## 4 Data Processing  
Raw data from the ergometer were obtained at a sample rate of 10 Hz. The mean tangential force F (N) at both pedals, averaged over one revolution, as well as the corresponding mean pedal rate PR (rpm) were derived from these values.  
From the first 3 s of the acceleration phase of the 60-s sprint, three or four cycles with linear decay in pedal force and one or two cycles at pedal rates above 160 rpm from the motoric test were evaluated to establish the fatigue-free F/v and P/v profiles. Since PR is directly proportional to the tangential speed of motion v at the pedal, they were based on the mean cadence PR and corresponding mean crank force F.  
The force-velocity continuum was subjected to linear (F/v profile) and non-linear (P/v profile) regression analysis. The function  
\[F(v)=a\cdot PR+b \tag{1}\]  
approximates the relationship between mean pedal force F and the movement velocity PR in the absence of fatigue.  
P(v) was calculated by multiplying F(v) by PR:  
\[P(v)=a\cdot PR^{2}+b\cdot PR \tag{2}\]  
For purposes of statistical analysis, the following indices of performance were calculated:  
theoretical maximal force Fmax= F(0)= b,theoretical maximal velocity of movement \(\text{PR}_{\text{max}}=-\text{b}\cdot\text{a}^{-1}\), optimal cadence \(\text{PR}_{\text{opt}}=-\text{b}\cdot(2\text{a})^{-1}\), and maximum power output \(\text{P}_{\text{max}}=-\text{b}^{2}\cdot(4\text{a})^{-1}\).  
To analyze the fatigue-induced performance loss during the 60s-sprint data, the slope derived from the fatigue-free force-velocity profile (1) was kept fixed, in accordance with [PERSON] and colleagues [17]. When a time domain was added and the exercise duration was included, the two-dimensional F/v profiles yielded a three-dimensional model F:\(\mathbb{R}^{2}\)\(\rightarrow\mathbb{R}\) in maximal sprints no longer than \(\sim\) 60 s in duration:  
\[F(v,t)=F(v)\cdot e^{\frac{-t}{\tau F}}+c \tag{3}\]  
where F(v) is the fatigue-free F/v profile and \(\tau\)F denotes the time constant of its decline due to fatigue. Since the results of [PERSON] and [PERSON] [20], among others, described a steady state of power output after an exponential drop, a limiting value c was included in the function to represent this steady state.  
As the F/v profile shifted, the characteristic parameters of the fatigue-free profiles also decreased, allowing the three-dimensional model to be reduced to a two-dimensional model by eliminating the dependence on cadence. The fatigue-induced changes in the F/v can, therefore, be characterized by the alteration of a single parameter such as the optimal cadence. The calculation of the instantaneous F/v profile and theoretical optimal cadence for each data point i was based on the associated cadence \(\text{PR}_{\text{i}}\) and corresponding mean crank force \(\text{F}(\text{PR}_{\text{i}})\):  
\[b_{i}=F(PR_{i})-a\cdot PR_{i}\Rightarrow PR_{opt,i}=-b_{i}\cdot(2a)^{-1} \tag{4}\]  
The decay in \(\text{PR}_{\text{opt}}\) during the sprint was approximated by an exponential function of time t:  
\[PR_{opt}(t)=A\cdot e^{\frac{-t+TD}{\tau}}+c \tag{5}\]  
Its parameters were determined by non-linear regression with \(\text{A}=\text{PR}_{\text{opt}}(0)\)-c, c as the limiting value, and \(\tau\) as the time constant.  
A time delay (TD) was used to compensate for any delays in the onset of fatigue.  
Theoretical mean power output \(\text{P}_{\text{mean}}(\text{PR}_{\text{opt}})\) was calculated as the mean power output that could be generated at the calculated mean optimal cadence.  
To investigate the meaning of the mean optimal cadence mathematically, the cadence that maximizes power output was determined as follows by non-linear optimization:  
\[\underset{v\in R^{+}}{\text{max}}P(v)=a\cdot PR^{2}+b\cdot PR \tag{6}\]