3.1. Time trial

The results from all individual TTs in Tour de France from 2010 to 2019 are presented in table 1. For the winner, the average velocity $\bar {V}_{TT}=D/T$, defined as the ratio between the distance $D$ of the TT and his time $T$, is indicated in column 9. Even if the distance of TT changes from $6.4\ \mbox {km}$ to $54\ \mbox {km}$ we observe that flat TTs (type FLT in blue) are covered with a mean velocity of the order of $52.5 \pm 2.5\ \mbox {km}\ \mbox {h}^{-1}$ while TTs in mountain regions (type MT in red) have velocities of the order of $35 \pm 2\ \mbox {km}\ \mbox {h}^{-1}$. We first address the limit of flat TTs and treat the general case in a second step.

Table 1.

Results from all individual TTs in Tour de France from 2010 to 2019. Flat TTs are indicated with (FLT), hilly time trials with (SMT) and those in mountains with (MT). The name of the winner is given together with his height $L_c$ and mass $M_c$. The distance $D$ of the TT is given together with the best time $T$. From these values, we calculate the actual average velocity $\bar {V}_{TT}=D/T$. The expected power is calculated using (3.3) with the time of the race $T$ and the mass $M_c$ of the winner. The predicted velocity $V_{TT}(T)$ is calculated using (3.5). The error between $\bar {V}_{TT}$ and $V_{TT}(T)$ is given in the last column.

3.1.1. Flat time trial

Since TT is observed to be a steady effort ($\textrm {d}/\textrm {d}t=0$), (3.4) states that, in the limit of FLT ($\alpha =0$), the mechanical power is mainly injected in aerodynamical friction so that the equation for flat TT reduces to $P_m\approx 1/2\rho SC_D V^3$. To maximize his velocity the rider will select $P_m=P_{max}$ and for long time efforts ($T\gg \tau$), the power $P_m$ will reach the quasi-plateau associated with aerobic efforts presented in figure 3(c). One thus deduces the characteristic velocity for flat TT

(3.5)\begin{equation} V_{TT}(T)=\left(\frac{2P_{max}(T)}{\rho SC_D}\right)^{1/3}. \end{equation}

Using $P_{max}(T)=\varPi M_c [1-\ln (T/\tau )^{\gamma }]$ with $\varPi =9.7\ \mbox {W}\ \mbox {kg}^{-1}$, $\tau =32$ and $\gamma =0.0886$ we evaluate $P_{max}(T)$ for each rider in table 1 (column 10).

Concerning the drag area, $SC_D$, recent experimental and numerical studies have revealed the complex structure of the flow around the cyclist (Crouch et al. Reference Crouch, Burton, Brown, Thompson and Sheridan2014; Hosoi Reference Hosoi2014). This complexity is illustrated in figure 4(a) using the isosurfaces of average streamwise vorticity (reproduced from Griffith et al. Reference Griffith, Crouch, Thompson, Burton, Sheridan and Brown2014). A summary of the values of $SC_D$ found in wind tunnels and reported in the literature is presented in figure 4(b). For the TT position we observe that $SC_D\approx 0.25\ \mbox {m}^{-2}$. Using $\rho =1.2\ \mbox {kg}\ \mbox {m}^{-3}$ we calculate $V_{TT}(T)$ with (3.5) and evaluate the error with the actual value $\bar {V}_{TT}$ in the last column of table 1. For all flat TT (FLT in blue) we observe that the error is smaller than $8\,\%$. For hilly (SMT) and mountain types (MT) the velocity $V_{TT}$ predicted by assuming $\alpha =0$ is unsurprisingly larger than the actual one and the discrepancy can reach 50 %.

Figure 4.

Isosurfaces of average streamwise vorticity behind a cyclist as revealed by Griffith et al. (Reference Griffith, Crouch, Thompson, Burton, Sheridan and Brown2014). (b) Reported drag area measurements from wind tunnel testing of cyclists in different positions and different studies: study 1 is Davies (Reference Davies1980), 2 is Kyle & Burke (Reference Kyle and Burke1984), 3 is Martin et al. (Reference Martin, Milliken, Cobb, McFadden and Coggan1998), 4 is Padilla et al. (Reference Padilla, Mujika, Angulo and Goiriena2000), 5 is Jeukendrup & Martin (Reference Jeukendrup and Martin2001), 6 and 7 are Garcia-Lopez et al. (Reference Garcia-Lopez, Rodriguez-Marroyo, Juneau, Peleteiro, Martinez and Villa2008), 8 is Gibertini & Grassi (Reference Gibertini and Grassi2008), 9 is Underwood et al. (Reference Underwood, Schumacher, Burette-Pommay and Jermy2011), 10 is Defraeye et al. (Reference Defraeye, Blocken, Koninckx, Hespel and Carmeliet2010), 11 is Barry et al. (Reference Barry, Burton, Sheridan, Thompson and Brown2014) and 12 is Barry et al. (Reference Barry, Burton, Sheridan, Thompson and Brown2015).

3.1.2. Non-flat time trial

While studying TT performance in § 3.1.1, we underlined that the velocity $V_{TT}$ predicted by (3.5) only holds in the limit of a flat road ($\alpha =0$). When this limit is not achieved, observations show that the average velocity is significantly reduced (table 1). For the case of the individual TT of the 18th stage of the Tour de France 2016, the road profile is clearly not flat, as illustrated in figure 5(a). As presented in table 1, Froome wins the stage with a mean velocity of $33.2\ \mbox {km}\ \mbox {h}^{-1}$, far below the $50\mbox { km/h}$ predicted by (3.5) for the flat TT limit.

Figure 5.

(a) Road profile for the mountain type individual time trial of stage 18 in tour de France 2016 between Sallanches and Megève (Photo credits: AFP/Presse Sport). (b) Table of comparison for the race of the winner C. Froome: $s$ is the distance from departure, $\alpha$ the slope of the road average over a km, $t_{meas}$ is the time measured during the race at 4 different locations (6.5 km, 10 km, 13.5 km and arrival), $V_{TTc}$ is the velocity computed using (3.7), $t_{calc}$ is the time calculated at each km. The last column compares $t_{calc}$ and $t_{meas}$.

To account for gravity, one needs to reconsider the equation of motion (3.4) without the unsteady term (TT is a steady regime) but with the gravitational contribution: $P_m=1/2\rho S C_D V^3+Mg\alpha V$. Using the flat limit expression $V_{TT}(T)=(2P_{max}(T)/\rho S C_D)^{1/3}$, this equation can be re-written as

(3.6)\begin{equation} \left[\frac{V_{TTc}(s)}{V_{TT}}\right]^3+F \alpha(s)\left[\frac{V_{TTc}(s)}{V_{TT}}\right]=1, \end{equation}

where $V_{TTc}(s)$ is the velocity at the location $s$ and $F=MgV_{TT}(T)/P_{max}(T)$. Using Viete's substitution, $V_{TTc}(s)/V_{TT}=Y-\alpha F/3Y$ (3.6) is transformed into the quadratic form $Z^2-Z-(\alpha F/3)^3=0$, where $Z=Y^3$. Finally, one gets the exact solution for the velocity as a function of the slope $\alpha$

(3.7)\begin{equation} \frac{V_{TTc}(\alpha)}{V_{TT}}=\left(\frac{1+\sqrt{1+4(\alpha F/3)^3}}{2}\right)^{1/3}-\frac{\alpha F/3}{\left(\dfrac{1+\sqrt{1+4(\alpha F/3)^3}}{2}\right)^{1/3}} . \end{equation}

In the small slope limit ($\alpha F/3\ll 1$), this expression reduces to $V_{TTc}(\alpha )=V_{TT}(1-\alpha F/3)$. The velocity decrease is thus directly proportional to the slope. It also depends on the athlete characteristics via the parameter $F=MgV_{TT}/P_m=(2M^3g^3/\rho S C_DP_m^2)^{1/3}$. The larger this parameter the larger the relative velocity decrease. In the case of Froome in this stage, one finds $P_{max}=409.3 \mbox {W}$, $V_{TT}=13.97\ \mbox {m}\ \mbox {s}^{-1}$ ($50.3\ \mbox {km}\ \mbox {h}^{-1}$) and $F=22.1$.

Using the slopes $\alpha (s)$ in the different sections of the stage presented in figure 5(a), one can then evaluate the corresponding velocity $V_{TTc}(\alpha (s))$ using (3.7) and deduce the time $t_{calc}(s)=\int _{0}^{s}\textrm {d}s'/V_{TTc}(\alpha (s'))$. This time is presented in the fifth column of figure 5(b). It can be compared to the 4 intermediate times measured during the race at the locations 6.5 km, 10 km, 13.5 km and 17 km. These times are reported in the third column $t_{meas}$. The error between the estimated and the actual time never exceed 3 % and the mean velocity we calculate is $33.4\ \textrm {km}\ \textrm {h}^{-1}$, very close to the $33.2\ \textrm {km}\ \textrm {h}^{-1}$ reported in table 1.

3.2. High mountains

The characteristics of some emblematic mountain climbs associated with Tour de France together with the records reproduced from the book of Vayer & Portoleau (Reference Vayer and Portoleau2001) are presented in table 2.

Table 2.

Some classic mountain climbs and their records reproduced from Vayer & Portoleau (Reference Vayer and Portoleau2001). The average slope is indicated in the second column, the length of the climb $D$ in the third and the record time $T$ in the fifth. From these values we calculate the average velocity $\bar {V}_{MT}=D/T$ (column 6). The record holder is given in 4 and his height and mass in 7 and 8. Using (3.3) we evaluate the power $P_{max}(T)$ in column 9. The velocity $V_{MT0}$ predicted by (3.8) is given in column 10. The corrected velocity $V_{MT1}$ accounting for the aerodynamic drag and rolling resistance is calculated with (3.9) in column 11. The error between the predicted velocity $V_{MT1}$ and the actual velocity $\bar {V}_{MT}$ is indicated in the last column.

The typical mean slope is $8\,\%$, the length varies from $5$ to $19\ \mbox {km}$ and the mean velocity for the fastest climbers is $22 \pm 2\ \mbox {km}\ \mbox {h}^{-1}$. At this velocity, the power consumed through aerodynamical friction is typically $20\text {--}30\ \mbox {W}$ while the power developed by the rider is still of the order of $400\ \mbox {W}$. We thus first neglect the aerodynamical drag in our analysis of mountain climb. In the steady regime, (3.4) reduces to the balance between the maximal generated muscle power $P_m=P_{max}(T)$ and the climbing term $Mg\alpha V$. This balance leads to the expression of the climbing velocity $V_{MT0}$:

(3.8)\begin{equation} V_{MT0}(T)=\frac{P_{max}(T)}{M g \alpha}. \end{equation}

Using (3.3) with $\tau =32\ \mbox {s}$, $\gamma =0.0886$ and $\varPi =9.7\ \mbox {W}\ \mbox {kg}^{-1}$ we estimate $P_{max}(T)$ for all the riders in table 2 (column 9) and deduce the velocity $V_{MT0}(T)$ (column 10) with (3.8) taking $M_b=10\ \mbox {kg}$. The predicted velocity is always larger than the actual velocity by typically 20 %–30 %.

To obtain a better prediction, one needs to account for the aerodynamic drag and for the rolling resistance. In the equation $P_m=1/2\rho S C_D V^3+Mg\alpha V+\mu Mg V$, the dominant term in climbing is the one associated with $\alpha$. Using a perturbative method, we get the correction associated with the extra two terms

(3.9)\begin{equation} V_{MT1}(T)=V_{MT0}(T)\left(1-\frac{\mu}{\alpha}-\frac{1}{2}\frac{\rho S C_D V_{MT0}^2}{Mg\alpha}\right) . \end{equation}

In the limit $\alpha \gg 1$ the corrective terms vanish and we recover $V_{MT1}=V_{MT0}$. With $\mu =0.0032$, $\rho =1.2\ \mbox {kg}\ \mbox {m}^{-3}$ and $S C_D=0.25\ \mbox {m}^{2}$, the corrected velocity $V_{MT1}(T)$ is calculated and the results are listed in column 11 of table 2. The error reported in the last column reveals that the corrected velocity is closer to reality (less than $10\,{\%}$).

3.3. The sprint

The last term of (3.4) is associated with the acceleration of the rider. This term has been neglected so far since we have only considered phases where the velocity remains mainly constant. This is no longer the case for sprints. Some characteristics of a road sprint are presented in figure 6: the power output recorded in a bunch sprint performed in a professional road cycling competition is presented in figure 6(b) together with the corresponding velocity (Menaspa Reference Menaspa2015). Interestingly, the power data were recorded during a successful sprint. In this example the duration of the final sprint is $11\ \mbox {s}$, and the mean power is $1020\ \mbox {W}$ (peak power $1248\ \mbox {W}$), with a maximal recorded speed of $66\ \mbox {km}\ \mbox {h}^{-1}$. The authors also report the intensity recorded before the sprint. The cyclist rode at an average power output of $490 \ \mbox {W}$ in the last 3 min. The data collected over a larger number of sprinters are gathered in the table presented in figure 6(c).

Figure 6.

(a) Example of final sprint in a road race. Photo credits: T. Vergouwen. (b) Example of power output and speed recorded at the end of a road race. The final sprint is highlighted in grey. Reproduced from Menaspa (Reference Menaspa2015). (c) Characteristics of road sprints in professional competitions; $\textrm {mean} \pm \textrm {standard deviation}$ (range). Reproduced from Menaspa et al. (Reference Menaspa, Quod, Martin, Abbiss and Peiffer2015).

Since the slope of the road $\alpha$ is small for sprints, their dynamics is described by a simplified version of (3.4): $\textrm {d}/\textrm {d}t(1/2MV^2)+1/2\rho S C_D V^3 = P_m$. This nonlinear equation can be turned into a linear equation in $V^3$ by replacing the time derivative term $\textrm {d}/\textrm {d}t(1/2MV^2)$ by its spatial equivalent $\textrm {d}(1/3MV^3)/\textrm {d}s$

(3.10)\begin{equation} \frac{\textrm{d} V^3}{\textrm{d}s}+\frac{V^3}{L_{sprint}}=\frac{3P_m}{M}, \end{equation}

where $L_{sprint}=2M/3\rho S C_D$ appears as the characteristic length scale of sprints. According to the data presented in figure 6, before launching the sprint the power is of the order of $P_{m0}\approx 490\ \mbox {W}$, which leads in the steady limit to the velocity before sprint $V_0\approx (2P_{m0}/\rho S C_D)^{1/3}$. With $SC_D\approx 0.25\ \mbox {m}^{2}$ this leads to $V_0=53\ \mbox {km}\ \mbox {h}^{-1}$. When the sprint is launched the power increases to $P_{sprint}\approx 1100\ \mbox {W}$ and the maximum velocity which could be reached in steady state is $V_{sprint}=(2P_{sprint}/\rho S C_D)^{1/3}$. The value of the aerodynamic coefficient in the sprint regular position is $0.3\ \mbox {m}^{2}$ (Blocken et al. Reference Blocken, van Druenen, Toparlar and Andrianne2019). We thus deduce $V_{sprint}\approx 66\ \mbox {km}\ \mbox {h}^{-1}$. The exact solution of (3.10) is

(3.11)\begin{equation} V(s)=V_{sprint}\left[1-\textrm{e}^{-s/L_{sprint}}+\left(\frac{V_0}{V_{sprint}}\right)^3\textrm{e}^{-s/L_{sprint}}\right]^{1/3}. \end{equation}

This exact solution reveals that the acceleration from $V_0$ to $V_{sprint}$ requires the characteristic length $L_{sprint}$. With $M=80\ \mbox {kg}$ and $SC_D\approx 0.3\ \mbox {m}^{2}$ one deduces that the sprints must start at least $150\ \mbox {m}$ before the finish line. At an average speed of $60\ \mbox {km}\ \mbox {h}^{-1}$, this distance is covered over $9\ \mbox {s}$, which is the characteristic duration of sprints (Menaspa et al. Reference Menaspa, Abbiss and Martin2013). The characteristic velocities $V_0$ and $V_{sprint}$ as well as the duration of actual sprints are thus correctly described by (3.10).

3.4. The descent

In a first approximation, we consider that, during descents, cyclists mainly rest ($P_m\approx 0$) and change their position in order to maximize their velocity. Different positions have been tested as illustrated in figure 7 reproduced from the detailed study of Blocken et al. (Reference Blocken, van Druenen, Toparlar and Andrianne2018a). Since the muscle power is null and the regime steady, the characteristic velocity in the descent $V_{D0}$ results from the balance between the propulsive gravitational power $Mg(-\alpha )V_{D0}$ and the resistive aerodynamical friction

(3.12)\begin{equation} V_{D0}=\sqrt{\frac{2Mg(-\alpha)}{\rho S C_D}}. \end{equation}

In their study, Blocken et al. (Reference Blocken, van Druenen, Toparlar and Andrianne2018a) considered the descent of C. Froome in stage 8 of Tour de France 2016. This stage ended with the descent of Peyresourde, which is steep with a regular slope ($\alpha =-8\,\%$) and is not characterized by sharp bends. On the day of the descent, the weather conditions were good and the road surface was dry. Near the very end of this stage, just before the top of Peyresourde, C. Froome accelerated and broke away from the group. During part of the descent, he adopted the position shown in figure 7(a) and achieved speeds up to $90\ \textrm {km}\ \textrm {h}^{-1}$. Since $M_c=66\ \mbox {kg}$ for Froome, we use $M=76\ \mbox {kg}$, $\alpha =-8\,\%$ and $SC_D=0.233$ in (3.12) to evaluate $V_{D0}=20.6\ \mbox {m}\ \mbox {s}^{-1}=74\ \mbox {km}\ \mbox {h}^{-1}$ which is smaller than the reported value. The difference is due to the injected power: the ‘Froome’ position is used instead of the ‘superman’ position since it allows for pedalling and C. Froome was indeed pedalling during the descent of Peyresourde (figure 8a).

Figure 7.

Four different positions used during descents with the corresponding static pressure coefficient $C_P=2(P-P_0)/\rho V_0^2$ on cyclist and bicycle surfaces. Here, $P$ is the static pressure, $P_0$ the atmosphéric pressure and $V_0$ the velocity of the centre of mass. The frontal area $S$ and the drag area $SC_D$ are also indicated for four significantly different positions: (a) the Froome position, (b) the Pantani position, (c) top tube 4 and (d) the superman position. This figure is reproduced from Blocken et al. (Reference Blocken, van Druenen, Toparlar and Andrianne2018a).

Figure 8.

(a) C. Froome in stage 8 of Tour de France 2016 pedalling during the descent of Peyresourde. Photo credits: France TV. (b) Impact of the pedalling on the speed during descent. The needed power (3.13) is presented in blue and the muscle power (3.14) is presented with a pink solid line.

To quantify the effect of pedalling during descent, one can rewrite (3.4) in the steady state limit of a descent in the form

(3.13)\begin{equation} P_m=\tfrac{1}{2}\rho S C_D V(V^2-V_{D0}^2). \end{equation}

When $P_m=0$ we recover $V=V_{D0}$ and when $P_m$ is positive, the velocity increases. This first relationship between the injected power and the velocity is plotted with a blue line in figure 8(b), using $SC_D=0.233$, $\rho =1.2\ \mbox {kg}\ \mbox {m}^{-3}$ and $V_{D0}=20.6\ \mbox {m}\ \mbox {s}^{-1}$. As already discussed, the injected power also depends on the pedalling rate $\dot {\theta }$ (which is related to the velocity by the relation $V=R G \dot {\theta }$, where $R$ is the radius of the wheel and $G$ is the gear ratio). Typically in mountain stages, professional cyclists have a maximal gear ratio $G=54/11=4.91$. This second relationship between the injected power and the velocity takes the form

(3.14)\begin{equation} P_m=8\varPi M_c \frac{V}{R G \dot{\theta}_{max}}\left(1-\frac{V}{R G \dot{\theta}_{max}}\right). \end{equation}

In this expression we have used the relationship $P_{max}=2\varPi M_c$ that applies for short efforts after rest (limit of (3.3) when $T=0$). This equation is presented with a pink solid line in figure 8(b), using $\varPi =9.7\ \mbox {W}\ \mbox {kg}^{-1}$, $M_c=66\ \mbox {kg}$, $G=4.91$, $R=0.334\ \mbox {m}$ and $\dot {\theta }_{max}=18.8\ \mbox {rad}\ \mbox {s}$ (which corresponds to $\dot {\theta }_{max}=180\ \mbox {r.p.m.}$). The first observation is that power can only be used if the maximal pedalling velocity $RG\dot {\theta }_{max}$ is larger than $V_{D0}$. This condition imposes a minimal gear ratio $G_{min}=V_{D0}/R\dot {\theta }_{max}$. In the case of Froome, we get $G_{min}=3.3$. The typical gear ratio used in mountain stages $G=54/11=4.91>G_{min}$ ensures that power can be injected during the descent. Once this condition is fulfilled, the effect of the injected power on the velocity is obtained by equating the needed power (3.13) and the injected power (3.14). This balance corresponds to the crossing point of the blue and pink lines in figure 8(b). One reads $V_{D1}=25.5\ \mbox {m}\ \mbox {s}^{-1}$ which is $91.8\ \textrm {km}\ \textrm {h}^{-1}$. A value much closer to the one observed in 2016 during the descent of Froome. One also observes in figure 8(b) that there exists a maximal velocity of descent $V_{Dmax}$ obtained when the injected power is maximal. In the case of Froome in Peyresourde we read $V_{Dmax}=27.4\ \mbox {m}\ \mbox {s}^{-1}=98.8\ \mbox {km}\ \mbox {h}^{-1}$. In order to achieve this maximal velocity of descent he should use a gear ratio such that $RG_{max}\dot {\theta }_{max}=2V_{Dmax}$ that is $G_{max}=8.7$. A huge modification (extra kg during the climb) for a modest gain. One major characteristic of descent is breaking, which is not addressed here but constitutes a perspective of this work.

4.1. What is a climber?

In the peloton, climbers are identified and usually associated with light weight together with a large power to mass ratio (Lucia et al. Reference Lucia, Joyos and Chicharro2000; Lucia, Hoyos & Chicharro Reference Lucia, Hoyos and Chicharro2001) such as Marco Pantani (1.72 m, 57 kg), Alberto Contador (1.76 m, 61 kg), Nairo Quintana (1.67 m, 58 kg), Egan Bernal (1.75 m, 60 kg).

On the physical side, one way to define a climber is to discuss the validity domain of (3.8) for $V_{MT0}$: in the steady regime, (3.8) only holds if the power dissipated by aerodynamical forces can be neglected. Since $V_{MT0}\propto 1/\alpha$ the smaller the slope the larger the velocity. This implies that (3.8) only holds above a critical slope $\alpha ^{\star }$ for which the gravitational power request $Mg\alpha ^{\star }V(\alpha ^{\star })$ exactly balances the power dissipated by the aerodynamical friction $1/2\rho S C_D V(\alpha ^{\star })^3$ (we neglect solid friction to simplify the discussion). This balance leads to the following expression:

(4.1)\begin{equation} \alpha^{\star}=\left(\frac{P_m}{M}\right)^{2/3}\left(\frac{\rho S C_D}{2Mg^3}\right)^{1/3}. \end{equation}

This angle separates the flat region ($\alpha <\alpha ^{\star }$) where the aerodynamics dominates gravity from the mountain region ($\alpha >\alpha ^{\star }$) where gravity dominates. The limit $\alpha ^{\star }=\infty$ corresponds to $g=0$ where the mountains do not influence the velocity anymore. The angle $\alpha ^{\star }$ also depends on the athlete characteristics ($P_m, M, SC_D$) and one deduces from the zero $g$ discussion that climbers have a large $\alpha ^{\star }$.

Equation (4.1) shows that, in order to achieve a large $\alpha ^{\star }$, one needs a large power to mass ratio (first term) but also a small mass (second term), thus recovering what has been reported for climbers by Lucia et al. (Reference Lucia, Joyos and Chicharro2000).

Since the power to mass ratio can be approached by the heuristic equation (3.3), one deduces that, for a climb, which lasts in general $T\approx 30\ \mbox {min}\gg \tau$, the power to mass ratio reduces to $P_m/M_c=\varPi [1-\ln (T/\tau )^{\gamma }]\approx 0.64\varPi M_c/M$, using $\gamma =0.0886$ and $\tau =32\ \mbox {s}$.

If $N$ stands for the number of cyclists of the Tour de France, one can define the climbers as the best $20\,\%$ of the peloton by a critical angle $\alpha ^{\star }_{80}$ such that $80\,\%$ of the peloton has a lower angle $\alpha ^{\star }$. One can then use a scale of $\alpha ^{\star }$ as the horizontal axis of the diagram presented in figure 9. The riders with a personal $\alpha ^{\star }$ larger than $\alpha ^{\star }_{80}$ are climbers.

Figure 9.

Phase diagram for road cycling. Photo credits: T. Vergouwen/AFP/Presse Sport.

4.2. What is a sprinter?

The first remark about sprinters is that they win lots of stages: the examination of the sprint results in several grand tours (2008–2011) indicates that 79 stages (31 % of 252 total number of stages) were won by only 24 sprinters. Five sprinters won 54 stages of which 1 sprinter won 30 stages (Menaspa et al. Reference Menaspa, Abbiss and Martin2013). Mark Cavendish (70 kg), Peter Sagan (73 kg), Erik Zabel (69 kg) are some emblematic figures of this discipline.

On the physical side, according to (3.10), the best sprinter is the one with the largest peak velocity $V_{sprint}=(2P_{sprint}/\rho S C_D)^{1/3}$. A sprinter is thus characterized by a large absolute power and does not depend on the power to mass ratio. What we know from figure 3(c) is that the maximal power is obtained over a short period corresponding to an anaerobic effort. Since sprints occur at the end of a race that last a few hours, a sprinter must be protected by his team in order to keep his energy for the last hundreds of metres. What we also learn from 3(c) is that the maximal value of the power depends on the cyclist. Always using (3.3), we can evaluate the power $P_{sprint}$ with $T=10\ \mbox {s}$ and we find $P_{sprint}=1.70\varPi M_c$. As we did for climbers, the associated velocity $V_{sprint}$ can be calculated for all the riders in the peloton in order to evaluate $V_{sprint}|_{80}$ such that $80\,\%$ of the peloton has a lower peak velocity. The sprinters can then be defined as the ones who verify: $V_{sprint}>V_{sprint}|_{80}$. The vertical axis of the diagram presented in figure 9 presents this classification.

4.3. The 3 jerseys

The phase diagram presented in figure 9 is thus composed of two axes, the horizontal one dedicated to climbers and the vertical axis to sprinters. This diagram defines four different regions and we need to discuss the associated physical properties to understand the origin of the three jerseys: to be on the right hand-side of the diagram, a cyclist must have $\alpha ^{\star }>\alpha ^{\star }_{80}$, which implies having a large $P_m^2/M^3=(P_m/M)^2/M$. In other words, a climber is defined by both a large power to mass ratio $P_m/M$ and a small mass.

On the other hand, to be on the upper part of the diagram, a cyclist must have a large power $P_{sprint}$ and a small drag area $SC_D$. Since the power scales with the mass, a large mass is expected to be in the upper part.

The upper-right part is populated with a category of cyclists who have both a large power to mass ratio and who are able to develop a large power. This rather rare combination defines TT specialists and very complete cyclists, which are the qualities recognized for Tour winners.

The phase diagram for road cycling thus has three optimal regions, occupied by three different physics and physiological characteristics and which are associated with three different jerseys: the green for the best sprinter (top left), the polka dot for the best climber (bottom right) and the yellow for the more complete one (top right).

Usually, the three jerseys are for three different cyclists but, depending on the route of the grand tour, one can have some overlap between these different regions. In 2019, the results presented in figure 1 reveal that the three jerseys winners were indeed different in Tour de France and in Giro but, for the Vuelta, the leader Primoz Roglic was also the best sprinter.

Since 1903, over the 106 Tour de France we observe only one exception, in 1969, where Eddy Merckx won the three jerseys (we thank P. Odier for this historical remark).

4.4. Phase diagram for the Tour de France 2017

To construct the phase diagram of Tour de France 2017, one needs to determine $\alpha ^{\star }$ and $V_{sprint}$ for each cyclist involved in the race. According to the discussions presented in §§ 4.1 and 4.2, one needs to determine the individual value of $\varPi$. We use the first stage of the Tour 2017, which was a flat time trial with $D=14\ \mbox {km}$. From the average velocity $\bar {V}_{TT}$ measured during the race for each rider, we calculate the individual value $2 \varPi =\rho S C_D \bar {V}_{TT}^3/[M_c (1-\gamma \ln (D/\tau \bar {V}_{TT}))]$. This expression is obtained using (3.5) and (3.3). This value is then used to evaluate the two characteristics

(4.2a,b)\begin{equation} \alpha^{\star}=\left(0.64\varPi\frac{M_c}{M}\right)^{2/3}\left(\frac{\rho S C_D}{2Mg^3}\right)^{1/3}\quad \mbox{and}\quad V_{sprint}=\left(\frac{3.4\varPi M_c}{\rho S C_D}\right)^{1/3}. \end{equation}

The mass $M_c$ and the size $L_c$ of each cyclist are listed in the Appendix. As we did throughout the article $S C_D$ is taken constant to $0.25\ \mbox {m}^{2}$, $\rho =1.2\ \mbox {kg}\ \mbox {m}^{-3}$ and $M_b=10\ \mbox {kg}$. From these values, one is able to evaluate $\alpha ^{\star }$ and $V_{sprint}$ for the whole peloton.

The cumulative distribution of $\alpha ^{\star }$ is presented in figure 10(a). One observes that $\alpha ^{\star }$ varies from $2.5\,\%$ to $4.2\,\%$ and we extract the value $\alpha ^{\star }_{80}=3.77\,\%$. Only $20\,\%$ of the riders have a higher $\alpha ^{\star }$. The cumulative distribution of $V_{sprint}$ is presented in figure 10(b). The maximum velocity ranges from 60 to 72 km/h with $V_{max}|_{80}=68.7\ \mbox {km}\ \mbox {h}^{-1}$. So that only $20\,\%$ of the riders have a higher $V_{max}$.

Figure 10.

Phase diagram for the Tour de France 2017: (a) determination of $\alpha ^{\star }_{80}$, (b) determination of $V_{max}|_{80}$, (c) construction of the phase diagram for the Tour 2017: each black square corresponds to a cyclist. The winner C. Froome is indicated with a large yellow square, the green jersey M. Matthews is indicated with a large green square and the polka dot jersey W. Barguil with a large white and red square. The large squares in grey are used for the final top 10.

Using these values, one can thus construct the phase diagram of the Tour 2017 which is presented in figure 10(c): the three jerseys of C. Froome (yellow square), M. Matthews (green square) and R. Bargil (white and red square) stand in the three different regions discussed in § 4.3. We also report, with grey squares, the locations of the riders who finished in the top 10 of the general classification (2-R.Urán, 3-R. Bardet, 4-M. Landa, 5-F. Aru, 6-D. Martin, 7-S. Yates, 8-L. Meintjes, 9-A. Contador, 10-W. Barguil). Clearly they are in the ‘climber’ zone, which underlines the fact that the Tour 2017 had a hilly design: 11 of the 21 stages were either medium mountain SMT (6) or high mountain MT (5).