On modelling bicycle power-meter measurements: Part II. Relations between rates of change of model quantities
OOn modelling bicycle power-meter measurements: Part II
Relations between rates of change of model quantities
Tomasz Danek ∗ , Michael A. Slawinski † , Theodore Stanoev ‡ May 12, 2020
Abstract
Power-meter measurements are used to study a model that accounts for the use of power bya cyclist. The focus is on relations between rates of change of model quantities, such as powerand speed, both in the context of partial derivatives, where other quantities are constant, andLagrange multipliers, where other quantities vary to maintain the imposed constraints.
Using power meters to study cycling performance allows us to gain quantitative information aboutrelations whose qualitative aspects are known based on observations; for instance, riding with a givenspeed with a tailwind requires less effort than riding with the same speed against a headwind. Thequantification of such a relation, however, is necessary to proceed with an optimization to achieve —under constraints imposed by the capacity of a cyclist — the least time to cover a distance that issubject to winds and contains flats, hills and descents.Many studies examine the physics of cycling. This article is the second part of Danek et al (2020),which also contains the pertinent bibliography. Herein — using a model relating power-meter mea-surements to the motion of a bicycle, examined by Danek et al (2020) — we formulate expressionsthat allow us to quantify relations between the rates of change of parameters contained within thismodel.We begin this paper by presenting the expression to account, by modelling, for the values measuredby a power meter. Using this expression and the implicit function theorem, we derive explicitexpressions of the rates between the model parameters. We complete this paper by interpreting andcomparing quantitive results based on power-meter and GPS measurements collected on a flat courseand an inclined course. Both are in Northwestern Italy; the former is between Rivalta Bormidaand Pontechino, in Piemonte; the latter is between Rossiglione and Tiglieto, in Liguria. In theappendices, we compare the flat-course results to optimizations based on the Lagrange multipliers,and comment on the bijection between the generated power and the bicycle speed. ∗ AGH–University of Science and Technology, Krak´ow, Poland, [email protected] † Memorial University of Newfoundland, Canada, [email protected] ‡ Memorial University of Newfoundland, Canada, [email protected] a r X i v : . [ phy s i c s . pop - ph ] M a y Formulation
To account for the cyclist’s use of the power measured by a power meter, we consider, P = F ← V → , (1)where P stands for the value of the required power, F ← for the forces opposing the motion, and V → for the ground speed of the bicycle.Explicitly, we assume that (e.g., Danek et al, 2020) P = mg sin θ + m a + C rr mg cos θ + η C d A ρ ( V → + w ← ) − λ (cid:124) (cid:123)(cid:122) (cid:125) F ← V → ; (2)herein, m is the mass of the cyclist and the bicycle, g is the acceleration due to gravity, θ is the slope, a is the change of speed, C rr is the rolling-resistance coefficient, C d A is the air-resistance coefficient, ρ is air density, V → is the ground-speed of the bicycle, w ← is the wind component opposing themotion, λ is the drivetrain-resistance coefficient, η is a quantity that ensures the proper sign for thetailwind effect, w ← < − V → ⇐⇒ η = − η = 1 ; throughout this work, η = 1 .To estimate quantities that appear on the right-hand side of equation (2) — specifically, C d A , C rr and λ — given the measurement, P , we write f = P − mg sin θ V → + m a V → + C rr mg cos θ V → + η C d A ρ ( V → + w ← ) V → − λ (cid:124) (cid:123)(cid:122) (cid:125) F ← V → , (3)and minimize the misfit, min f , as discussed by Danek et al (2020). We seek relations between the ratios of quantities on the right-hand side of equation (2). To doso — since f , stated in expression (3), possesses continuous partial derivatives in all its variables atall points, except at λ = 1 , which is excluded by mechanical considerations, and since f = 0 , as aconsequence of equation (1) — we invoke the implicit function theorem to write ∂y∂x = − ∂f∂x∂f∂y =: − ∂ x f∂ y f , (4)where x and y are any two quantities among the arguments of f ( P, m, g, θ, a, C rr , C d A , ρ, V → , w ← , λ ) . (5) To use formula (4), in the context of expression (3), we obtain all partial derivatives of f , withrespect to its arguments. ∂ P f = 12 m f = − ( a + g (C rr cos θ + sin θ )) V → − λ∂ θ f = − m g (cos θ − C rr sin θ ) V → − λ∂ a f = − m V → − λ∂ C rr f = − m g cos θ V → − λ∂ C d A f = − η ρ ( V → + w ← ) V → − λ ) ∂ ρ f = − η C d A ( V → + w ← ) V → − λ ) ∂ V → f = − m a + η ρ C d A ( V → + w ← ) (3 V → + w ← ) + 2 m g (C rr cos θ + sin θ )2 (1 − λ ) ∂ w ← f = − η ρ C d A ( V → + w ← ) V → − λ∂ λ f = − (cid:16) m a + η ρ C d A ( V → + w ← ) + 2 m g (C rr cos θ + sin θ ) (cid:17) V → − λ ) In accordance with the definition of a partial derivative, all variables in expression (5) are constant,except the one with respect to which the differentiation is performed. This property is apparent inAppendix A.3, where we examine a relation between differences and derivatives.
To use the partial derivatives stated in Section 2.3, we consider measurements collected duringtwo rides. The flat-course measurements correspond to a nearly flat course. The inclined-coursemeasurements correspond to an uphill with a nearly constant inclination. For both the flat andinclined course, we let m = 111 and g = 9 .
81 . Other values are stated in Sections 2.4.2 and 2.4.3.
According to Danek et al (2020), the flat-course input values are as follows. The average measuredpower, P = 258 . ± . V → = 10 . ± . d A = 0 . ± . rr = 0 . ± . λ = 0 . ± . w ← = 0 = ⇒ η = 1 and ρ = 1 . ± . θ = 0 . ± . a = 0 . ± . θ = a = 0 .The corresponding values of partial derivatives, formulated in Section 2.3, are listed in the left-handcolumn of Table 1. For consistency with power meters, whose measurements are expressed in watts, which are kg m / s , we use the SI units for all quantities. Mass is given in kilograms, kg , length in metres, m , and time in seconds, s ; hence, speed isin metres per second, change of speed in metres per second squared, force in newtons, kg m / s , and work and energyin joules, kg m / s ; angle is in radians. artial derivative Flat course Inclined course ∂ P f ∂ m f − . ± . − . ± . ∂ θ f − . ± . − . ± . ∂ a f − . ± . − . ± . ∂ C rr f − . ± . − . ± . ∂ C d A f − . ± . − . ± . ∂ ρ f − . ± . − . ± . ∂ h f . ± . . ± . ∂ V → f − . ± . − . ± . ∂ w ← f − . ± . − . ± . ∂ λ f − . ± . − . ± . Table 1:
Values of partial derivatives — following formulæ in Section 2.3 — with the power-meter and GPSmeasurements collected on a flat and inclined courses igure 1: Left-hand plot: ground speed, from GPS measurements, V → = 4 . ± . P = 286 . ± . Figure 2:
Optimal values; left-hand plot: C d A = 0 . ± . rr = 0 . ± .
011 ;right-hand plot: λ = 0 . ± . Following the method outlined by Danek et al (2020), we calculate the inclined-course values. Thedata are grouped in eleven speed intervals whose centres range from 3 . . . V → = 4 . ± . P = 286 . ± . d A = 0 . ± . rr = 0 . ± .
011 , λ = 0 . ± . θ = 0 . ± . . ◦ and 4 .
60% . Thiscourse is known among cyclists of the region as a particularly constant incline. The change ofspeed throughout the ride is a = 0 . ± . θ = 0 . a = 0 . Since the change of altitude isnegligible — in the context of air density — we set ρ = 1 . ± . w ← = 0 = ⇒ η = 1 .The corresponding values of partial derivatives, formulated in Section 2.3, are listed in the right-handcolumn of Table 1. 5 igure 3: Misfit of equation (3): left-hand plot: flat course, f = 0 . ± .
321 ; right-hand plot: inclinedcourse, f = 2 . ± . As required by the implicit function theorem and as shown in Figure 3, f = 0 , in the neighbourhoodof the maxima of the distributions, for both the flat and inclined courses. Also, as required by thetheorem, in formula (4), and as shown in Table 1, ∂ y f (cid:54) = 0 , in the neighbourhoods of interest, foreither course.Notably, the similarity of a horizontal spread for both plots of Figure 3 indicates that the goodnessof fit of a model is similar for both courses. The spread is slightly narrower for the inclined course;this might be a result of a lower average speed, V → , which allows for more data points for a givendistance and, hence, a higher accuracy of information. The misfit minimization of equation (3), min f , treats C d A , C rr and λ as adjustable parameters.The values in Table 2 are the changes of C d A due to a change in C rr or λ ; in either case, the otherquantities are kept constant. Let us examine the first row.For the flat course — in the neighbourhood of V → = 10 .
51 and P = 258 . d A = 0 . rr = 0 . ∂ C rr C d A = − . ∂ C d A C rr = − . d A) = ∂ C d A ∂ C rr d(C rr ) = − . rr )and d(C rr ) = ∂ C rr ∂ C d A d(C d A) = − . d A) ;in other words, an increase of C rr by a unit corresponds to a decrease of C d A by 16 . d A by a unit corresponds to a decrease of C rr by 0 . d A)C d A (cid:12)(cid:12)(cid:12)(cid:12) C d A=0 . = 10 . , artial derivative Flat course Inclined course ∂ C rr C d A − . ± . − . ± . ∂ λ C d A − . ± . − . ± . Table 2:
Model rates of change following formula (4) and values in Table 1 which is an increase of about 384% , corresponds tod(C rr )C rr (cid:12)(cid:12)(cid:12)(cid:12) C rr =0 . = − . . , which is a decrease of about 2644% .For the inclined course — in the neighbourhood of V → = 4 .
138 and P = 286 . d A =0 . rr = 0 . ∂ C rr C d A = − .
754 and its reciprocal is ∂ C d A C rr = − . d A by about 1 / . rr by about 0 . / . d A by 1% corresponds to a decrease ofC rr by 6 .
89% , for the flat course, and a decrease of only 0 .
19% , for the inclined course. This resultquantifies that the dependence between C d A and C rr , within adjustments of the model, is morepronounced for the flat course than for the inclined course, as expected in view of expression (2),whose value—for the inclined course—is dominated by the first summand in the numerator, whichincludes neither C d A nor C rr . This result provides a quantitative justification for the observationthat the dependance of the accuracy of the estimate of power on the accuracies of C d A and C rr varies depending on the context; it is more pronounced on flat and fast courses.Similar evaluations can be performed using the values of derivatives contained in the second row ofTable 2. Therein, an increase in λ results in a decrease of C d A , with different rates, for the flat andinclined courses.
Physical inferences — based on minimization of expression (3) — are accurate in a neighbourhoodof V → and P , wherein the set of values for C d A , C rr and λ is estimated, since, as discussed inSection 3.1, these values — in spite of their distinct physical interpretations — are related amongeach other by the process of optimization of the model.In view of expression (2), and as illustrated in Figure 4, power as a function of ground speedis a cubic. The inflection point of the curve corresponds to the speed for which there is no airresistance, since the ground speed is equal to the tailwind, V → = − w ← . At that point, P is thepower to overcome the rolling and drivetrain resistance, only. To the left of that point, the empiricaladequacy of expression (2) is questionable. However, for the results presented in this article, weconsider the cases of w (cid:29) − V → , which are well to the right of the inflection point.The values in Table 3 are the changes of ground speed due to a change in power, mass, slope andwind; in each case, the other quantities are kept constant. These values allow us to answer suchquestions as what increase of speed would result from an increase of power by 1 watt? To answerthis question, let us examine the first row. 7 igure 4: Power as a function of speed ceteris paribus
For the flat course — considering the neighbourhood of V → = 10 .
51 and P = 258 . ∂ P V → =0 . ∂ V → P = 56 . P = ∂P∂V → d V → = 56 . V → ;in other words, an increase of V → by a unit requires an increase of P by 56 . . For the inclined course — considering the neighbourhood of V → = 4 .
138 and P = 286 . ∂ P V → =0 . ∂ V → P = 74 . . V → V → (cid:12)(cid:12)(cid:12)(cid:12) V → =10 . = 110 . , (6)which is a increase in speed of about 9 .
5% , requiresd PP (cid:12)(cid:12)(cid:12)(cid:12) P =258 . = 56 . . , which is an increase in power of about 22% . For the inclined course, d V /V = 1 / .
138 and d
P/P =74 . / . .
3% , for the flat course, and an increase of only about 1 .
1% , for the inclined course.This result provides a quantitative justification for a time-trial adage of pushing on the uphills andrecovering on the flats, to diminish the overall time.Since, as illustrated in Figure 4, the slope of the tangent line changes along the curve, the valueof expression (4) corresponds to a given neighbourhood of pairs, V → and P . Our interpretation is ∂P/∂V can be also found by differentiating expression (2) with respect to V → . However, the implicit functiontheorem allows us to obtain relations between quantities, without explicitly expressing one in terms of the other.Also, in accordance with the inverse function theorem, ∂V/∂P = 1 / ( ∂P/∂V ) , which is justified by the fact that, forexpression (2), ∂P/∂V → (cid:54) = 0 , in the neighbourhood of interest, as required by the theorem. However, expression (4),which states the implicit function theorem, provides a convenience of examining the relations between the rates ofchange of any two quantities without invoking the inverse function theorem and requiring an explicit expression foreither of them. artial derivative Flat course Inclined course ∂ P V → . ± . . ± . ∂ m V → − . ± . − . ± . ∂ θ V → − . ± . − . ± . ∂ w ← V → − . ± . − . ± . Table 3:
Physical rates of change following formula (4) and values in Table 1 tantamount to comparing the slopes of two such curves — one corresponding to the model of the flatcourse and the other of the inclined course — at two distinct locations, ( V → , P ) = (10 . , .
8) and( V → , P ) = (4 . , .
6) . Even though the slope of the tangent line changes, it is positive for allvalues. This means that the function is monotonically increasing, even though it is a third degreepolynomial. In other words, the relation of power and speed is a bijection, as illustrated in Figure 4and as discussed in Appendix B.Let us examine the second row of Table 3. For the flat course, ∂ m V → = − . ∂ V → m = − .
936 . This means that an increase of speed by 1 metre per second — dueonly to the loss of mass — requires a decrease of mass of about 229 kilograms. For the inclinedcourse, ∂ m V → = − . ∂ V → m = − . . m/m = 228 . /
111 , which is a decrease of mass of about 206% . For the inclined course,d
V /V = 1 / .
138 ; hence, an increase in speed by about 24% requires d m/m = 30 . /
111 , whichis a decrease of mass of about 27% . Remaining within a linear approximation, for the flat course,an increase of speed by 1% requires a decrease of mass of about 22% , and for the inclined course, itrequires a decrease of mass of only about 1% .This is supportive evidence of an empirical insight into the importance of lightness for climbing; incontrast to flat courses, in the hills, even a small loss of weight results in a noticeable advantage.Also, this result can be used to quantify the importance of the power-to-weight ratio, which playsan important role in climbing, but a lesser one on a flat.Similar evaluations can be performed using the values of derivatives contained in the third and fourthrows of Table 3. In both cases, the sign is negative; hence, as expected, the increase of steepness orheadwind results in a decrease of speed. These rates of decrease, which are different for the flat andinclined courses, can be quantified in a manner analogous to the one presented in this section.
The results derived herein are sources of information for optimizing the performance in a time trialunder a variety of conditions, such as the strategy of the distribution of effort over the hilly and flatportions or headwind and tailwind sections. For instance, examining ∂ w ← V → , for a flat course, wecould quantify another time-trial adage of pushing against the headwind and recovering with the9ailwind, to diminish the overall time, under a constraint of cyclist’s capacity; such a conclusion isillustrated in Appendix A. A further insight into this statement is provided by the following example.Let us consider a five-kilometre section against the headwind, w ← = 5 , and, following a turnaround,the same five-kilometre section with the tailwind, w ← = − P = 258 . V → = 8 . V → = 14 . V → = 8 . V → = 13 . V → = 11 . P = 531 . (cid:29) . not too much , because going too far slows you down again as youapproach the equal-speeds scenario.A quantification of this, and another, rule of thumb of Anton (2013) is presented in Appendix A,where we question their generality.Also, results derived in this paper allow for a quantitative evaluation of the aerodynamic efficiencyand — for team time trials — of the efficiency of drafting. Under various conditions, there are differentrelations between the rates of change of quantities in question. In this paper, as a consequence of theimplicit function theorem, relations between the rates of change of all quantities that are includedin a model are explicitly stated, and each relation can be evaluated for given conditions.Furthermore, the derived expressions allow us to interpret the obtained measurements in a quanti-tative manner, since the values of these expressions entail concrete issues to be addressed for a givenbicycle course. The reliability of information — which depends on the accuracy of measurements andthe empirical adequacy of a model — is quantified by a misfit and by standard deviations of modelparameters. Also, using partial derivatives listed in Section 2.3, we can write the differential of P ,and, hence, estimate its error inherited from the errors of other quantities,d P = ∂ m P d m + · · · + ∂ λ P d λ , where, in accordance with equation (4) and in view of ∂ P f = 1 , ∂ m P = − ∂ m f, . . . , ∂ λ P = − ∂ λ f .10 igure A1: Joseph-Louis (Giuseppe Luigi) Lagrange examining his optimizations based on the fact that,in general, in accordance with expression (2), headwinds, w ← > w ← < w ← ≮ − V → . A Time minimization with Lagrange multipliers
A.1 Preliminary remarks
Consider a flat course of length d , whose one half is covered against the wind, as illustrated inFigure A1, and the other half with the wind. To minimize the time, t , we need to maximize theaverage speed, V → = dt = dd V U + d V D = 2 V U V D V U + V D , (A.1)where V U and V D are the speeds on the upwind and downwind sections, respectively. The maximumof this function occurs for all values along V U = V D . To get a pair of values that correspondsto a realistic scenario, we invoke the method of Lagrange multipliers and find the maximum ofspeed (A.1), subject to constraints. To do so, we state the problem as a Lagrangian function of twovariables with n constraints, L ( V U , V D ) = V → ( V U , V D ) + Λ Γ ( V U , V D ) + · · · + Λ n Γ n ( V U , V D ) , (A.2)where Λ i , with i = 1 , . . . , n , is a Lagrange multiplier. The optimization is achieved at the stationarypoints of function (A.2), which we find by solving the system of equations, ∂L∂V U = 0 , ∂L∂V D = 0 , ∂L∂ Λ = 0 , . . . , ∂L∂ Λ n = 0 , (A.3)whose solution is the pair, V U , V D , that extremizes expression (A.1) and satisfies the constraints, Γ i ,where i = 1 , . . . , n , within the physical realm. 11 .2 Constraint of total work Let us impose a constraint in terms of the amount of total work, W = W U + W D , to be done bya cyclist on the upwind and downwind sections, whose proportions of length are stated in expres-sion (A.1),Γ W = W U (cid:122) (cid:125)(cid:124) (cid:123) C rr m g + C d A ρ ( V U + w ← ) − λ (cid:124) (cid:123)(cid:122) (cid:125) F ← d W D (cid:122) (cid:125)(cid:124) (cid:123) C rr m g + C d A ρ ( V D − w ← ) − λ (cid:124) (cid:123)(cid:122) (cid:125) F ← d − W = 0 . (A.4)Herein, we assume W = C rr m g + C d A ρ V → − λ (cid:124) (cid:123)(cid:122) (cid:125) F ← d to be the total amount of energy available to the cyclist, which corresponds to the work done on thesame course, with a maximum effort — with no wind, w ← = 0 — resulting in a given value of V → .We write function (A.2) as L W = V → + Λ W Γ W . (A.5)Considering d = 10000 , model parameters stated in Section 2.4.2, namely, m = 111 , g = 9 .
81 , ρ = 1 . d A = 0 . rr = 0 . λ = 0 . V → = 10 .
51 , we obtain W = 205878 . To minimize the traveltime with w ← = 5 , we write system (A.3), in terms offunction (A.5), ∂L W ∂V U = 2 V D ( V D + V U ) + Λ W (1627 . V U + 8138 .
32) = 0 ,∂L W ∂V D = 2 V U ( V D + V U ) + Λ W (1627 . V D − .
32) = 0 ,∂L W ∂ Λ W = 813 . (cid:0) V U + V D (cid:1) + 8138 .
32 ( V U − V D ) − . (A.6)Solving system (A.6) numerically, we obtain a single physical solution, V U = 8 . V D = 11 . V → = 9 . w ← = 0 , namely, V → = 10 .
51 . This quantifies an adage that ridingwith the wind does not compensate for the speed lost by riding against the wind. The loss is dueto the dissipation of energy due to the air, rolling and drivetrain resistances, which are present onboth the upwind and downwind sections.
A.3 Constraint of average power
Let us impose a constraint in terms of the value of average power, P , maintained by a cyclist on theupwind and downwind sections. In contrast to work, power is not a cumulative quantity. Hence, the12istance does not appear explicitly in a constraint, and we require constraints for both the upwindand downwind sections, Γ P U = P U (cid:122) (cid:125)(cid:124) (cid:123) C rr m g + C d A ρ ( V U + w ← ) − λ (cid:124) (cid:123)(cid:122) (cid:125) F ← V U − P , (A.8)Γ P D = P D (cid:122) (cid:125)(cid:124) (cid:123) C rr m g + C d A ρ ( V D − w ← ) − λ (cid:124) (cid:123)(cid:122) (cid:125) F ← V D − P . (A.9)Herein, we assume P = C rr m g + C d A ρ V → − λ (cid:124) (cid:123)(cid:122) (cid:125) F ← V → to be the average power available to the cyclist, which corresponds to the average power achievedon the same course, with a maximum effort — with no wind, w ← = 0 — resulting in a given value of V → . Function (A.2) is L P = V → + Λ P U Γ P U + Λ P D Γ P D . (A.10)For m = 111 , g = 9 .
81 , ρ = 1 . d A = 0 . rr = 0 . λ = 0 . V → = 10 .
51 , weobtain P = 216 .
378 . To minimize the traveltime with w ← = 5 , we write system (A.3), in terms offunction (A.10), ∂L P ∂V U = 2 V D ( V D + V U ) + Λ P U (cid:0) . V U + 3 . V U + 6 . (cid:1) = 0 ,∂L P ∂V D = 2 V U ( V D + V U ) + Λ P D (cid:0) . V D − . V D + 6 . (cid:1) = 0 ,∂L P ∂ Λ P U = 0 . V U + 1 . V U + 6 . V U − .
378 = 0 ,∂L P ∂ Λ P D = 0 . V U − . V U + 6 . V U − .
378 = 0 . The single physical solution is V U = 7 . V D = 13 . , (A.11)which both maximizes expression (A.1) and satisfies constraints (A.8) and (A.9). The correspondingaverage is V → = 9 . A.4 Relation between differences and derivatives
To conclude this appendix, let us comment on difference ∆ V → / ∆ w ← , discussed herein, in the contextof ∂ w ← V → , whose value is presented in Table 3. Partial derivatives correspond to a tangent to acurve at a point, and the differences to a secant over a segment of the curve. Also, partial derivativesare obtained under the assumption that all other quantities are constant.13he latter requirement is satisfied in Appendix A.3, where∆ V → ∆ w ← = V U − V D w ← − ( − w ← ) = 7 . − . − . , which agrees with ∂ w ← V → , in Table 3, to two decimal points. For w ← = 0 .
05 , we obtain V U =10 . V D = 10 . V → / ∆ w ← = − . ∂ w ← V → to sixdecimal points. In general, lim ∆ w ← → ∆ V → ∆ w ← = ∂V → ∂w ← , as expected, in view of a secant approaching a tangent.The requirement of constant quantities is not satisfied in Appendix A.2, since P is allowed to varyto maintain the imposed value of W . In Appendix A.3, W varies to maintain the imposed valueof P , but W is not a variable in function (5), used in partial derivatives.As shown in this appendix, properties of partial derivatives need to be considered in examiningtime-trial strategies. In contrast to common optimization methods, partial derivatives correspondto a change of a single variable, only. A.5 Closing remarks
Let us examine the constraints discussed in this appendix in terms of required powers. For the workconstraint, the average speed is V → = 9 . P U = 365 .
537 and P D = 59 . P = 173 .
316 . Thus, P U is significantly greater than P .For the power constraint, with P = 216 .
378 , the average speed is V → = 9 . v . [ . . . ] Endeavor to ride at v ∼ = v + w/ v ∼ = v − w/ V → = 10 .
51 =: v to be a target speed, with w = 5 , speeds (A.11), which result fromthe power constraint, are less congruent with this rule than speeds (A.7), which result from the workconstraint, yet—according to the present analysis—speeds (A.11) appear to be preferable. This isan indication of further subtleties that need to be considered in developing a time-trial strategy. B One-to-one relation between power and speed
Proposition 1.
According to model (2), with a = 0 and η = 1 , the relation between the measuredpower, P , and the bicycle speed, V → , is one-to-one.Proof. It suffices to show that ∂P/∂V → > V → ∈ (0 , ∞ ) . Since ∂P∂V → = (C rr cos θ + sin θ ) g m + C d A ρ V → − λ , λ (cid:28) ∂P/∂V → > ∂ P/∂V → > η = ± V → ∈ (0 , ∞ ) . Another consequence is that — ceteris paribus — the increase of speed requires increase of power, and an increase of power resultsin an increase of speed. As illustrated in Figure 4, this remains true even for η = − Acknowledgements
We wish to acknowledge Len Bos, for fruitful discussions, David Dalton, for his scientific editingand proofreading, Elena Patarini, for her graphic support, and Roberto Lauciello, for his artisticcontribution. Furthermore, we wish to acknowledge Favero Electronics for inspiring this study bytheir technological advances and for supporting this work by providing us with their latest model ofAssioma Duo power meters.
Conflict of Interest
The authors declare that they have no conflict of interest.