In competitive cycling, cyclists typically ride for large portions of the race as a group, or peloton. This has the advantage of significant reductions in aerodynamic drag. However, to win a race, a cyclist must choose to break away from the peloton at some strategic position during the race. To execute a successful breakaway, cyclists must balance energy expenditure, aerodynamic drag and the inherent risk of crashing, often under complex and dynamic race conditions. To date, existing research has primarily focused on isolated aspects of racing, such as drag reduction through drafting [1], nutrition or the biomechanics of power generation [2,3]. However, exploration of the specific interplay between these factors in breakaway situations is limited. Understanding this interaction is essential for optimizing race strategy, particularly for athletes seeking to achieve a tactical advantage.

Aerodynamic drag plays a significant role in determining the energy expenditure of a cyclist, especially when drafting within the peloton or executing a breakaway. As previous studies have shown, aerodynamic resistance accounts for up to 90% of the overall resistance a cyclist experiences at race speeds [4,5]. While drafting can significantly reduce drag—by as much as 80% for cyclists in a peloton [6,7]—a breakaway rider is faced with the challenge of overcoming this loss of aerodynamic advantage.

Gaul et al. [8] considered the problem of breakaway optimization by focusing on cyclist exhaustion and stamina, but did not consider the peloton structure or the important role crashing plays in their optimization strategy.

Crashing also contributes substantially to strategic decision-making. Even a minor crash with no injuries can mean a rider loses around a minute, in addition to the extra energy expenditure required to catch up with the peloton. In a sport where victories are achieved with razor-thin margins, a single crash can make or break a contender’s attempt. Moreover, a single major crash can lead to severe injuries, ultimately forcing a contender to abandon an event.

Crashes are influenced by environmental variables, race conditions and cyclist behaviour [9–11]. In peloton settings, a higher density of riders increases the likelihood of collisions, which can play a tactical role, as riders often seek to escape this risk by moving to the front or breaking away entirely. Breakaway riders, however, encounter different risk factors, such as fatigue-induced impairments and exposure to crosswinds that may affect stability.

In this article, we aim to uncover the important role crashes play in deciding racing outcomes, and how competitive cyclists can make informed strategic decisions. We present a comprehensive mathematical approach that integrates energy consumption, drag reduction and crash probability to determine the optimal position and timing for initiating a breakaway attempt.

We begin in §2 by presenting the governing equations for our model. These comprise equations of motion for the riders and equations that capture the crashing probability. The equation of motion for each rider is based on Newton’s second law, incorporating aerodynamic drag, power generation and losses owing to rolling resistance and drivetrain friction. As we are interested in the role of attacking and escaping from the peloton, we incorporate a position-dependent drag coefficient, which depends on how far the rider is from the head of the peloton when riding in a group, and takes the same value as a rider at the front of the peloton when the rider is cycling alone. We introduce an objective function that balances the time advantage of winning a race with the probability of crashing.

To facilitate analytical progress, we choose to write the system in terms of the position of just two riders: the rider at the front of the peloton, and a special rider who will break away from the peloton at some point during the race.

We first consider a simple case of a flat course with no crashes in §3, to demonstrate the methodology. We extend this by incorporating the probability of a crash, in §4 . Here, we show how we may obtain an analytic expression for our objective function, which allows for easy optimization.

We develop the model further in §5, to account for rider fatigue. We also briefly discuss more realistic course profiles with variable elevation in appendix C and outline the numerical techniques required to analyse these more complex equations. In §6, we draw conclusions from the analysis conducted. We use this to make relevant predictions to real-world cycle races, and we discuss extensions that would provide valuable further input for the optimization procedure.

To illustrate the mathematical model, we begin in this section by considering a race in which no crashes take place. We suppose that a single rider sits at position i in the peloton until making a breakaway attempt at some attack point along the course, which we denote as xa. By our choice of non-dimensionalization, the peloton moves at unit dimensionless speed, and hence xp(t)=t. Next, we assume the power exerted has the following form. Before the attack, the rider is in the peloton at position i and so the power exerted is Pi=Cd,i, using equation (2.9). We denote Pl,i=Cd,i as the lurking power of the rider at position i in the peloton. Note that owing to the choice in non-dimensionalization, Pl,i for i>1. After the rider makes a breakaway move at, say, t=ta, we suppose that they provide a constant power Pa>Pl,i until the end of the race. The total energy expenditure is then simply

As the peloton moves with unit speed, the attack position is related to the attack time by xa=ta (neglecting terms in ζi as noted earlier). After the attack, the time left to complete the course is tf−ta=(1−xa)/va where va≡dxi/dt=(Pa/Cd,1)1/3, using equation (2.9) and the fact that the drag on a solo rider is Cd,1. We can thus find the earliest attack location for a given attack power, Pa, that a rider can provide and their total energy reserve, E∗:

Here, we have introduced the max⁡ operator to ensure the attack position is in the domain, i.e. xa>0. Since we require va>1 for an attack to be successful, this gives a minimum attack power, Pamin=Cd,1 and thus an associated earliest attack time

using equation (3.2). For the rider to have a chance of winning, we must have xamin<1, and so this means that the minimum energy of the rider for an attack to be successful is Emin∗=Cd,i.

In the left-hand panel of figure 2, we plot the dependence of xa on the attack power for different values of E∗, given by equation (3.2). For small energy budgets, the earliest attack position is close to the finish line, but for higher energy budgets, the attack position is closer to the start line. In particular, if the energy budget is high enough (that is, if E∗>Cd,1), then the earliest attack position xa≤0. In this case, this means that a rider has sufficient energy to make a break at the beginning of the race (and still have surplus energy at the end of the race).

Alternatively, a racer may have identified a particular place where they wish to break away. In this case, we can rearrange equation (3.2) for Pa to determine the required power for the optimal breakaway that uses all their energy reserves by the end of the race:

The dependence of Pa on the attack position xa is plotted in the right-hand panel of figure 2. We show the minimum attack power required to escape from the peloton (Pa>Cd,1 from va>1), as a dashed grey line.

We may also solve for the finish time ahead of the peloton for the breakaway rider in terms of Pa and E∗:

or in terms of xa and E∗,

We may see how the finish time ahead of the peloton, Δt varies with either the attack power or the breakaway position in figure 3. In the left-hand panel, we observe that the time gap initially increases as the breakaway power rises, but eventually, it reaches a maximum before falling again. Similarly, the time gap initially grows with attack position before reaching a maximum and then falling as the attack position gets closer to the end of the course with Δt=0 at xa=1. Another feature we highlight from the plots for Δt(xa) is the discontinuous derivative of the function at the minimum attack position. Breakaways attempted before this position lack the power necessary to overcome the increased drag when riding in isolation, and the rider simply returns to the peloton and finishes the race with the main group, with Δt=0.

A key observation from figure 3 is the existence of an optimum that maximizes the time difference between the breakaway rider and the peloton for a given attack power Pa or attack position xa, when taking account of the available energy reserves. In the following section, we study the dependence of this optimum on the parameters and consider how this is affected by the introduction of a non-zero crashing probability.

Although the model introduced in §3 can provide team directors with useful reference points for deciding team strategy, such as the minimum energy expenditure, and a good estimate for the optimal attack position, it assumes an unrealistic evolution of the power developed by a rider attempting a breakaway. Once an athlete exceeds their anaerobic limit, their maximum power output will fall. This short-term fatigue effect is commonly attributed to lactic acid build-up as a consequence of anaerobic respiration. However, another cause of fatigue is the net movement of potassium ions (K⁣+) out of contracting skeletal muscle [15]. There are contrasting opinions on whether lactic acid or potassium has the larger effect on force output (e.g. [16]). Here we choose to model the short-term fatigue in a general manner that may be attributed to lactic acid build-up or the movement of potassium ions or other physiological effects.

In this article, we have developed a mathematical framework that captures the key trade-offs faced by elite cyclists: balancing aerodynamic efficiency, energy expenditure and the risk of crashing. By incorporating both physiological limits and probabilistic crash modelling, our approach extends earlier models from [8] to more realistically capture the strategic considerations involved in breakaway attempts.

One of the key strengths of our model lies in its relative simplicity and analytical tractability. Despite this, it successfully quantifies several intuitive yet previously unmodelled strategic insights. For example, it confirms that breakaways near the end of a race are more energy-efficient, while earlier attacks can result in larger time gains, but at a higher energetic cost. Similarly, spending more time near the rear of the peloton reduces drag [6] but significantly increases the risk of being caught in a crash. These results not only validate widely held beliefs among athletes and team directors but also allow for rigorous quantitative evaluation of such strategies.

A particularly novel feature of this work is the inclusion of crash dynamics as a fundamental part of strategic planning. By treating the probability of crashing as a dynamic penalty, we show that risk management is as crucial to success as fitness or power output. This aligns with real-world race scenarios, where riders must weigh the dangers of remaining in a densely packed peloton against the energetic demands of escaping it. In elite competition, where physical capabilities are often comparable among top athletes, it is these marginal strategic decisions—timing of attacks, positioning and risk tolerance—that frequently determine the outcome.

The model is also versatile enough to accommodate different risk profiles via the parameter β, representing a rider’s willingness to trade risk for performance. This provides a useful lens to interpret decisions made by different types of riders: for instance, a General Classification (GC) leader may choose a conservative strategy, while a stage hunter might favour risky strategies.

While the current model assumes a flat course and relatively simple power profiles, extensions are straightforward. In appendix C, we outline a framework for incorporating elevation changes, showing that our methods readily generalize to more complex terrains. Though analytical results are harder to obtain in this setting, standard numerical solvers can be used to explore optimal strategies in varied topography. In appendix D, we show how the crash model could be generalized to account for more realistic crash propagation patterns and environment-dependent risks such as road conditions, weather or race congestion.

The model presented in this article relies on a number of simplifying assumptions and therefore has some important limitations. For instance, our assumption that the peloton power is constant over the race rarely holds, with teams increasing the pace close to the finish line either to drop riders from the peloton (known as ‘making a selection’) or to secure better positions for their sprinting specialists. The constant or exponentially decaying functional forms we used for the attack power are pragmatic choices to allow for analytical progress. A more realistic choice that teams could employ is a ‘tank’ model, where different amounts of energy are available to be consumed at different power rates [19,20]. However, such models quickly increase the number of parameters in the model, potentially leading to parameter identifiability issues and overfitting to a particular cyclist.

Our use of the total energy expenditure as a control variable is also a simplification of the real racing scenario. However, regardless of how complicated and time-dependent a power profile may be during a stage, the total energy expenditure is always a scalar quantity that is readily available to cyclists from their GPS/cycling computers (and often quoted by trainers and fitness trackers). This makes it a suitable candidate to be used in a user-friendly mathematical model, as it is already widely used in the athletic world, and its significance is well understood by cyclists. Mathematically, the use of this scalar quantity as a constraint has the complication that it behaves singularly as the attack position nears the finish line, xa→1. A finite amount of energy must be used within a vanishingly small amount of time, which leads to unrealistically large power profiles. Although sprinters can achieve power outputs close to 2500 W [21] (approximately 17 times our estimated 150 W lurking power [18]), when the attack location is chosen closer to the finish line, the attack power exceeds these amounts. Therefore, our model should not be used to inform strategies where breakaway attempts occur near the finish line and power outputs exceed the maximum achievable by the breakaway rider. This would correspond simply to truncating figure 2 at the maximum power of the cyclist.

Breakaways in cycling are seldom performed by a single cyclist, and future models should account for multi-rider breakaways. In particular, as cycling is a team sport, having supportive teammates in the breakaway usually brings a significant advantage and an increased likelihood of winning [22]. This is true even when the breakaway rider is not aiming to win the stage, but is instead offering assistance to the team leader, usually a GC contender, who might be chasing behind. Cycling teams could therefore benefit from extensions of the model presented here that account for multi-rider breakaway attempts.

There are other promising directions for future work. One is to include dynamic chasing behaviour, where the peloton adjusts its speed in response to breakaways, and how this affects the viability of a breakaway. Another is to model physiological decay more precisely by linking the power decay rate μ to underlying rider attributes or empirical data (e.g. from power meters or lactate threshold tests [23]). Similarly, incorporating nutrition and hydration strategies, particularly over multi-stage races such as the Tour de France, could provide insights into optimal energy allocation over days rather than single stages [24].

Another future direction for this work could be to explore velocity-dependent crash probabilities, which would reflect the reduced time to react to a crash at higher speeds. Moreover, integrating spatial features of the course such as narrow roads, roundabouts or downhill corners would offer a more nuanced crash model. These refinements could also be informed by data analytics from race telemetry and crash reports.

A third future direction is to understand how headwinds, tailwinds and crosswinds can shape racing strategies, particularly when the weather forecast is different for different stages of a race. This is because strong headwinds should lead riders to prioritize staying inside the peloton owing to higher drag at the same ground speed, but tailwinds effectively reduce drag and hence the advantages of staying in the peloton. Moreover, crosswinds can lead to the formation of echelons [25], where riders who are not well positioned at the front of the peloton may be easily dropped [26], giving rise to exhilarating racing conditions.

Our work fits more generally in the field of athletic decision making, such as withdrawal from endurance events. Withdrawal is common in Grand Tours; on average, one in every four participants abandons the race before the twenty-first stage [27]. As the model presented in this work can quantitatively predict successful outcomes in racing, it may enhance our understanding of the reasons participants choose to withdraw. Previous work has focused mainly on environmental conditions [28], but it is natural to ask whether participants in endurance events have an intuitive ‘failure compass’ that allows them to decide if withdrawal is the best option. If this is the case, it would be interesting to ask whether their intuitive predictions match the quantitative predictions.

Finally, we emphasize that elite racing is not solely a test of physical ability, but of strategic timing and decision-making. As athletic performance nears physiological limits [29,30], small tactical choices, such as when and where to initiate a breakaway, can determine the outcome of a race. Our framework provides a structured way to analyse these choices, translating race-day intuition into quantifiable insights. The analytical clarity that our mathematical model provides makes this a valuable tool for interpreting rider behaviour and guiding tactical planning in competitive settings.