calculus. ↑
- Since the blade is rigidly affixed to the shaft, if you know the shaft angle you know the blade angle. ↑
- Pressure is a scalar (and isotropic), with units of force over area. The pressure over the blade is integrated over its area to yield the magnitude of the propulsive force. The center of pressure is determined via a centroid calculation of the pressure distribution over the blade. Once the center of pressure is known the propulsive force vector is assumed to be directed from that location, perpendicular to the blade’s power face. Fun! ↑
- Nicholas Caplan, “The Influence of Paddle Orientation on Boat Velocity in Canoeing,”
*International Journal of Sports Science and Engineering*, Vol. 3, No. 3, pp. 131-139 (2009). ↑ - Star Wars reference, anyone? ↑
- With a caveat: Ultimately you have to take into account stroke cadence, and the relative amount of time you spend in the power phase vs. the recovery phase. Which will be the topic of a later Science of Paddling article. ↑

I was recently asked to develop C4 Youth paddler time adjustments for the NECKRA Flatwater Points Series. This led me to dig into how the Points Series is computed since it was put together before C4s started showing up at races. What I came to understand are the limitations of using additive time bonuses for “leveling the field.” They work, and pretty well at that. But after a while physics and the Series’ mathematical model diverge.

In this installment of The Science of Paddling we’ll explore how the NECKRA Points Series’ model reflects the underlying physics of paddling canoes and kayaks. There’s a fair amount of math involved, but it’s mostly algebra. I hope that, among other things, the analysis provides a view into how engineers use approximations to model complex physical systems, and gain insight.

THE POINT SERIES

As you may know, the NECKRA time bonus / penalty system takes into account paddler gender, age, and hull type so that racers may compare their results to their fellow competitors. As noted on the NECKRA website www.neckra.org:

“The NECKRA Points Series provides:

- A fun way to track your racing
- Encouragement to participate in more races
- Competition on a more leveled playing ground

Our handicapping system has proven to be a good leveler of gender, age and boat type”

The NECKRA handicapping system adjusts finishing times based on paddler age, boat type (for example, C2 vs. C1, pro boat vs. 16% Recreational tandem, etc.), and sex. For flatwater canoes, the system is based on the performance of a hypothetical 3×27 Pro Boat paddled by two men between the ages of 19 and 59.

Let’s consider a simple example of how the Point Series works. I’ll use a flatwater canoe example; the same process applies to downriver and kayak paddlers. If a C2 Pro Boat paddled by a team consisting of (1) a 25-year-old man, and (2) a 30-year-old woman finishes a race in exactly 1 hour, in the Point Series their adjusted time would equal their finishing time (1 hour) minus 2 minutes (the female paddler time bonus):

If instead this C2, again finishing in 1 hour, was paddled by a 71-year-old man and a 25-year-old woman, for the Point Series their adjusted finishing time would equal their finishing time minus 2 minutes (the female paddler time bonus), minus another 2 minutes (the 70-74 year-old paddler time bonus):

Naturally, if the finishing time was not exactly one hour a fractional time adjustment is made. A 90-minute finishing time would entail multiplying the time adjustments by 1.5 (since 90 minutes is 1.5 times the one hour basis for the time bonuses), etc.

As you can see from these examples, *time adjustments are applied additively ^{[1]}*. For each time adjustment, be it for hull type, sex, or age, each bonus category entails an additive adjustment, and

Okay, got it? The Points Series is based on adjusted times computed via the sum of prescribed time bonuses in order to “level the field” for all participating paddlers.

So now someone like me comes along and asks, “Is this correct?” More specifically, does adjusting finishing times via the simple addition of a series of time bonuses reflect the underlying physics of how a hull is propelled through the water?^{[2]} Short answer: It’s pretty good; better than I thought it should be. But it has limits. And that’s where we get into the fun part. So first, let’s explore the assumptions and limitations of the Points Series by once again reviewing the physics of paddling.

PHYSICS AND THE SINGLE BONUS

From “The Science of Paddling, Part 1: Tandem vs. Solo,” we learned that a tandem canoe of length *L* having two identical male paddlers that each supply an average paddling power *P _{men}* will produce an average, steady-state racing speed

The factor *C*_{2}’ smooshes together all of the hull’s hydrodynamics into one term; a major simplification, but you have to start somewhere. As we learned in “The Science of Paddling, Part 8: Leveling the Field” one can scale the power term *P* with a multiplicative factor to represent a paddler having a smaller average paddling power. We’ll then write

I’ve chosen to express the power scaling factor as (1 – 2), where 0 ≤ ≤ 0.5, rather than as just a simple multiplier. This looks like an odd way to introduce a power scaling factor, but down the road writing things this way will simplify the mathematics quite a bit^{[3]}. The term in parenthesis will get smaller as grows, and when * *= 0 you recover the tandem men’s result. Just remember: when * *= 0 you recover the men’s power *P _{men}*, and when grows it represents a paddler with less power.

Now in high school algebra^{[4]} we learned that distance *D* equals velocity *V* times time *T*. Since all hulls in a given race will be racing over the same course, the distance traveled by these boats will all be the same; the only thing that differs will be the finishing times. Consequently,

where *T*_{2,}* _{men}* and

Combining terms, and simplifying a bit, this expression for the “non-adjusted” tandem finishing time *T*_{2} reduces to

As we see, if = 0 (tandem men’s case), *T*_{2} = *T*_{2,}* _{men}*; check

Sure it does! Subtracting a time bonus adjustment from the finishing time *T*_{2} is the same as multiplying that finishing time by some number. That is,

where is the NECKRA time adjustment, which has units of minutes *per hour*. And in this hypothetical case, *T _{2,adjusted}* should exactly equal the men’s finishing time

This takes the same general form – a multiplicative term – as we derived from the physics model, e.g., the non-adjusted tandem finishing time *T*_{2} is equal to the men’s tandem finishing time *T*_{2,}* _{men}* multiplied by a scaling term:

So for this case, with *one and only one time adjusting factor*, the NECKRA Points Series does indeed reflect the underlying physics of how a hull is propelled through the water. A judicious choice of the time bonus will represent the impact of the factor ; the physics are just smooshed together in the inverse cube root term. You may ask yourself, well, what about this cube root? The point is, in this case (repeat: in this case), we don’t care. The inverse cube root term is *fixed* for a given , hence the fraction is fixed as well. In both equations the non-adjusted finishing time is proportional to the men’s finishing time multiplied by a number. Works for me.

ANOTHER APPROACH

There is another way to approach this, and in doing so we’ll be prepared to take on the subsequent analysis where *multiple* time adjustment factors are introduced. Think of it as a math HIT session. We begin by revisiting the expression for the non-adjusted finishing time expressed in terms of the factor . We’ll employ a bit of math called a binomial series expansion to represent the cube root term. Don’t worry; it won’t hurt a bit. Our ultimate goal is to determine whether or not additive corrections for paddlers or paddling teams having more than one Points Series time bonus can be derived from the underlying physics.

The binomial series expansion allows us to represent the cube root as a sum of more familiar algebraic terms. In fact, it’s an infinite sum. But a further assumption will allow us to simplify the binomial expansion: we assume that the factor is much smaller than one^{[6]}. Mathematically, we’ll write this requirement as << 1. So, regarding that cube root, its binomial expansion is

The ellipsis indicates that the expansion has many more terms involving higher powers of . Recall that we assumed is much smaller than one. When you take a number that’s much smaller than one and square it, the result is much, much smaller than one. (If you cube it the result is even smaller than that; ditto for higher powers.) As a result, we can ignore the squared and other higher order terms in the expansion above and write

This means we can approximate our equation for the non-adjusted time *T*_{2} as

That is, the non-adjusted time *T*_{2} is greater than the men’s tandem time *by an additive term*. To make the adjusted time equal the men’s tandem time, just subtract this term from the non-adjusted finishing time. Which sounds just like the NECKRA Points Series formula. Check.

As you see, for this approximation the time series “bonus” is linearly expressed in terms of the factor . But recall that our factor was chosen such that when it has smaller values the scaled power is closer to that of the hypothetical male paddler, so when = 0 (no power scaling) *T*_{2} reduces to *T*_{2,}* _{men}* in the equation above. Check.

WHAT ABOUT MULTIPE TIME BONUSES?

This brief side trip allows us to now consider the case when multiple time adjustments are applied. Assume we have two paddlers, each of whom has an associated factor _{1} and _{2}, in our “non-adjusted hull” vis

We can express the non-adjusted time *T*_{2} then as

Combining terms, and doing a little simplifying,

This looks just like our single time adjustment result, except now the argument in the denominator’s cube root includes two factors _{1} and _{2}. In order to expand this in a binomial series we now require that the *sum* of the factors _{1} + _{2} be <<1. Making this assumption, we then derive a familiar form for the non-adjusted time:

It is familiar in that the non-adjusted time *T*_{2} is greater than the men’s tandem time by *two* *additive terms*. To make the adjusted time equal the men’s tandem time, just subtract *both* of these terms from the non-adjusted finishing time. Which sounds just like the NECKRA Points Series formula. Check… at least for the moment.

In order to cover our bases we’ll consider one final example: the case of a tandem where one of the paddlers has two power adjustments, say a first for gender and a second for age. For these two paddlers in our non-adjusted hull, again using two factors _{1} and _{2},

Since we assumed that the sum of the factors _{1} and _{2} was much smaller than one, and each factor has a positive value, then each factor alone is much smaller than one. The product term _{1}_{2} in this equation will thus be much, much smaller than one, and hence can be neglected. The above equation, using distance equals velocity multiplied by time, and the requirement that all hulls travel the same length course, then yields

This is the exact same result as before where each paddler had their own power scaling. And, and consistent with previous assumptions, the non-adjusted time can be approximated using a binomial series expansion:

By assuming that the factors _{1} and _{2} are small the product term _{1}_{2} vanished, effectively “decoupling” the two factors and yielding a simple *additive form* for the time adjustments. One can continue to introduce multiple factors _{i} to both paddlers, and the same type of result will ensue *as long as your analysis is consistent these mathematical assumptions*. Indeed, from the analysis above one can show that, for a tandem where each paddler has two time bonus adjustments^{[7]},

LIMITS OF THE MODEL

Now if you’ve been following the assumptions inherent in this analysis, you’ll start to see where the NECKRA Points Series assumption of independent time bonus corrections might break down. The factors are assumed to be “small.” So, what is “small”? Let’s look at a concrete example. From “Leveling the Field” we learned that a female paddler on average produces about 82% of the power of an average male paddler, per the NECKRA Points Series time bonuses. This means that a 2-minute time bonus implies

using the expression above for the power multiplier (1 – 2). Is 0.09 “small” compared to 1? OK, I’ll accept that. Besides, we’re using an approximation. What about two paddlers, each having a 2-minute time bonus? Since (per the analysis above) a 2-minute time bonus means = 0.09, then a 4-minute time bonus – consistent with the same analysis – is twice this, or 0.18. Is 0.18 “small” compared to 1? Well, not really… unless you squint really hard. Even if you decide to incorporate higher-order terms (squares, cubes, and so on) in the binomial series expansion you’ll have to start retaining the cross-terms involving _{1}_{2}, etc., which *couple* the various power scaling terms. That more accurate approach would reveal that these are not independent of each other.^{[8]}

At its heart, both the simplified analysis based on the binomial series expansion, and NECKRA’s additive time bonuses, assume that each of the various time bonuses are independent of each other. But consider a model of a tandem hull’s average velocity in terms of paddlers who have multiple scaling power factors *f _{i}*, which for the sake of illustration are expressed in a slight simpler form than before:

Here, we’ve substituted *f _{i}* for each (1 – 2

Without the simplifying assumptions we’ve made in the binomial series expansion analysis, is there any hope of expressing the fraction in the equation above as a sum of terms, each of which independently depends on one and only one scaling power factor *f _{i}* which may or may not be “small”? Short answer: No. Instead, if

So what to do? Remember that for hulls (and racers!) having a single time adjustment we showed that using an additive bonus time adjustment is spot on with our physics model; each bonus should work well as long as it was well chosen. A second adjustment for a hull? Yeah, you can probably live with that in light of our analysis. But as you start piling on multiple time adjustments, or even a large time adjustment for age along with some other time bonus, realize that the underlying assumptions of an additive model start to break down. But like Winston Churchill once noted, “Democracy is the worst form of government, except for all the others.” The NECKRA Points Series works well enough. Be happy. Don’t worry. And just paddle!

© 2018, Shawn Burke, all rights reserved.

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- And yes, subtraction is “additive”: you’re just adding a negative number. ↑
- Equally as important, why would anyone ask this? Short answer: I’m an engineer! ↑
- Isn’t this the kind of thing that drove you nuts in college? Yeah; me too. ↑
- We’re actually using speed here, rather than velocity; velocity is a vector quantity, which means distance is as well. This way I can avoid dot products and vector notation. Apologies for the conflation. ↑
- We’re using the positive real result of the cube root throughout. Just being rigorous. ↑
- We’ll get in to what “much smaller” means later on. ↑
- It’s easy to derive – go ahead! ↑
- If you did this you’d also have to consider the convergence properties of the binomial expansion, which means lots of obscure math if you want to be rigorous. ↑
- And for extra credit, is the physics model itself accurate? Is multiplying a baseline power
*P*by a power scaling factor*f*an accurate representation of our paddling biomechanics and physiology? As Morpheus asked in The Matrix, do you choose the red pill, or the blue pill? ↑

Some of us augment our race season prep with strength training; many of us do not. I’ve felt that strength training has helped me with starts and jumps. But that assessment is purely qualitative; my stroke feels “stronger” if I’ve been lifting conscientiously, but what does “stronger” mean? I know lots of excellent paddlers who routinely crush me out on the course once we’re past the start. How much has strength training benefited me downstream? Is the time I spend on strength training providing enough benefit to keep doing it? Equally as important, am I doing it “right”? And for paddlers, what is “right”?

In this installment of The Science of Paddling I’ll make the case for an approach to strength training that aligns with the physics of paddling. This is a work in progress. I don’t know yet whether or not it is “right.” But the underlying philosophy is grounded in both physics and exercise physiology, and (so far) is borne out by my own experiments. In order to make the case we’ll first review more conventional ideas of the functional benefits of strength training. Then, we’ll add a pinch of physics and see what happens.

ON “CLASSIC” STRENGTH TRAINING

My introduction to strength training came in college. I knew lots of jocks, mostly football players. I would diligently follow them to the gym and muck about with free weights and machines. The gospel was “three sets of ten reps^{[1]}” for each exercise. I certainly gained muscle doing this, and got my scrawny upper body more in balance with my lower body. And that was about it.

When I decided to get serious with off-season training about a decade ago I researched strength training for paddlers. Mostly what I found were prescriptions for specific exercises. Then, in Tudor Bompa’s *Periodization*, as well as his *Periodization Training for Sport*, I found detail; lots of detail. Bompa laid out the benefits of lifting weights at specific fractions of one’s single repetition maximum^{[2]} (“1RM”) for particular exercises in light of what function(s) one wished to train. An example of this breakdown appears in Fig. 1. Note that we’re not yet talking about any particular exercise; that will come later. For the moment let’s consider any exercise, generically.

Fig. 1: Functional benefits of weight training vs. percentage of 1RM (after Bompa).

Through this figure Bompa conveyed his (and others’) notion of the functional benefits from performing an exercise at some fraction of that exercise’s 1RM. Lifting a load from 80-100% of 1RM benefited one’s maximum strength for that exercise. From 30% to 80% of 1RM benefited “Power,” which he linked to various forms of muscle endurance: acyclic movements (single or infrequent, like jumping or lunging during a game of basketball) and cyclic movements (frequent, regular movements… like paddling). Further, from Bompa I first learned about the relationship between % of 1RM and how many repetitions one could do in a single set of a particular exercise before one couldn’t do another, aka “reps to failure.” This is illustrated in Fig. 2 below.

*Fig. 2: Repetitions to failure for loads that are a percentage of 1RM.*

In this Figure we see that you can lift 100% of your single repetition maximum, well, once. You can lift 90% of your 1RM four times; 72% of your 1RM ten times; and so forth. And when I say “four times,” that means you can’t perform that same exercise a fifth time within the same set. Ditto for ten repetitions at 72%; if you can do an eleventh repetition you’re lifting less than 72% of your 1RM.

I found that the relationship between the number of repetitions I did in a single set to failure vs. weight lifted tracked this curve remarkably well down to about 50% of 1RM. Below that, not so much. I did lifts at 80-90% to increase maximum strength; 70-80% for “Power,” and so forth. I got stronger, and a little bigger. But aside from feeling “stronger” it was unclear how this work benefited my performance on the water.

After doing this for 4 years whatever nebulous benefits I reaped from horsing around all that weight plateaued. My muscles didn’t feel as tired late in races as they once did (though the rest of me did!) I was looking for something different. My friend and fellow paddler Tom Walton pulled me aside after our first General Clinton Canoe Regatta and said, “I’ve got the workout for you.” This consisted of four (mostly) paddling-specific exercises, done using body weight only, performed as a circuit^{[3]}, running through the circuit 3-5 times. The exercises? Pullups, ab wheel rollouts, inverted rows^{[4]}, and pushups. Doing this workout 3-5 days per week helped me, and the workout only took 20-25 minutes to boot. I think a good deal of the benefit came from how these exercises emphasized core engagement while doing functional exercises.^{[5]}

I thought that was it. And yet something was still niggling at me. I’m an engineer; I’m driven to ask “Why?” I understood that the exercises were paddling specific. But aside from feeling a further benefit on the water, could I get even more bang for the buck if I did these exercises… differently?

TIME FOR SOME SCIENCE

For those of you who suffered through this long preamble, let’s get down with our nerdy selves. I’ve embarked on an experiment this off-season inspired by two things. First, in putting together the Science of Paddling website I re-read all of my past articles.^{[6]} In the first installment we learned that the steady-state cruising speed *V* of a canoe or kayak is proportional to the cube root of the average power *P* provided by the paddlers:

Note that this equation does ** not** show that speed is proportional to strength, which would be represented by just the

Power *P* is the rate of doing work, or the amount of energy transferred per unit time. Mathematically, power is equal to the product of the applied force *F* times the velocity *v* at which this force is applied^{[7]}:

Since velocity *v* is distance *d* divided by time *t*, we see that power can also be expressed as

Now force times the distance over which the force is applied is defined as work. This reinforces the definition of power as work per unit time.^{[8]}

In plain terms “large” power is the application of a force quickly, and in the context of paddling doing this over and over (and over and over…). So which do you train: for more force, or for more velocity? What if you could actually train power directly rather than deconstructing it into its component parts?

This is where the “second thing” comes in. I was reading one of my favorite tech blogs (*Ars Technica* – highly recommended) when I saw a review for a new exercise gizmo that measures the velocity^{[9]} of your movements during exercise. And, since for a particular exercise you know the weight you’re moving, you’ll know the power exerted during that exercise. This gizmo is the “Beast” sensor, made by Beast Technologies S.r.l. in Italy^{[10]}. The product comprises a small Bluetooth 4.1-connected sensor that can be placed atop a weight stack, inserted in a wrist band (for bar-based lifts), or worn on a body mount (for pushups, pullups, inverted rows and the like) to capture movement and thus velocity. The photo below shows the sensor atop the stack of a multi-station cable weight machine.

*Fig. 3: Beast sensor atop weight stack.*

The Beast sensor sends average velocity data to a connected smartphone app. This app lets you select an exercise, enter the weight you’ll be lifting, and then presents a “Start” button you press when you’re ready to perform the exercise. For each and every lift you can have it report either the velocity of your lift (in m/s) or the power of your lift (in Watts). And you can watch it on the screen during each rep along with a bar graph of each rep’s velocity. Here’s an example:

*Fig. 4: Beast sensor app display.*

OK; great. There’s a gizmo. Engineers like gizmos. Now what?

VELOCITY BASED TRAINING

In researching the Beast sensor I learned about a branch of strength training called “Velocity Based Training” or “VBT.” VBT’s thesis is that there are certain training ranges based on moving weights at velocities that correspond to specific functional benefits. This sounds a bit like Bompa’s training ranges based on percentages of single repetition maximum lifts, and in a way it *is* similar. But VBT differs in a key way.

In “classic” strength training one works out, determines their 1RM, and lifts at some fraction of 1RM until it improves, or they get bigger, or achieve some other goal. And so on. But – and here’s the big caveat – on any given day an athlete doing strength training can have a different 1RM. Hard training over previous days, irregular sleep patterns, stress and the like can actually lower 1RM; 1RM is not an inviolate quantity. This means that on a given day prescribing 3 sets of 10 reps at 70% max may actually result in the athlete lifting 80% of their 1RM, and the desired training effect will not be achieved. Do this enough and injury or overtraining can follow. Or prescribing lifts at 70% of max may actually correspond to 60% of 1RM on a given day, and again the desired training effect would not be realized.

Instead, if one trains by moving a weight at a specific target *velocity*, and adjusts the weight so that a maximal effort results in that desired velocity for a series of reps, then you will always be training “correctly” for a desired functional result even when your 1RM changes (up or down). There are now generally-accepted velocity ranges that target specific functional benefits. These are shown in the Fig. 5 below.

*Fig. 5: Velocity ranges and associated functional benefits.*

In this Figure^{[11]} slower velocity ranges are to the right, while faster velocity ranges are to the left. There are two particularly interesting velocity ranges: Speed-Strength, and Strength-Speed. Mann^{[12]} defines Strength-Speed as “moving a moderate weight as fast as possible, and has a higher rate of force development compared to accelerative strength.” Mann further defines Speed-Strength as “moving a lighter weight as fast as possible,” and notes that “[t]his trait has the second highest rate of force development of all the traits.” Paddlers know that we’re exerting a force quickly through the water, so it would seem that doing strength training somewhere in this velocity range would be relevant for paddlers based on these descriptions. I set out to do just that.

Using the Beast sensor I performed the following exercises: Lat pulldown, inverted row, pushup, and overhead cable press. My goal was an average velocity of 0.75 m/s over a set, which corresponds to the lower velocity end of the Strength-Speed range above. I chose this as a starting point since it led me to select weights that provided enough resistance to make the lifts “interesting.” Faster velocities like 1.0 m/s just felt too easy. For the overhead cable press and the lat pulldown I experimented so as to determine what weight I could move at 0.75 m/s velocity. And by “could move” I mean pop the exercise at the highest average velocity I could attain, using good form, so that for each lift I achieved as close to 0.75 m/s as I could and felt that I could not have moved the weight any faster. No cheating! The good news is that the app displays the average velocity you achieve for each and every rep, as shown in Fig. 4. This instant feedback is both fascinating and encouraging; you know exactly where you’re at.

As to how many repetitions per set, and how many sets, well, that involves a bit of art that is informed by science. Others may disagree with my reasoning, and that’s fine. Those of you who dug into the details of the Science of Paddling post “What Fuels You” will recall that the phosphate system, which ultimately fuels the muscles, has a store of “free” energy in the form of Adenosine Triphosphate (ATP) stored in the muscles that can be expended over the first 10-12 seconds of exercise, say in a sprint, or… lifting weights! After that the other metabolic systems take over. It takes about 2-3 minutes of rest for this store of “free” ATP energy to be replenished in the working muscles (not all muscles; just the working muscles are depleted). I reasoned, then, that I should perform a specific exercise for 10-12 seconds^{[13]} at the maximum speed I could for a given weight, then move on to the next exercise (and different working muscles group) so that the ATP store would (1) be mostly depleted over the set, and (2) subsequently replenished before starting the next set. I rested between circuits to ensure my working muscles were ready for the next one. This 10-12 second bound for a set almost always resulted in 8 reps for the 0.75 m/s target velocity, and I would usually see the velocity of each rep start to slowly decline over a set, as shown in Fig. 6. I often found that my “best” rep was the second or third rep. Not sure why.

*Fig. 6: Degradation of velocity over a set.*

I interpret this decline in velocity as a sign of depletion of the ATP stores, as well as other neuromuscular and biochemical factors coming in to play that limit the muscle group’s ability to continue doing work at that rate. It’s also great feedback – if you’re not hitting your target average velocity over a set you may have to adjust the weight up or down to get the specific training benefit you’re looking for. After the first few trial sessions I established 8 reps/set as my goal at 0.75 m/s. Again, *these are not sets done to failure* like in “classic” strength training. They are done within a defined time range for reasons noted above, with an eye to monitoring degradation of average rep velocity over the course of the set. VBT is about quality, not quantity. I’m currently doing 3 sets/session, with a goal of increasing that gradually to 5 sets/session *as long as I can hit the target velocity in each set, with no loss in form*.

As to the body weight exercises – pushups and inverted rows – I’ve been able to do sets at the target average velocity of 0.75 m/s for inverted rows, and about 0.6 – 0.65 m/s for pushups. For those of you who have done plyometric pushups, e.g. pushups where you clap your hands at the top of every repetition, this velocity range feels a lot like those during the concentric muscle contraction phase. As I progress with inverted rows I’ll add a weighted vest to increase resistance; ditto with pushups when I get there. Again, the goal is to achieve a target average velocity, not a particular weight. The weight you move will systematically increase over time. If you’re systematically exceeding your target average velocity, it’s time to increase the resistance.

Over a month of VBT sessions I saw modest but consistent, measurable gains in weight moved at 0.75 m/s. That in itself was encouraging. So naturally, as an engineer, I wondered if there is a way to determine experimentally whether or not 0.75 m/s is the “right” average velocity to *maximize *power. Remember, the cruising speed of a hull is proportional to the cube root of power. I could almost hear Captain Kirk yelling, “Scotty, I need more power!”

Recall that power is the product of weight (force) times velocity. And I noticed that for light weights I could easily exceed my 0.75 m/s velocity target, while for heavier weights I slowed down to as little as 0.28 m/s, e.g. the “Absolute Strength” velocity range in Fig. 5. While it’s nice to work at a target velocity prescribed in a book or technical paper, I became curious as to whether there was a relationship between velocity and power, or weight lifted and power, and if this relationship would indicate an “optimum” velocity that maximized power. In other words, if I varied *both* velocity and force, what combination yields the most? Fortunately, Mann’s book provided a clue. In it, he showed what is called a “power curve.” He had constructed exemplary power curves from his own bench press workout data. It took several reads through his book to untangle the implications of the power curve, but eventually, I got it.

You see, if you can move a light weight very fast, you might be generating less power than moving a somewhat heavier weight a little less fast. And certainly, lifting a heavy weight slowly might generate even less power than moving a light weight fast. In order to ground this speculation in fact, I constructed a power curve of me performing lat pulldowns. First, I used a digital weight scale to calibrate my cable lat machine to make sure I knew the correspondence between the number of plates and the actual weight moved at the hand grips. Then, I performed lat pulldowns in sets of 8 reps (see above), progressively adding 10 lb between sets, starting at 120 lb and going up to 210 lb (the limit of my machine). At each weight and for each rep I performed the exercise as fast as I could, while maintaining good form. I did no other exercises during this session. Further, between each set I rested for 3 minutes (see above). For the highest weights I limited myself to 4-6 repetitions since I could see my velocity start to drop off with each rep; one VBT rule of thumb is that you should stop an exercise if your reps fall more than 10% below your best rep’s velocity during a set. While tiring, this workout was fun because… data!

After I was done I logged on to the Beast website and recorded the data from my workout; the app automatically uploads your data when you’re finished. I then plotted average power for each set, e.g. at each weight, vs. average velocity. This appears in Fig. 7 below.

*Fig. 7: Power curve for lat pulldowns – power (Watts) and **weight lifted (pounds) vs. average velocity (m/s).*

The abscissa (sometimes referred to colloquially as the “x-axis” or “horizontal axis”) in this plot is velocity in m/s. Consequently, my first sets, which results in the highest average velocity with the lowest weight, are on the right side of the plot. The slower average velocity data points on the left correspond to the later, higher-weight sets. If you like, I was working from right to left in the plot. It was fascinating and gratifying to see that the plot has a maximum at around 0.84 m/s, which was achieved using 140 lb of weight plates. This is my average velocity for lat pulldowns that generates the maximum average *power*. Yay! It’s not necessarily the velocity that generates maximum power for anyone else. And it’s not necessarily the velocity that generates maximum power for me on any other exercise. To determine that I’ll need to generate a power curve for each exercise. But by varying *both* velocity and weight I was able to find my velocity sweet spot to maximize power for this exercise. And the literature suggests that while the weight I can move at this velocity should increase over the coming months with training, the velocity at which I can generate maximum power should stay the same. That’s the underlying benefit of VBT. You know what to target for each workout, and you know where you’re at with each rep.

It was reassuring to note that my max power velocity of 0.84 m/s for lat pulldowns lies within the Strength-Speed range illustrated in Fig. 5. And, if I may be allowed to extrapolate, I’ll contend that exercises like lat pulldowns, overhead cable presses, inverted rows, and cable rows are paddling-specific exercises. If I train to maximize my ability to generate power with these exercises using a velocity sensor, rather than to maximize hypertrophy (building muscle mass) or maximum strength (e.g., force generation) using conventional training prescriptions, I should – emphasis on *should* – see some benefit on the water when Spring ice-out arrives.

The question now is should I train only at 0.84 m/s, sometimes at 0.75 m/s or 1 m/s, or some other velocity? I don’t know. I’m an experiment; I’ll find out. It’s science!

Now should you follow this path, and try VBT? First off, for liability reasons I have to advise you to consult your doctor before embarking on any exercise plan. At a minimum you should be healthy and not have any muscle, joint, or mobility issues. Everybody runs a high risk of injury or even death doing strength training, including VBT. So just don’t do it. OK?

But for me, I’ve made a few more observations about how my body is responding to VBT that give me some guidance about how to proceed. As they say, “your mileage may vary.” I do three VBT sessions a week. I would absolutely *not* try VBT unless I had been doing strength training to some significant degree for at least a year or more. And by strength training I mean doing the types of paddling-specific exercises described above, or whatever paddling-specific exercises one might intend to perform. The reason for caution? I see that it would be really easy to injure my shoulders and elbows popping weights so fast. I also worry about developing tendinitis in my elbows, something that I’ve suffered from in the past. I have a base in each exercise from the past year of working out; I think that’s helpful. I’m very, very, VERY careful to pre-load each exercise. In other words, I engage the working muscle chain with the weight (or the grip for cable machines) prior to moving it. This prevents me from putting a jerk load (jerk is the time rate of acceleration) on my joints at the start of a rep. Jerking a load can be destructive to joints and connective tissue, both of which take a long, long time to heal. I’m careful to use excellent form, and strongly engage my core muscles before the start of each set and keep the core engaged throughout. And I will terminate a set if any part of my body experiences even the slightest twinge.

Like anything new I’m cautiously optimistic about VBT. It aligns with the physics of paddling. The data I see is indicative of underlying elements of exercise physiology. Both of these features address the “why am I doing this?” question for me, and I love that. More importantly, I’m having fun. And I can’t wait for Spring!

ADDENDUM – Determining Resistance for Pushups and Inverted Rows

Using a velocity sensor like Beast requires that you enter a weight corresponding to each exercise you do in order to accurately compute power. The little sensor does the velocity part. Now it may be tempting to just use your body weight as input for pushups. But this would be incorrect. First, when you’re standing, you have a single point of contact (your feet) between you and the ground. (OK, you have two feet; let’s just consider them together.) That’s why a weight scale works: there is no other path for weight (which is a force) to “pass through” to the ground. But when you’re doing a pushup you make contact with the ground not only through your feet, but through your hands as well. This means there is a force being transmitted through two points of contact. (I know; it’s really four. But work with me here.) And since the total force being applied to the ground must equal your body weight, the force your moving when doing a pushup must be less than your total body weight. So how do you determine what this weight is? Well, an engineer might be tempted to construct a free body diagram and impose a static equilibrium requirement. But the easier way is to take out that weight scale, get on the floor like you’re going to do a pushup, and put both of your hands atop the scale. Take the reading. That’s the weight you’re moving with each pushup.

As to inverted rows, again you have two points of contact: through your heels to the ground, and through your hands to a bar above you (which is ultimately connected to the ground as well, unless you have a sky hook). And since the inverted row is in most ways just an upside-down pushup, the weight you’re moving with each repetition is essentially the same as what you move doing a pushup. I could prove it with a free body diagram, but just take my word for it.

- For the uninitiated, “rep” is short for “repetition,” a single instance of an exercise movement such as moving a weight. A “set” is a number of consecutive repetitions of a single exercise. ↑
- Single Repetition Maximum is the maximum amount of weight one can lift once and only once for a given exercise. ↑
- A circuit comprises performing one set of an exercise, immediately followed by a set of a different exercise, and so on until all exercises have been performed. ↑
- An inverted row is sometimes called a reverse pushup. It’s like a bench row, but with no bench, and upside down. Heels on the floor, body in a plank, back on or just above the floor, arms extended upwards to a bar (or TRX). Hold the plank and pull, pivoting at the heels. Lower yourself down. Repeat. ↑
- “Traditional” strength training often comprises exercises that isolate muscle groups. Contrast a bench press with a pushup. The former, while a great overall power lifting move, primarily works the triceps, anterior deltoids, and pectoral muscles with secondary activation of surrounding muscles in the upper body. A pushup does that as well but actively engages the core, and much of the biomechanical chain from heels to shoulders. ↑
- And I didn’t fall asleep once! ↑
- While power is a scalar quantity, meaning it doesn’t have a direction, both force and velocity are vectors. This means that force and velocity are directional. For the sake of simplicity we’ll assume that force and velocity are directionally aligned and thus may be thought of as scalar variables. Or if you prefer to be more mathematical, note that the expression for power is a dot product, not a scalar multiplication, and the dot product of two vectors yields a scalar. Your choice. ↑
- And… work has units of energy, like Joules, or Calories. Very convenient. ↑
- Actually, it measures the components of velocity using a MEMS accelerometer and reports back speed in the direction of travel. You know; velocity is a vector, speed is a scalar. But in this article we’ll occasionally conflate the two. Sorry! ↑
- Full disclosure: I bought a Beast sensor with my own funds, and Beast played no role whatsoever in the writing or preparation of this article. ↑
- Note that Fig. 5 also includes indications of % 1RM based on research by Gonzalez-Badillo and Marques that links velocity ranges with training intensity. Just FYI. ↑
- Mann, Bryan,
*Developing Explosive Athletes: Use of Velocity Based Training in Training Athletes*, 3^{rd}Edition, Ultimate Athlete Concepts, Michigan USA (2016). ↑ - Yes, I used a timer. Yes, I’m an engineer. ↑

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Copyright (c) 2017, Shawn Burke. All rights reserved.

]]>One of the joys of canoeing is seeing so many different people and boats at a race. Young and old; men and women; sleek hulls and… less sleek hulls. We each have our own paddling preferences, and many of us when given the chance will race C1 at one event; C2 Mixed (one man, one woman) at another, and so forth. The sport is very social, and it’s always fun to mix things up. But we are, after all, racing, and that competitive spirit bubbles up when we look at race results to evaluate our performance. We evaluate our results not only to see how we did compared to our own past performances, but how we did compared to our fellow competitors.

But how does one compare their race result vs. competitors paddling different hull types, say C2 vs. C1? Or paddlers of different ages? Or paddlers of different genders? It turns out that you can apply time adjustments or “handicaps” based on these differences, and evaluate performances across all paddlers competing in different classes. In this installment of The Science of Paddling we’ll consider how one racing organization does this, and uses its approach to rank *all *paddlers for annual awards. Then we’ll use physics – naturally – to fill in one gap in their handicapping method when a statistical approach cannot be employed, to level the field.

The New England Canoe and Kayak Racing Association (“NECKRA”) has a Downriver and a Flatwater “Points Series” where paddlers’ race results are tallied over a season. At the end of the season Series winners receive awards. As noted on the NECKRA website www.neckra.org:

**A fun way to track your racing****Encouragement to participate in more races****Competition on a more leveled playing ground**

Our handicapping system has proven to be a good leveler of gender, age and boat type”

Their handicapping system adjusts finishing times based on paddler age, boat type (for example, C2 vs. C1, pro boat vs. 16% Recreational tandem, etc.), and gender.

Time adjustments are applied to each race’s results. A Men’s 3×27 tandem canoe (aka a “pro boat”), where both paddlers are under 50 years of age, receives no time adjustment in the NECKRA system. This is because an idealized Men’s pro boat performance is the basis for the time handicaps. In a hypothetical one-hour-long race, the Point Series assumes that a pro boat paddled by two young men would finish in one hour, a Mixed tandem in 1 hour 2 minutes, a Women’s tandem in 1 hour 4 minutes, etc. This is the basis for the NECKRA Point Series handicapping system. Some will paddle faster than this ideal; others slower. The goal of the NECKRA Points Series is to foster “Competition on a more leveled playing ground.”

Let’s consider a concrete example of the NECKRA Points Series. For each paddler between 60 and 69 years of age 1 minute *per hour* is subtracted from their adjusted finishing time. If a tandem canoe had one paddler aged 65 years and a finishing time of 1:30 (e.g., 1.5 hours), then 1.5 hours times 1 minute per hour = 1.5 minutes would be subtracted from their 1:30 race time to yield an *adjusted time* of 1:28:30. If this tandem was paddling a NYMCRA Stock canoe a further 3.5 minutes per hour (based on the original 1:30 finishing time) would be subtracted from their adjusted finishing time, or 1.5 hours times 3.5 minutes per hour = 5.25 minutes, or 5 minutes 15 seconds (aka 0:05:15). So this tandems adjusted time then becomes 1:30:00 (their unadjusted race time) – 0:01:30 (age adjustment) – 0:05:15 (hull type adjustment) = 1:23:15. Analogous time adjustments are applied to the finishing times of all racers who are NECKRA members. Similar computations are performed for all kayak finishers using kayak-specific handicaps; kayaks have their own category in the overall NECKRA Points Series.

Once all adjusted finishing times for a given race have been computed these times are sorted in order. The fastest adjusted time is awarded 50 Points Series points for their performance; the second fastest adjusted time is awarded 49 points, etc. Note that the first boat to cross the finish line isn’t necessarily awarded 50 points. For example, a Men’s tandem may have had a Mixed (one man, one woman) tandem on their stern in the final sprint to the finish, and – everything else being equal – the time adjustment would award more Points to the Mixed hull.

I was the “Points Person” for NECKRA from 2008 through 2012, computing the Points Series results for all of their flatwater canoe and flatwater kayak races. True to the statement on their website, the handicapping system proved to be a good leveler of gender, age and boat type. But there were a few exceptions. In 2009 I was asked to consider whether or not their C1 handicaps were fair; at the time the Mens’ C1 handicap was -3 minutes per hour, while the women’s C1 handicap was -8 minutes per hour. It turned out that all of the handicaps applied up to that point were based on experience from past races, having been set by NECKRA’s Competition Committee at their annual end-of-season meeting several years before. I did a brief statistical analysis that showed the C2 Mixed handicap of -2 minutes / hour was reasonable, as was the C2 Women’s handicap of -4 minutes / hour. To do this I compared the median finishing positions of C2 Men’s, C2 Mixed, and C2 Women’s hulls in the first 15 races of the 2009 season based on adjusted finishing times, e.g. after applying the Mixed and Women’s handicaps to respective finishers. I used the median since race results – especially for long races, or races with many participants – tend to have a cluster of top finishers, then a long-ish “tail” of finishers thereafter. A plot of the resulting finishing positions vs. races 1 – 15 appears in Fig. 1.^{[1]}

*Figure 1: Normalized finishing position by Race for various C2 teams.*

Since each race in the Points Series had a different number of NECKRA participants I normalized the median finishing positions by the number of C2s in each race. As you can see in Fig. 1 there are variations race-to-race, and in some races there were no Women’s tandems. But over the course of these 15 races the average normalized median finishing position using the NECKRA C2 gender handicaps was 0.43 (Men), 0.44 (Women), and 0.4 (Mixed). This quick-and-dirty analysis showed that the NECKRA Mixed handicap of -2 minutes / hour, and the Women’s handicap of -4 minutes / hour, were both reasonable. For these racers NECKRA’s flatwater Points Series handicaps had leveled the field. Parenthetically, I never received any complaint about these handicaps in my 5 years of scoring the Points Series.

My analysis did show, however, that the C1 Men’s handicap of -3 minutes / hour wasn’t as reasonable. Race results from the 2009 season showed that, using this handicap, the median finishing position in a hypothetical race comprised of a mix of ten C1 and C2 hulls was 8^{th}. The flatwater Points Series was skewed toward C2s. I subsequently proposed a C1 Men’s handicap of -4.5 minutes / hour. This handicap has been in use since the 2010 season, and while it may merit re-visiting in light of a larger sample of recent race results it seems to be working^{[2]}.

But what about the Women’s C1 handicap? Was it fair? There were so few Women’s C1 finishers in the 2009 season (or in the 2008 season, where I also had all the race results) that there was no way to draw a statistically meaningful conclusion either way. All I could offer NECKRA was, well, keep on doing what you’re doing in the absence of more information.

But wait, what about science? Isn’t that why we’re all reading this article? Well yes, we can use physics to level the field, and determine a fair and reasonable handicap for C1 Women paddlers. And to do so we’ll leverage the C2 and C1 handicaps discussed above, and a model developed in Part 1 of our series that relates paddler power to steady-state hull speed.

From “The Science of Paddling, Part 1: Tandem vs. Solo,” we learned that a tandem canoe of length *L* having two identical paddlers that each supply paddling power *P* will produce a steady-state racing speed *V*_{2} in quiet water (e.g., no current) via

We’ve introduced a constant *C*_{2} that incorporates all of the hydrodynamics of the tandem hull, thus greatly simplifying the analysis. Now if the two paddlers are a mixed team we can denote the power supplied by the man as *P _{m}*, and that supplied by the woman as

Similarly, for a men’s tandem

Now in high school algebra^{[3]} we learned that Distance equals Velocity times Time. Since all hulls will be racing on the same course the distance traveled by these canoes will all be the same; all that differs will be the finishing times. Consequently,

where *T*_{2,}* _{men}* and

Substituting the expressions above for the men’s and mixed steady state racing speed allows us to relate the men’s and women’s paddling power *P _{m}* and

If we assume that the men’s paddling power Pm is one – as you’ll see, we’re only interested in the relative paddling powers, not their absolute values – then

This means that the NECKRA mixed C2 handicap implies women exert about 81% of the power men do when paddling^{[4]}. Is this consistent with the Women’s C2 handicap? To answer we’ll do a quick version of the analysis. As above, the steady state speed for a women’s tandem is

For the 1-hour race the time adjustment for a men’s tandem is zero; for a women’s tandem it is -4 minutes, which means that in the absence of a time adjustment the women’s tandem will finish in 1:04, or 1.0666 hours, or… 1.0666 times the finishing time of the hypothetical men’s tandem. Thus

If we again assume that the men’s paddling power *P _{m}* is one, then

This means that the NECKRA women’s C2 handicap implies women exert about 82% of the power men do when paddling. The difference between the paddling powers derived using the two handicaps – 0.8126 and 0.824, respectively – is about 1.4%. That’s remarkably consistent!

So what do we do with this information? We use it to level the playing field for Women C1 paddlers, of course! We’ll take a similar tack to what we did above, but now use the NECKRA Men’s C1 handicap to derive a Women’s C1 handicap using the power proportionality factor above. Proceeding as before, but now with a C1,

where *V*_{1} and *C*_{1}’ are the speed and “hull constant” for a C1. We have assumed that the tandem and solo hulls are the same length *L*; if not any difference can be accounted for in the analysis^{[5]}. Similarly, for a Women’s C1,

Again, we note that Distance equals Velocity times Time. Since all hulls will be racing on the same course the distance traveled by a Men’s C1 and a Women’s C1 will be the same; all that differs will be the finishing times. Consequently,

Using our expressions for the Men’s and Women’s C1 speeds in terms of their respective paddler’s powers, this becomes

Assuming *P _{w}* = 0.824

Here, we’re interested in determining the relative finishing times of the Men’s and Women’s C1s; we already determined the relative paddling powers. Using the NECKRA Men’s C1 handicap of -4.5 minutes / hour, the finishing time for a Men’s C1 (in a hypothetical race where a Men’s C2 finishes in 1 hour) is 1:04.5, or 1.07666 hour. Solving the equation above for the Women’s C1 finishing time yields

Consequently, the NECKRA Flatwater Points Series Women’s C1 handicap should be -8.9 minutes (8 minutes 54 seconds) per hour based on the analysis above^{[6]}. And what is the current handicap? -8 minutes per hour. Considering the machinations we went through to derive a handicap, the two numbers are pretty consistent, differing by approximately 10%.

Where did the difference come from? NECKRA established the -8 minutes / hour time adjustment for Women’s C1 paddlers some time ago because (as I understand it) this number seemed reasonable. In light of the analysis above, it is – keeping in mind that “reasonable” is based on an analysis that relies on other NECKRA handicaps which also seem “reasonable.” We’ve merely used science to show that the NECKRA Flatwater Points Series is self-consistent, which is all you can ask for. And that physics models can extend established systems to make up for a lack of a statistically-significant sample of race results.

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Copyright (c) 2017, Shawn Burke. All rights reserved.

- Note that I considered time-adjusted finishing positions in my analysis, as opposed to just the adjusted finishing times. I did this because the Points Series is scored based on finishing position. I have since checked my analysis using adjusted finishing times and my conclusions remain unchanged. ↑
- One complaint I did sometimes receive about the NECKRA Points Series is that the handicaps should be based instead on race results of top finishers, rather than on median statistics, since the same few paddlers won the series year after year. Now you can base time adjustments on any criteria you like. But basing them on the results of individual performers lacks rigor, and (dare I say it) verges on the political. There; I said it. ↑
- We’re actually using Speed here, rather than velocity; velocity is a vector quantity, which means distance is as well. Apologies for the conflation. ↑
- Note that this does not mean women exert 81% of the paddling power men do when paddling! In order to make any such determination one would have to recruit a statistically meaningful sample of men and women paddlers, put them on an ergometer, and do testing. What this result means is that the NECKRA Points Series handicap, using the physical model outlined above, yields the 81% figure. When reading any technical article, always keep the model inputs and underlying assumptions in mind when interpreting any result! ↑
- The exercise is left to the reader. ↑
- If the Men’s C1 handicap were reduced then the Women’s C1 handicap would be reduced as well using this analysis since one depends upon the other. ↑

So which is faster: paddling an out-and-back course without current, or with current? You lose time going upstream into current, but you regain it going downstream… don’t you? A round trip should take the same amount of time, right? Well, read on.

Consider an out-and-back course that first travels upriver, then back downriver to the starting point. If the one-way distance to the turnaround buoy is *L*, then the round-trip distance is twice this, or 2*L*. When there is no current, and you paddle this course at speed *V* (and, unlike me, could magically make the buoy turn in zero time), then your round-trip paddling time *T _{still}* would equal

Now let the river have everywhere a uniform current, with speed *U*. This means your upstream speed will now equal *V* – *U*, e.g., the current slows you down. As a result, your time to the turnaround buoy *T _{up}* is now

As you can see, since your upstream speed (*V *– *U*) is less than your still water speed *V*, your time to the turnaround buoy *T _{up}* is now greater than in still water due to the river’s current. Nothing new there, eh? Good.

With current, your downstream speed will now equal the sum of your quiet water paddling speed *V* plus the speed of the current, *V* + *U*. As a result, your time from the turnaround buoy back to the start, *T _{down}* , will equal

As expected, since your downstream speed (*V *+ *U*) is greater than your still water speed *V* your downstream time will be faster than if there was no current. Again, nothing new.

With current, your round-trip time is the sum of your upstream time *T _{up}* and your downstream time

Using our expressions for *T _{up}* and

This relationship is plotted in the Figure below for *V* = 6.4mph, and 2*L* = 4.95 miles^{[1]}. The no-current round-trip finishing time is 0:46:24. Note that if the river had no current, *U *= 0 and *T _{current}* =

If the current is much slower than your quiet water paddling speed, *U* << *V*, the term *U*^{2}/*V*^{2} in the denominator of equation (7) is much smaller than 1. In that case you can use a binomial expansion to approximate equation (7) as

For non-zero current speed *U*, the term *U*^{2}/*V*^{2} is greater than zero, which means 1 + *U*^{2}/*V*^{2} is greater than 1. Your round-trip time will always be slower in the presence of current, *no matter what*. Going faster on the downstream run because of current will not make up for going slower on the upstream run. It’s like death and taxes: you can’t win, you can’t break even, and you can’t get out of the game.

There are two other limiting cases of interest: If the current is running as fast as you can paddle, e.g., *U *= *V*, then *T _{current}* becomes infinite (see the fifth equation above); you never get to the turnaround buoy… or anywhere else for that matter, as you’d just be sitting still in the Earth’s reference frame. If the current is going faster than you can paddle, e.g.,

Is the fact that you “lose” round-trip time when there is current all that surprising? Not really, if you change your perspective. We habitually think of the round-trip transit problem in terms of distance, since we remember that distance equals velocity times time, or in terms of speed. For a fixed time, if we double the speed, we go twice as far. But for a fixed *distance* course, time and speed have an inverse relationship. As a result, even when there is no current the relation between round-trip course time and boat speed is not a simple linear proportionality, as shown in the Figure below.

This curve “flattens out” at higher boat speeds, reflecting a more linear proportionality the faster you go. A faster boat speed means the effect of current on round-trip course time is reduced, not eliminated.

The only way to make up the time deficit is to paddler faster than the still water speed *V* on one of the legs. If you’re racing you’d naturally pick up the pace over the last downstream mile or so, right? But, alas, paddling faster on one of the legs violates a key assumption of this analysis; we assumed that *V* would be *constant*. This is a nice example of how one must always keep the underlying assumptions in mind when considering the conclusions of any study or article such as this one.

**AFTERWORD**

While writing this article I was reminded of a problem noted at the Rio 2016 Olympics. It turned out that the swimming pool may have had some current in the high numbered lanes. (See, for example, https://www.theguardian.com/sport/2016/aug/18/olympic-pool-current-swimming-results-rio-2016.) Swimming events are timed no more precisely than 0.01sec (ten milliseconds) because pool walls are only so flat; these irregularities in lane length correspond to about 5 milliseconds of time at 50m freestyle swimming speeds. In light of the analysis above, a slight bit of current in the pool could have had as large an impact on timing round-trip swimming events 100m in length or longer as irregularities in pool shape.

Copyright (c) 2017, Shawn Burke. All rights reserved.

- A distance familiar to those of us who used to paddle Wednesday nights on the Charles River. ↑

I’ve been curious about why canoes do what they do for some time. And in training for downriver and flatwater races I have also become curious as to why my body does what it does. In particular, why am I training in any particular way? The common wisdom is that one needs to do overdistance paddling, intervals, and (maybe) tempo work to prepare for racing season. But… Why? Why not just go out for a pint and be done with it? What goes on inside your body when you train or race? And does this relate in any way to how you (should) train?

It turns out that your working muscles are fueled by three distinct metabolic energy systems (sometimes called “pathways”): the *aerobic system*, the *anaerobic* or *lactate system*, and the *phosphate system*. Each plays a role corresponding to the intensity and duration of your paddling. The efficiency of these systems helps determine your paddling endurance capacity.

The following overview of these controlling metabolic systems will provide background and context for the “curious paddler” who wonders how and why the body performs and adapts to training. And don’t worry. You needn’t to be an exercise physiologist or biologist to read on; I’m not!

**The Aerobic System**

The aerobic system, also called the “oxygen” system, utilizes fats (in particular, triglycerides) and sugar (in particular, glycogen) to fuel your muscles. It does so by producing a compound called *adenosine triphosphate* (ATP). ATP fuels muscle contractions, as will be explained below. The aerobic system is idealized in Figure 1.

While this figure may look like biochemistry gobbledygook, consider that it summarizes much of what happens in our muscles when we paddle. We noted that there are two raw fuel sources for the aerobic system: fats and sugars. Your body stores fats from your diet in the form of triglycerides. These are processed during exercise in several intermediate steps. In the first step triglycerides are combined with water to yield fatty acids and glycerol, as shown in the Figure. The fatty acids are then processed in the muscles via a process called *beta oxidation*.

Carbohydrates in the form of glycogen, a long-chain sugar molecule, are stored in the liver and muscles. During exercise glycogen is processed in the muscles via a process called *glycolysis* to produce ATP. This ATP may be used directly by the muscles; glycolysis is a fast reaction compared to beta oxidation. Note that the glycerol released during fatty acid production is also processed to produce ATP via glycolysis as shown in the Figure.

Both beta oxidation and glycolysis produce intermediate compounds that are necessary for muscle metabolism. These compounds enter structures within the muscles’ cells called *mitochondria*. Mitochondria function as “engines” that produce ATP at the cellular level through a combination of the *Krebs Cycl*e and the *Electron Transport System*. In the electron transport system the muscle metabolism byproduct *adenosine diphosphate* (ADP) is combined with phosphorous to produce the muscle contraction compound ATP. As Figure 1 indicates, the aerobic system is where oxygen is “burned,” combining with free hydrogen ions to produce water as shown in the lower right hand corner of the block diagram.

The muscles are continuously processing ATP to release energy for muscle contractions. This energy is produced when a phosphorous atom is released from ATP (the “tri”-phosphate), producing ADP (the “di”-phosphate). The precursor compound ADP is always present in the muscles, waiting to be recycled in the electron transport system to produce ATP. All that is required is fuel, either in the form of fats or carbohydrates, to be processed through the linked pathways of beta oxidation, glycolysis, and the combining Krebs cycle. Now as with any oxidative process there are byproducts, just as your car produces emissions. The byproducts of muscle respiration, *carbon dioxide* (CO_{2}) and *water* (H_{2}O), are removed by the lungs and carried through the bloodstream, respectively.

*Figure 1: Combined energy pathways.*

The store of carbohydrates in the body that fuel glycolysis is limited, whereas the store of fats to feed beta oxidation is comparatively unlimited, no matter how lean you are. The “fat” and “glucose” systems work simultaneously, but their contributions to the energy supply are different, and depend greatly on your level of exertion as well as how you train. Fats are metabolized primarily during low-intensity exercise. For example, as you sit and read this article your body is mainly burning fat as its energy source. As exercise intensity increases, carbohydrate oxidation via glycolysis becomes the dominant energy source, as idealized in Figure 2.

The body can store enough carbohydrate in the muscles and the liver to provide about 90 minutes of energy for exercise by glycolysis alone. When your carbohydrate stores are depleted, fat burning increases and the ability to sustain your level of effort decreases – fat oxidation always requires a small amount of carbohydrate to “keep the fires lit.” That is why a marathoner can “hit the wall” at around the 90-minute mark unless they replenish their glycogen stores during the race. Glycolysis can be prolonged by drinking carbohydrate drinks and eating modest amounts of easily-digestible high-carbohydrate foods during long-duration training and racing^{[1]}.

*Figure 2: Aerobic fat and glucose utilization during steady exercise.*

What is of particular interest is that through exercise you can train the muscles’ metabolic systems to use energy more economically. With appropriate training the number and density of mitochondria in the muscles increase, providing more “engines” to burn fat and glucose. The metabolic pathways become more efficient as well; a well-trained athlete can oxidize fat for a longer time, shifting the curves in Figure 2 to the right, and thus save (some) glycogen for high-intensity efforts like jumping wakes and finishing sprints.

In addition to metabolic adaptations, aerobic exercise will increase the number and density of the tiny blood vessels within the muscles called *capillaries*. This change can be seen under a microscope after the first few weeks of aerobic training – just don’t try this at home! As a result, the heart will be able to supply more oxygen-rich blood to the muscles, and more efficiently remove the chemical by-products of exercise including carbon dioxide, water, lactic acid, and free hydrogen ions. A long-term program of aerobic exercise will also increase the heart’s strength and stroke volume. Consequently, the well-trained heart can pump more blood with each beat, with a corresponding lower heart rate for the same level of effort, and a lower resting heart rate.

A few other tidbits about the aerobic system that may shape your training:

- As you begin to exercise, it takes about 2 to 3 minutes before the heart, lungs, and circulatory system fully function, and for the aerobic system to be fully engaged. This is one reason why you need to warm up prior to training or racing.
- It takes as long as 48 hours to replenish glycogen stores in the liver after long, strenuous exercise.
*Muscle*glycogen, however, can be replenished in a few hours. Consequently, long over-distance workouts (approximately 2 hours or longer) should be followed by a recovery day of light exercise, or a strength-training day. That said, many ultra-long distance racers will perform “sandwich” aerobic workouts on consecutive days, say three hours one day and five hours the next, to acclimate the body to partial recovery and how it feels during very long races like the Clinton or the AuSable. - Aerobic energy is synonymous with oxygen uptake capacity and mitochondria density. The body adapts to the stress of training over time, and will increase the number and density of aerobic muscle mitochondria from 3.5% of muscle mass to as much as 5% of muscle mass. Extensive aerobic training can increase aerobic endurance capacity by approximately 50%. This adaptation takes about 4 years to plateau. Sorry, but great endurance athletes aren’t created overnight!
- Increases in mitochondria number and density, and capillarization, occur
*in the muscles that are stressed*in aerobic exercise. Consequently, to develop aerobic capacity for paddling, you should paddle. This phenomena is called*training**specificity*. Running and cycling, for example, won’t by themselves enhance paddling aerobic fitness, but they will generically increase cardio-pulmonary efficiency. You are what you eat; you can best do what you train for.

**The Anaerobic System**

Glycolysis produces a metabolic byproduct called *lactic acid*, a feature we conveniently ignored when discussing Figure 1. Lactic acid is reprocessed in the liver via the *Cori Cycle*, producing glucose to fuel the muscles. The Cori Cycle is depicted conceptually in Figure 3. Glycolysis and the Cori cycle do not require oxygen to produce ATP to fuel muscle contractions, hence the Cori Cycle is “anaerobic,” or functioning without oxygen. Glycolysis and the Cori cycle produce ATP more quickly than the aerobic system, since the anaerobic pathway entails only about a dozen chemical reactions.

*Figure 3: The Cori Cycle.*

At the start of exercise, irrespective of its intensity, the energy supply is anaerobic, fueled first by the phosphate system (which we’ll discuss later), and then by the anaerobic system. For exercise lasting around 1 to 3 minutes, such as 400m and 800m runs, the energy supply is predominantly anaerobic– recall that it takes 2 to 3 minutes before the aerobic system “warms up” completely and provides ATP. The anaerobic system also supplies the energy during increases of pace (surges, finishing sprints) beyond the aerobic level in an otherwise aerobic paddling session. During steady-state exercise the Cori Cycle is always operating in the background, even at very low levels of exercise intensity. It is important to note that the Cori Cycle can recycle lactic acid completely *below a blood lactate concentration of around 4.0 millimoles per liter*. Lactic acid is providing fuel for your working muscles.

However, as the level of exertion increases this recycling process cannot keep up with lactic acid production. This *lactate threshold* occurs at a blood lactate concentration of approximately 4.0 millimoles per liter. As lactate concentration increases to around 7.0 millimoles per liter, beta oxidation (fat metabolism) essentially stops, and the muscles are increasingly fueled by recycling lactate into glucose via the Cori Cycle. At extremely high levels of exercise intensity blood lactate concentration can peak at approximately 25 millimoles per liter.

Lactic acid production ultimately limits anaerobic power. Lactate accumulation increases muscle acidity; this state of muscle acidity is called *acidosis*. Acidosis damages the walls of the muscle cells, causing leakage through the cell wall into the blood. Acidosis also interferes with and damages the system of aerobic enzymes within the cells, which decreases aerobic endurance capacity. And to add insult to injury, muscle contractions at very high lactate concentrations become more difficult because of a lack of ATP. Anyone who has done an all-out 400-meter finishing sprint will be familiar with this state; it feels like a bear has jumped on your back at about the 300-meter mark.

With increasing acidosis an enzyme produced by glycolysis blocks lactic acid production. If not for this enzyme, further increases in muscle acidity would damage muscle protein and mitochondria. Consequently, the body does not learn to “tolerate” high lactic acid production; instead lactic acid production is self-limited. What the body must learn to do is adapt itself so that you can train and compete within tolerable lactate levels, e.g. increase the level of exercise intensity where *your body is still able to recycle lactic acid*. This level of exercise intensity is called the *lactate threshold*, and ultimately is the determinant of race performance.

After hard anaerobic exercise it can take days before the body recovers sufficiently to regain full aerobic capacity. When exercise is repeatedly too intense, e.g. without sufficient recovery between bouts of hard anaerobic exercise, aerobic and anaerobic endurance capacity decrease considerably, leading to a number of problems referred to collectively as *overtraining*. Recovery time, from 24 hours to as much as 96 hours, must be provided after hard anaerobic workouts before muscle cells normalize. This should be taken into account when developing a training program that includes anaerobic intervals, high intensity “threshold” exercise or tempo training, or racing.

A few other tidbits about the anaerobic system that may shape your training:

- When the body is at rest it takes about 25 minutes to remove half the accumulated blood lactate that results from maximum exertion. 95% of lactate is removed after about 1 hour and 15 minutes of rest.
- Lactate is removed from the blood and muscles much more quickly when you perform light, continuous exercise at the end of a workout, rather than complete rest. This “active recovery” is best for your cool-down following a race or hard workout. A cool-down of light paddling will remove 90% of blood lactate in approximately 20 minutes. Yeah, that’s a long cool down, but at least 5 minutes if not 10 should be possible.
- It is a common assumption that lactic acid is responsible for increased recovery times after intense workouts. However, free hydrogen ions and other byproducts of anaerobic exercise also delay recovery.

**The Phosphate System**

The phosphate system supplies energy directly to the muscles, and is fed by the aerobic and anaerobic pathways as described above. By itself the phosphate system also drives fast, powerful muscle contractions. The phosphate system does not use oxygen. This is fortunate, since very intensely contracting muscles squeeze off their own (oxygen-rich) capillary blood supplies. The phosphate system does not use glucose, or produce lactic acid, hence it supplies *anaerobic alactic* energy. It is the fastest muscle energy pathway because it requires the fewest chemical reactions.

Muscle contractions are controlled locally by our friend adenosine triphosphate (ATP), which is broken down into adenosine diphosphate (ADP) and energy:

ATP –> ADP + energy.

Prior to exercise the muscles themselves have a store of ATP that can be called upon for short bursts of maximum effort, up to approximately 3 seconds. The phosphate system then kicks in, and can provide energy for maximum effort for another 10 seconds, such as a sprinter would expend in a 100-meter dash. Thus, the phosphate system is of particular importance during short-duration sprints, race starts (hey; free energy!), and while strength training.

A compound called creatine phosphate (CP) provides the necessary phosphorous (P) to drive the phosphate system, re-synthesizing ATP in the muscles once their ATP stores are exhausted. The phosphate system replenishes 70% of ATP and CP within 30 seconds; these compounds are 100% replenished in 3 to 5 minutes. The phosphate system is trained by hard, short sprints, alternated with periods of *complete* rest. Think weight training if you’re lifting heavy – this relies solely on the phosphate system. Rest periods should be long enough for ATP and CP to re-synthesize. It is important to note that the replenishment of CP is delayed by high lactic acid concentrations. Thus, one should not combine anaerobic training sessions with pure sprint speed workouts.

The muscles’ ATP and CP stores increase by 25% to 50% after several months of sprint training. However, the ultimate speed one can achieve in sprints is not limited by the store of ATP or the efficiency of its re-synthesis. Speed is also function of technique, and technique is determined by the central nervous system and your neuro-muscular pathways.

**How Muscle Metabolism Delineates Training**

The body employs all three energy systems described above during exercise. Yes, all of them. Their *proportional* engagement depends on the length and intensity of effort. The relationship between the duration of exercise and the percentage share of the various energy systems is conceptualized in Figure 4. Note that each bar on the plot represents a maximum effort for the corresponding time interval on the bottom axis. As seen in the Figure, short sprints are anaerobic and alactic, primarily engaging the phosphate system. Assuming a complete warmup, a starting sprint will slightly engage the aerobic system. However, energy supply then is still dominated by phosphates and the anaerobic system. For longer races the aerobic system becomes the dominant source of energy.

*Figure 4: Energy utilization vs. time (after Janssen).*

This figure suggests that you should train each of the energy systems in proportion to how it is engaged for a goal race’s length. In other words, a paddler training for a long race should spend most of their training volume developing their aerobic system, especially to improve the efficiency of fat oxidation, and far less (but not zero) training their anaerobic system. For these paddlers, the phosphate system is far less relevant to train in isolation – except for starts. As the season progresses, and this paddler begins to train for shorter races, training can and should incorporate more anaerobic and sprint training as required by new goal race distances. Racers training for shorter events will already be doing so. And Figure 4 shows why.

Copyright (c) 2017, Shawn Burke. All rights reserved.

**References **

Anderson, Owen, “Periodization Training Technique: If you want to improve your performance, you can’t train the same way all the time,” http://www.pponline.co.uk/encyc/0147.htm.

Bishop, D., Bonetti, D., & Dawson, B., “The effect of three different warm-up intensities on kayak ergometer performance.,” *Medicine and Science in Sports and Exercise*, Volume 33, pp. 1026-1032.

Croston, Glenn, “Beta oxidation of fatty acids,”http://www.biocarta.com/pathfiles/betaoxidationPathway.asp.

Croston, Glenn, “Feeder pathways for glycolysis,” http://www.biocarta.com/pathfiles/feederPathway.asp.

Higdon, Hal, “Ultramarathon training,” http://www.halhigdon.com/ultramarathon/ultramarathon2000.htm, 2000.

Isaka, T., & Takahashi, K. “Effects of off- and pre-season training on aerobic and anaerobic power of kayak paddlers,” *Medicine and Science in Sports and Exercise*, Volume 29, Number 5, Supplement abstract 1242, 1997.

Janssen, Peter, *Lactate Threshold Training*, Human Kinetics, Champaign, IL, 2001. Jenkinson, Dave, “USACK guide to shoulder exercises,”

Maffetone, Philip, *Training for Endurance*, David Barimore Productions, Stamford NY, 1996.

Noakes, Tim, *Lore of Running*, 4th Edition, Human Kinetics, Champaign, IL, 2003.

Nolte, Volker, *Rowing Faster*, Human Kinetics, Champaign, IL, 2005.

Pfitzinger, Pete, “How to speed up recovery from racing,” http://www.copacabanarunners.net/i-recovery-racing.html, 2004.

Pfitzinger, Pete, “Finding your optimal training / recover ratio, ”http://www.copacabanarunners.net/i-training-recovery.html, 2004.

Schulman, Deborah, “Fuel on fat for the long run,” http://www.marathonguide.com/training/articles/MandBFuelOnFat.cfm, 2000.

Sleamaker, Rob, and Browning, Ray, *SERIOUS Training for Endurance Athletes*, 2nd Edition, Human Kinetics, Champaign, IL, 1996.

Szanto, Csabo, “Daily training program for advanced athletes, ”http://www.canoeicf.com/default.asp?Page=1605&MenuID=Development%2F1012%2F0.

http://www.nismat.org/physcor/max_o2.html, “NISMAT Exercise Physiology Corner: Maximum Oxygen Consumption Primer,” 2005.

http://www.bbc.co.uk/scotland/education/bitesize/higher/biology/cell_biology/respiration2_rev.s html, “Stages of aerobic respiration.”

http://biocarta.com/pathfiles/h_etcPathway.asp, “Electron transport reaction in mitochondria.”

- There are other approaches to training and racing based on low-carb diets and fueling, but this is outside the scope of the current article. ↑

When you paddle, each stroke propels your hull forward against our familiar nemesis, drag. But why does the canoe move forward, and not in some other direction? And more fundamentally, why does it move at all? The concept of momentum provides the answer. And it all can be understood by considering … space boots.

When I was in 6^{th} grade I bought a collection of science fiction stories at a garage sale. One of the tales centered on an astronaut marooned outside his spacecraft; how he became marooned, I don’t recall. Fortunately, he was wearing a space suit – it’s pretty cold in outer space. Unfortunately, the space suit had no thrusters for propelling him back to his craft. And he had a limited supply of air left to breath. How did he get back inside his rocket ship before running out of air? By using science!

Our stranded astronaut recalled that when you fire a rifle, there is recoil. The rifle “throws” a bullet in one direction, and in response the rifle moves in the *opposite* direction. The astronaut realized that if he threw something in the direction opposite his spacecraft, his body would “recoil” in response and move toward the spacecraft. Knowing that he couldn’t throw his oxygen tanks (gotta breathe), his space helmet (thus avoiding the worst case of brain freeze ever), or his space gloves (he needed functioning hands to open the hatch back into his spacecraft), he took off his space boots and threw them away from his rocket ship as hard as he could. He had a pretty good aim because the rest of him drifted slowly toward the space ship’s hatch, whereupon he was able to climb back inside. All because he remembered physics, and the concept of *conservation of momentum*. And, of course, because he had space boots.

We’ve all heard the word “momentum,” but what does it means in scientific terms, and how does it apply to paddling? Momentum can be thought of as “mass in motion.” A canoe and its paddler(s) have mass, so if a canoe is moving then it has momentum. An object’s momentum depends upon its mass and its velocity. Formally, momentum – typically written in shorthand as “*P*” – is equal to the mass of the object times its velocity:

This equation shows that momentum is linearly proportional to an object’s mass, as well as linearly proportional to its velocity. And by “linearly” we mean that neither quantity appears as a square, cube, square root, etc., but only as itself. This means that if you throw a baseball at 45mph, and a major league pitcher throws the same baseball at 90mph (e.g., twice as fast), then the baseball thrown by the major leaguer has twice as much momentum as when you threw it.

So now consider a canoe paddled by a solo paddler. At the end of each stroke their hull has a certain amount of momentum *P _{hull}*, which is quantified as the total mass of the hull

With each stroke, the paddle blade is accelerating a volume of water. It may seem that your paddle is planted in concrete at the catch, but in truth there is slippage; if you watch the strokes of adjacent paddlers you’ll see little pools of swirling, aft-moving water behind their blades at the conclusion of their power phase. This slip is illustrated in the sequence shown in Figs. 1-5 showing a side view of the power phase. The paddle rotates at and after the catch, and *slips sternward* in the inertial (Earth-based) reference frame as the water entrained by the blade accelerates.

*Figs. 1-5: Power phase, side view, with both hull and fixed fiducials.*

Your paddle stroke has imparted momentum *P _{water}* to this mass of water

The minus sign indicates that the water’s momentum is in the *opposite* direction of the hull’s momentum. This is because in physics velocity is a vector quantity, which means velocity has both a magnitude (referred to as “speed”) and a *direction* as well. The minus sign here indicates that the momentum of the slug of water is opposite to the momentum of the hull since the water and the canoe move in opposite directions.

Physics teaches us that the momentum of a system is conserved. Here, the system is the canoe and paddler, *as well as the slug of water* “thrown” by the paddle. Consequently, at the conclusion of each stroke the total momentum of the system – here, the canoe/paddler(s) and the water – must equal zero. This may be written as,

As a result, the momentum of the canoe plus paddler, plus the momentum of the slug of water, must be equal. This means

So to move the hull, you impart momentum to the plug of water entrained by your paddle. And the hull will travel in the direction opposite the plug of water that you have “thrown.” This should be familiar to anyone who has paddled a canoe; the hull doesn’t move in the same direction as your paddle blade moves in the water, nor does it move perpendicular to the path of the paddle. And perhaps the most interesting take away, particularly relevant to our discussion, is that the momentum of the water *equals* the momentum of you and your canoe *at the end of each stroke*. It’s exactly like our astronaut floating freely in space – to move toward his spacecraft, the astronaut threw his space boots in the direction opposite his ship, and he zoomed off. If he had heavier space boots, or threw them faster, he would move faster in response.

So what does this tell us about moving the canoe faster? Equation (4) shows us how. One can express equation (4) in terms of hull speed *v _{hull}* as

To increase hull speed, we see from this equation that there are several options:

- For a fixed paddle size and power phase stroke finishing speed, the numerator in equation is a constant. So to go faster our paddler should get a lighter hull, lose weight, or both since the only remaining variable in this case is the combined mass of the canoe and paddler.
- For a fixed power phase stroke finishing speed, our paddler could move a bigger slug of water by using a paddle with a bigger blade in order to entrain more water with the paddle stroke.
- For a fixed paddle size, our paddler could move the slug of water faster with a faster stroke power phase finishing speed,
*or*use a smaller paddle but have an even faster power phase finishing speed. - Or, our paddler could lose weight, buy a lighter hull, get a paddle with a bigger blade, and move that blade faster. Phew! I’m tired already, not to mention broke from the cost of all the new gear.

Practically speaking, each option will appeal to specific paddlers based upon energy, paddle force, and power considerations. We’ll address energy issues below; paddle force and power will be addressed in a subsequent Science of Paddling article.

Now what if the canoeist isn’t paddling, but instead is poling? As you may (or may not) know, canoe polers “stand tall in their canoes” and propel themselves using a pole of wood or aluminum, pushing the pole into the riverbed and pushing off as illustrated by 12-time ACA poling champion Harry Rock:

*Fig. 6: Poling up a drop.*

Assuming the pole is well planted and doesn’t slip, the poler is pushing against the mass of the Earth. In this case our equation for momentum conservation takes a particularly interesting form:

So each time the poler plants their pole and propels their canoe forward, they are changing the momentum of the Earth, pushing it in the opposite direction!

As noted above, a canoe can attain a particular speed by using the paddle to move a larger mass of water slowly, or a smaller mass of water more quickly. The choice is yours as long as total momentum is conserved, per Equation (4). The difference between these two scenarios becomes more apparent when we consider the *energy* expended in each scenario, e.g. the cost in energy to *you*, the paddler.

At the end of each stroke, the energy in the canoe/paddler/water system is due in the motion of the canoe/paddler(s) and the motion of the water. This “motion energy” is called *kinetic* energy. Kinetic energy is linearly proportional to mass, and proportional to the square of velocity^{[2]}. The total kinetic energy of the system is then expressed as the sum of the kinetic energy of the hull and paddler, and the kinetic energy of the water:

Let’s consider the impact of paddle blade size for a given hull speed *v _{hull}*. We’ll consider two cases: a “smaller” paddle blade with an area represented by

For the “smaller blade” paddle, the total system kinetic energy at the end of the stroke is then

For our “bigger blade” paddle, the kinetic energy totals

The ratio of these two energies, e.g. the ratio of how much energy is expended by using a smaller paddle vs. using a larger paddle to attain the same hull speed, is

Now it turns out that the second term in both the numerator and the denominator in this equation is larger than the first term^{[3]}. As a result, the ratio can be approximated as

One next uses the momentum conservation equation for each scenario to solve for *v _{water}*

Since the “larger” blade’s area *A*_{2} is always bigger than the “smaller” blade’s area *A*_{1}, the ratio *A*_{2} / *A*_{1} will always be greater than 1. The approximation above shows that, to first order, you will always expend more energy paddling with a smaller blade than paddling with a larger blade in order to attain the same hull speed. The cost of using a smaller paddle to achieve the same hull speed is greater energy expenditure – by you. Surprise! In both scenarios the mass and speed of the canoe and paddler at the end of the stroke stays the same, hence their contribution to kinetic energy is fixed. But while the mass of water moved by the smaller paddles is less, the water’s kinetic energy is only *linearly* proportional to this reduction, and this gain is more than offset by the *squared* dependence of kinetic energy *on the velocity*. Since the smaller slug of water moved by the smaller blades must be moved faster to achieve a given hull speed because of momentum conservation, what you appear to gain in moving a smaller mass of water you more than lose in energy expenditure by having to move it with a faster power phase speed. So if you prefer a smaller blade, use it briskly, and have a well-trained aerobic/anaerobic engine.

I’ll be the first to admit that I was initially puzzled by these results; smaller blades are the trend among many local paddlers. But thinking back to my own experiences in 2008, I realized that I had actually lived these equations. That Summer I purchased a new bentshaft paddle with a wider blade; bend angle and shaft length were the same as my other paddles. It took me a while to zero in on how take advantage of the new paddle, since I seemed to be pulling harder but not going any faster, and getting more tired in the process. Things got better when I spent some time paddling C-2 with a much more experienced partner. In doing so I learned to accelerate the blade more smoothly, and to employ a more “complete” power phase by not recovering before the wider blade’s velocity had reached steady state. In other words, a slightly longer stroke than the “choppy” stroke I had been using. A few sessions later, and suddenly my C-1 race pace improved, with a slightly lower cadence *and with a lower heart rate*, e.g. less energy expenditure. Granted, I’m mostly made of slow-twitch muscle fiber; I’m what you’d call a “grinder” when it comes to canoe racing. But I was able to grind at a faster race pace, using less energy, once I tamed the bigger blade.

And I owe it all to space boots.

- This is a simplification, since momentum is imparted to a mass of water entrained by the paddle that both translates and circulates. ↑
- Formally, since velocity is a vector this should be expressed as a dot product, but for simplicity we are only considering motion in one direction. Bonus points if you saw this. ↑
- As many authors have noted in the past, “The proof is left to the reader.” It’s not hard; give it a try. ↑

Copyright (c) 2017, Shawn Burke. All rights reserved.

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]]>OK, so we all know that paddling in shallow water is a pain. You trim aggressively bow down to keep from experiencing that “sinking feeling” in the stern, and paddle like there’s no tomorrow. Yet when you hit the shallows it still feels like you’ve hit a wall. Waves become steeper; your hull speed falls. And heaven forbid if you’re on someone’s inside as you enter a shallow water turn. Hello, shoreline!

So what is it that makes shallow water so challenging to paddle? Why does your hull slow down? Why do waves break toward shallow water and then steepen? More fundamentally, why are the physics governing hulls in shallow water any different than deep water? After all, it’s still just water; right?

Yes, it’s all water. But we inhabit a very particular environment when we paddle: the water’s surface. It is at this interface between air and water that our challenges arise. The culprit is waves. We see waves all around us when we’re on or around water. We watch expanding wave patterns created by a pebble cast into the water; we slowly rise and fall in the long-wavelength swells created by a powerboat on a lake’s far shore; we take a breather by surfing the waves behind another boat during a long race. What dominates paddling resistance at race speeds are the surface waves caused by varying pressures around our hull as it moves through the water. And the fundamental character of these waves begins to change when the depth of water beneath your hull is less than about half your boat’s length.

Now that we’ve let the cat out of the bag, you can proceed to the next article. But if you’re curious about the *why*, consider waves moving over the surface of water shown in Figure 1. The water

*Figure 1: Water and wave geometry.*

has average depth *D*. The distance between two wave crests is defined as the wavelength, represented by the Greek symbol λ (“lamda”). We know from experience that waves such as this don’t just sit there, but travel at a certain speed which we represent by *c _{p}*. There is a relationship between this “phase speed”

In this equation *g* is the gravitational acceleration constant and *π* is the constant “pi” (which equals 3.1415…), and tanh is the hyperbolic tangent function. No need to worry about the hyperbolic tangent, as we’ll be looking at deep and shallow water cases where this function reduces to something very simple. As you can see, the water wave’s speed is a function of water depth and the wavelength – plus a few constants – and nothing else. In other words, wave motion is a *property of the water having a free surface*, whether there is a boat there or not. And this wave motion must satisfy that relationship: for a given wave speed, there is a corresponding wavelength. The only thing a boat does is provide what mathematician’s call a “forcing function” to the water’s surface, and thus *initiate* the waves.

When the water is deep, e.g. the depth *D* is much greater than the wavelength *λ*, the hyperbolic tangent function reduces to 1. The equation above for phase speed is then greatly simplified:

Deep water waves, then, do not depend on the depth *D*, but only upon the wavelength *λ*. For the next bit we’ll use this deep-water approximation to guide our analysis.

We’ve all seen the wave train spreading out beside and behind a hull moving across deep water, as depicted in Figure 2. These so-called “Kelvin Waves” – the same Kelvin after whom the unit of temperature is named – comprise divergent waves spreading out from the hull, and transverse waves that spanning the divergent waves. The wave pattern depends on boat speed. This is because the waves created by the canoe satisfy the deep water phase speed equation above, which relates wavelength λ and the wave’s phase speed *c _{p}*. As the hull speed increases, the waves created by the hull follow the hull at that same speed. And those waves,

*Figure 2: Kelvin waves.*

This leads to some interesting wave patterns alongside the hull, as depicted in Figure 3. This shows a profile view of a USCA C-1, paddled at speeds such that the transverse bow wave’s wavelength is in various proportions to the C-1’s length. In Figure 3(a) the bow wavelength is half the hull length; this corresponds to a paddling speed for an 18’6” C-1 of about 4.7 mph. As the speed increases to about 5.4 mph the wavelength is now about two-thirds as long as the hull. Then, when the paddling speed reaches about 6.7 mph, the wavelength equals the hull length. Throughout, the transverse bow wave and stern wave interact. A phenomenon called superposition causes these waves to combine constructively and destructively, leading to the wave pattern shown. There is a trough in the wave pattern around and just aft of midships. As a result, the stern begins to sink a bit, since there is less water to support the rear half of the hull. This is why you need to trim a racing hull slightly bow down even in deep water; when the stern sinks like this the hull becomes less hydrodynamically efficient. (Canoe designers tend to move the hull’s widest point aft of midships to address this shift in buoyancy, as well as to address other hydrodynamic effects.) Finally, as the hull speed surpasses 7 mph, as shown in Figure 3(d), the superposed wave pattern causes the stern to sink further, requiring significant bow down trim to maintain hydrodynamic efficiency – plus a lot of hard paddling to maintain this sprint speed in a C-1!

*Figure 3: Hull in its own wave train at increasing speeds (not to scale).*

The speed where the water wavelength equals the hull length is sometimes called the *hull speed*. The hull speed, described by Froude in 1868, is a rule of thumb describing the approximate maximum efficient speed for a so-called displacement hull. Since racing canoes and kayaks are (at worst) semi-displacement hulls, this hull speed is not a speed limit, but rather an indication of when wave drag begins to dominate the total drag on the hull. A C-1 can be paddled well beyond 6.7 mph; it just takes a fair amount of fitness to maintain that speed. We can characterize this drag effect using a non-dimensional ratio called the *Froude Number*. The Froude Number *F _{r}* is the ratio of hull speed

As a rule of thumb, the higher the Froude Number, the greater the wave resistance because of the increasing amount of energy transferred into the divergent waves. The case depicted in Figure 3(c) corresponds to *F _{r}* = 1. When

So why does a hull moving through water cause waves in the first place? Perhaps if we could somehow prevent wave creation, we’d escape the effects of wave drag. Unfortunately, like death and taxes, you can’t get around the laws of physics. Waves arise because the shape of the hull – *any* hull – deflects water. Consider again a USCA C-1 moving at speed *V*_{0} as shown in Figure 4.

*Figure 4: Plan view of flow speeds adjacent to hull (not to scale), and flow speed plot.*

Once again, we adopt a paddler-centric reference frame. This means that to an observer sitting in the boat the water appears to approach the hull at speed *V*_{0} as they paddle. The water that impinges upon the side of the canoe near the bow slows down a bit to speed *v*_{1}. As the water continues to flow past the hull, it accelerates a bit around the location of maximum waterline width, reaching a speed *v*_{2} that is a bit higher than your paddling speed *V*_{0}. And finally, as the water flows toward the stern it slows to speed *v*_{3}, which is a bit slower than the speed adjacent to the bow since some of the water’s energy is spent creating a wake. This variation in flow speed along the side of the hull is depicted conceptually by the dashed line plotted in the lower half of Figure 4.

The water impinging on the bow has a certain amount of *kinetic* *energy*, e.g. energy due to motion. The kinetic energy of a mass of incoming water is expressed by

where *m* is the mass of water. Since energy must be conserved, the kinetic energy at location 1 with water speed *v*_{1} must equal this “incoming” energy. However, since *v*_{1} is less then *V*_{0}, and energy is conserved, this decrease in kinetic energy at location 1 must be augmented by an increase in *potential* *energy* there. Potential energy can be thought of as stored energy. When you throw a ball into the air, the instant it leaves your hand it has a certain amount of kinetic energy. As it rises above you the ball slows, until its vertical speed equals zero. At this precise moment the baseball has only potential energy, which exactly equals the amount of kinetic energy you gave it the moment it left your hand, all because of energy conservation. As the ball falls to earth its potential energy is converted back into kinetic energy, and the ball picks up speed. It will return to your hand at the same speed as when you initially threw it; when it lands it will have the same amount of kinetic energy as when you threw it as well because… energy is conserved.

Like the ball, water can have potential energy if you lift it up in the air. So the energy at location 1 is written as the sum of the local kinetic and potential energy, which must equal the initial amount of kinetic energy:

The height *h*_{1} is the displacement of water about the average water depth *D as *illustrated in Figure 5. Conservation of energy requires that the water height *h*_{1} be greater than the zero, e.g. the water is displaced above the average depth. A similar result occurs at location 3; since the local water speed *v*_{3} is a bit less than *v*_{1}, the height *h*_{3} is a bit lower than *h*_{1} yet still higher than the average depth *D*. Conversely, at location 2 the local water speed *v*_{2} is greater than the incoming water speed *V*_{0}. This means that the

*Figure 5: Profile view of hull with hull speed wave train (not to scale).*

kinetic energy at location 2 is greater than the incoming kinetic energy. Since energy is conserved, the potential energy at location 2 must offset this increase in kinetic energy, so that

In this case, the height *h*_{2} lies *below* the average depth *D*; the water is pulled down to satisfy energy conservation.

So what does this water profile along the hull in Figure 5 look like? That’s right: waves. Because your hull deflects and slows water, then accelerates it, then slows it again you create a disturbance in the water. And, since a property of the water is that it “likes” to support and sustain waves, these local variations in water height caused by your hull become the aforementioned forcing functions that drive wave creation. Now designers of modern racing hulls have deduced many crafty ways of minimizing wave creation and wave drag. And these very high performance hulls allow you to paddle well beyond the theoretical hull speed. But the underlying principles still apply: water is still water, your hull displaces it when under way, and the laws of physics still hold.

Now all of this wave stuff is great, but, when does water become *shallow*? Isn’t that the point of the article? Indeed it is; we just had to cover the preliminaries. A lot of preliminaries! Time to return to the general phase speed equation for surface waves. Previously we had concerned ourselves with the deep water limit, where the hyperbolic tangent function reduced simply to 1. This holds true when the argument of the hyperbolic tangent equals about 3 (you can check for yourself). This means water is considered “deep” when

Equating the wavelength λ to the hull length *L*, this means that water is “deep” when it is deeper than about half the hull’s length. Again, for an 18′ 6″ racing hull this means water is “deep” when it is more than 9′ in depth.

In the shallow water case, where the depth *D* now becomes much less than the hull length *L*, the general expression for the surface wave phase speed simplifies to

In other words, in very shallow water the phase speed has nothing to due with the surface wave’s wavelength. It only depends on the gravitational constant *g* and the depth *D*. And as the depth decreases, the phase speed decreases as the square root of depth. Things have changed! But what happen in between? Is the transition from deep water behavior to shallow water behavior abrupt or smooth?

We can compute the phase speed vs. depth to answer these questions. We shall assume that the wavelength in the general phase speed equation corresponds to the length of an 18’6” C-1 running at 6.65 mph. The resulting data is plotted in Figure 6. As you can see, when the depth equals the boat length, or even about half the length, the phase speed equals the hull speed – the plot stays flat. The phase speed begins to decrease when the depth is about half the hull’s length – as expected give our analysis – and starts to drop significantly when the water depth is approximately one one quarter to one third of the hull’s length. This corresponds to what paddler’s refer to as “concrete water,” about 4’ to 5’ deep. As the water depth decreases further, the phase speed drops precipitously. What this means is that *the water inherently supports more slowly-moving waves as it becomes shallow*. If you try to paddle faster than the phase speed you are providing a forcing function that tries to make the surface waves faster than they want to be, and you pay the price in wave resistance.

*Figure 6: Phase speed vs. depth for an 18’6” hull.*

One can appreciate this increase wave drag effect by plotting the Froude Number vs. depth, as shown in Figure 7. Recall that the Froude Number expresses the ratio of hull speed to the phase speed, and is indicative of increasing wave resistance. Again, we consider the case of an 18’6” long C-1 traveling at 6.65 mph. This suggests that when the water depth drops below 2’, wave drag forces on the hull begin to skyrocket. Sound familiar?

Other factors come into play in shallow water besides decreasing surface wave phase speed. The water beneath the hull becomes “squeezed” between hull and bottom, which are now in close proximity to each other. This causes the water beneath and around the hull to accelerate, leading to a downward suction force on the hull because of Bernoulli’s Principle. This downward suction force is equivalent to having the paddlers suddenly gain weight, as the boat is pulled lower in the water. And as we learned in The Science of Paddling, Part 1, a heavier team is a slower team, everything else being equal, because now you are pushing a larger wetted hull area through the water than before you hit the shallows. Further, in shallow water surface waves grow in height, and eventually nonlinear effects take over – our analysis has been based upon the so-called “small wave” approximation, which soon fails as the wave height is no longer small compared to the water’s depth. At least it was fun while it lasted.

*Figure 7: Froude Number vs. depth.*

And finally, about those inside turns. Water tends to shallow as you get closer to shore, especially on the inside of river turns. Shallow water effects become more pronounced, then, closer to shore. We have seen in Figure 6 that the phase speed decreases as water becomes more shallow. One can construct a “map” of wave patterns over the width of a river using this information, as suggested in Figure 8.

*Figure 8: Wave refraction, (a) top view, and (b) river cross section.*

In Figure 8(a) the arrows represent the phase speed at the depth corresponding to their location. The shore is shown in green; the water, in blue. A longer arrow represents a greater phase speed; they also span the wavelength of one surface wave cycle. As the water becomes more shallow, the phase speed decreases, as does the wavelength. In order that there be no discontinuities in the wave pattern, the velocity vector must turn toward the shore. Thus the surface waves themselves *turn toward the shore* as the bottom becomes more shallow there. This phenomenon is called *refraction*, and is analogous to the bending of light waves at an air/water interface – viewing a pencil in a glass half full of water is a familiar example of this wave bending phenomena, albeit with light waves rather than water waves. So, as the surface water waves bend toward shore, they want to take you with them: toward the shore. And as if refraction wasn’t enough, because the wavelength shortens, the wave heights must increase in order to satisfy conservation of energy. Which is why you want to lead the way into a shallow inside turn, rather than follow on a leading boat’s inside shoulder. Surf’s up!

Copyright (c) 2017, Shawn Burke. All rights reserved.

(v4.1)

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