Part 10: Linearizing the Field

by Shawn Burke, Ph.D.

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):

${{T}_{adjusted}}=1\text{ hour}-2\text{ minutes}=58\text{ minutes}$

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):

${{T}_{adjusted}}=1\text{ hour}-2\text{ minutes}-2\text{ minutes}=56\text{ minutes}$

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 all bonuses are added together to determine the adjusted finishing times for the Points Series. Once computed, adjusted times are then sorted, Points awarded, the crowd cheers, etc.

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 V_2,men in quiet water (e.g., no current) via

${{V}_{2,men}}={{C}_{2}}^{\prime }\sqrt[3]{{{P}_{men}}+{{P}_{men}}}\centerdot {{L}^{-\frac{2}{3}}}$

The factor C₂’ 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

${{V}_{2}}={{C}_{2}}^{\prime }\sqrt[3]{{{P}_{men}}+\left( 1-2\delta \right){{P}_{men}}}\cdot {{L}^{-\frac{2}{3}}}$

I’ve chosen to express the power scaling factor as (1 – 2 $\delta$ ), where 0 ≤ $\delta$ ≤ 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 $\delta$ grows, and when $\delta$ = 0 you recover the tandem men’s result. Just remember: when $\delta$ = 0 you recover the men’s power P_men, and when $\delta$ 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,

$D={{V}_{2,men}}\cdot {{T}_{2,men}}={{V}_{2}}\cdot {{T}_{2}}\text{ }\to \text{ }{{T}_{2}}=\frac{{{V}_{2,men}}}{{{V}_{2}}}\cdot {{T}_{2,men}}$

where T_2,_men and T₂ denote the men’s finishing time and the finishing time of some other tandem hull, respectively. Note that the analysis that follows can be applied to C1s, kayaks, C4s, etc.; I’m just using the tandem canoe case as an example. Substituting for the velocities in terms of paddler power,

${{T}_{2}}=\frac{{{C}_{2}}^{\prime }\sqrt[3]{{{P}_{men}}+{{P}_{men}}}\cdot {{L}^{-\frac{2}{3}}}}{{{C}_{2}}^{\prime }\sqrt[3]{{{P}_{men}}\left( 1-2\delta \right){{P}_{men}}}\cdot {{L}^{-\frac{2}{3}}}}\cdot {{T}_{2,men}}$

Combining terms, and simplifying a bit, this expression for the “non-adjusted” tandem finishing time T₂ reduces to

${{T}_{2}}=\frac{\sqrt[3]{2{{P}_{men}}}}{\sqrt[3]{2{{P}_{men}}}\cdot \sqrt[3]{\left( 1-\delta \right)}}\cdot {{T}_{2,men}}=\frac{1}{\sqrt[3]{\left( 1-\delta \right)}}\cdot {{T}_{2,men}}$

As we see, if $\delta$ = 0 (tandem men’s case), T₂ = T_2,_men; check^[5]. And as $\delta$ grows the fraction gets larger, meaning that a tandem with less average paddling power will be slower than one with more average paddling power; check. But does this look anything like the formula for time adjustment used by NECKRA? I mean, with the cube root and all?

Sure it does! Subtracting a time bonus adjustment from the finishing time T₂ is the same as multiplying that finishing time by some number. That is,

${{T}_{2,adjusted}}={{T}_{2}}-\Delta {{T}_{2}}={{T}_{2}}\left( 1-\Delta \right)$

where $\Delta$ 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 T_2,_men – that’s the goal of the Point Series time adjustments. In other words,

${{T}_{2,adjusted}}={{T}_{2}}\left( 1-\Delta \right)={{T}_{2,men}}\text{ }\to \text{ }{{T}_{2}}=\frac{1}{\left( 1-\Delta \right)}\cdot {{T}_{2,men}}$

This takes the same general form – a multiplicative term – as we derived from the physics model, e.g., the non-adjusted tandem finishing time T₂ is equal to the men’s tandem finishing time T_2,_men multiplied by a scaling term:

${{T}_{2}}=\frac{1}{\sqrt[3]{1-\delta }}\cdot {{T}_{2,men}}$

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 $\Delta$ will represent the impact of the factor $\delta$ ; 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 $\delta$ , 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 $\delta$ . 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 $\delta$ is much smaller than one^[6]. Mathematically, we’ll write this requirement as $\delta$ << 1. So, regarding that cube root, its binomial expansion is

$\frac{1}{\sqrt[3]{1-\delta }}=1+\frac{\delta }{3}-\frac{2{{\delta }^{2}}}{9}+\ldots$

The ellipsis indicates that the expansion has many more terms involving higher powers of $\delta$ . Recall that we assumed $\delta$ 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

$\frac{1}{\sqrt[3]{1-\delta }}\simeq 1+\frac{\delta }{3}\text{ for }\delta \ll 1$

This means we can approximate our equation for the non-adjusted time T₂ as

${{T}_{2}}\simeq \left( 1+\frac{\delta }{3} \right)\cdot {{T}_{2,men}}\simeq {{T}_{2,men}}+\frac{\delta }{3}{{T}_{2,men}}\text{ for }\delta \ll 1$

That is, the non-adjusted time T₂ 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 $\delta$ . But recall that our factor $\delta$ was chosen such that when it has smaller values the scaled power is closer to that of the hypothetical male paddler, so when $\delta$ = 0 (no power scaling) T₂ 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 $\delta$ ₁ and $\delta$ ₂, in our “non-adjusted hull” vis

${{V}_{2}}={{C}_{2}}^{\prime }\sqrt[3]{\left( 1-2{{\delta }_{1}} \right){{P}_{men}}+\left( 1-2{{\delta }_{2}} \right){{P}_{men}}}\centerdot {{L}^{-\frac{2}{3}}}$

We can express the non-adjusted time T₂ then as

${{T}_{2}}=\frac{{{C}_{2}}^{\prime }\sqrt[3]{{{P}_{men}}+{{P}_{men}}}\cdot {{L}^{-\frac{2}{3}}}}{{{C}_{2}}^{\prime }\sqrt[3]{\left( 1-2{{\delta }_{1}} \right){{P}_{men}}+\left( 1-2{{\delta }_{2}} \right){{P}_{men}}}\cdot {{L}^{-\frac{2}{3}}}}\cdot {{T}_{2,men}}$

Combining terms, and doing a little simplifying,

${{T}_{2}}=\frac{\sqrt[3]{2{{P}_{men}}}}{\sqrt[3]{2{{P}_{men}}}\cdot \sqrt[3]{1-{{\delta }_{1}}-{{\delta }_{2}}}}\cdot {{T}_{2,men}}=\frac{1}{\sqrt[3]{1-\left( {{\delta }_{1}}+{{\delta }_{2}} \right)}}\cdot {{T}_{2,men}}$

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

${{T}_{2}}\simeq \left( 1+\frac{{{\delta }_{1}}}{3}+\frac{{{\delta }_{2}}}{3} \right)\cdot {{T}_{2,men}}={{T}_{2,men}}+\frac{{{\delta }_{1}}}{3}{{T}_{2,men}}+\frac{{{\delta }_{2}}}{3}{{T}_{2,men}}\text{ for }\left( {{\delta }_{1}}+{{\delta }_{2}} \right)\ll 1$

It is familiar in that the non-adjusted time T₂ 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 $\delta$ ₁ and $\delta$ ₂,

${{V}_{2}}={{C}_{2}}^{\prime }\sqrt[3]{{{P}_{men}}+\left( 1-2{{\delta }_{1}} \right)\cdot \left( 1-2{{\delta }_{2}} \right)\cdot {{P}_{men}}}\cdot {{L}^{-\frac{2}{3}}}={{C}_{2}}^{\prime }\sqrt[3]{{{P}_{men}}+\left( 1-2{{\delta }_{1}}-2{{\delta }_{2}}+4{{\delta }_{1}}{{\delta }_{2}} \right)\cdot {{P}_{men}}}\cdot {{L}^{-\frac{2}{3}}}$

Since we assumed that the sum of the factors $\delta$ ₁ and $\delta$ 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 $\delta$ ₁ $\delta$ 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

${{T}_{2}}=\frac{1}{\sqrt[3]{1-\left( {{\delta }_{1}}+{{\delta }_{2}} \right)}}\cdot {{T}_{2,men}}$

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 $\delta$ ₁ and $\delta$ ₂ are small the product term $\delta$ ₁ $\delta$ ₂ vanished, effectively “decoupling” the two factors and yielding a simple additive form for the time adjustments. One can continue to introduce multiple factors $\delta$ _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],

${{T}_{2}}\simeq \left( 1+\frac{{{\delta }_{1}}}{3}+\frac{{{\delta }_{2}}}{3}+\frac{{{\delta }_{3}}}{3}+\frac{{{\delta }_{4}}}{3} \right)\cdot {{T}_{2,men}}\text{ for }\left( {{\delta }_{1}}+{{\delta }_{2}}+{{\delta }_{3}}+{{\delta }_{4}} \right)\ll 1$

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 $\delta$ 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

$0.82=\left( 1-2\delta \right)\text{ }\to \text{ }\delta =0.09$

using the expression above for the power multiplier (1 – 2 $\delta$ ). 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 $\delta$ = 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 $\delta$ ₁ $\delta$ 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:

${{T}_{2}}=\frac{{{C}_{2}}^{\prime }\sqrt[3]{2{{P}_{men}}}\cdot {{L}^{-\frac{2}{3}}}}{{{C}_{2}}^{\prime }\sqrt[3]{{{f}_{1}}{{f}_{2}}{{P}_{men}}+{{f}_{3}}{{f}_{4}}{{P}_{men}}}\cdot {{L}^{-\frac{2}{3}}}}\cdot {{T}_{2,men}}$

Here, we’ve substituted f_i for each (1 – 2 $\delta$ _i) for notational simplicity.

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 f₁ represents a power scaling factor for female paddlers, and f₂ represents a power scaling factor based on age, this model suggests that you’d have to have a separate time bonus for each combination of gender and age, and each combination of these that comprise a canoe’s crew, to have the additive model work. For a C4 this means over 32,000 separate bonus time adjustments. And that’s just silly.^[9]

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!

v1.0

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? ↑