Impact of mass gain, tailwind and age on the performance of Usain Bolt from Beijing 2008 to Rio 2016

It is known that the effectiveness of a high-performance athlete first increases with age up to a maximum level, and from there on starts to diminish. It seems that there is no way to determine at what age such a maximum will be reached, because of the many factors (genetic, training technique, reaction time, possibility of disqualification, self-confidence, etc) involved when trying to find out. The recent results at the 2016 Olympic Games (OG) and the 2017 Australian Open (AO) are good examples in which 'old men' beat younger ones: Michael Phelps winning 5 golds and a silver medal at 32 years of age, and the AO final match between Rafael Nadal (31 years old) and Roger Federer (36 years old), won by the latter. The unbeatable athletic trajectory of Usain Bolt (UB) in World Championship Athletics (WCA) and at the OG since he broke his own 2008 Beijing OG 100 m sprint world record at the 2009 Berlin WCA, leads us to ask whether he will be able to do it again. The purpose of the analysis in this work is to investigate which factors could have influenced, in our opinion, the fact that he has not been able to set a new world record for such a distance again. A question naturally arises: was 2009 the year when UB (23 years old) reached his maximum effectiveness? Incidentally, his 200 m sprint world record is from that same year.

Throughout the years, several kinematic (Krzysztof and Mero 2013), physiological (Charles and Bejan 2009) and psychological (Varlet and Richardson 2015) analyses have been published analysing some of these factors. However, the ways in which they are reflected in the performance of a runner are not easy to quantify. A more trustworthy approach is through the theoretical simulations (Vaughan 1983a, 1983b) of mechanical models in which several variables can be taken into account. Among the many models in the literature (Furusawa et al 1927, Fenn 1930, Keller 1973, Shanebrook and Jaszczak 1975, Holmlund and von Hertzen 1997, Wagner 1998, Alexandrov and Lucht 1981, Pritchard 1993, Helene and Yamashita 2010, Barbosa et al 2016, Janjić et al 2017), in this paper we will concentrate on the successful mechanical model developed by Hernández-Gómez et al (2013) for the 100 m sprint, which we now apply to understanding Bolt's performance from Beijing 2008 to Rio 2016. To address the question of why he has not been able to break the world record he set at Berlin 2009, we evaluated the relevance of parameters such as the mass and age of the runner, as well as the sprint tailwind. We carefully show how to apply the model to obtain substantial predictions on the subject, so that any undergraduate student will be able to obtain a physical insight into the physics of the 100 m sprint, as well as into the application of mathematical models of real physical conditions. To do so, we have structured this paper as follows: in section 2 we show the theoretical application of the referred model to other sprint events, along with a practical example of its application at the 2016 Rio OG; here we also analyse the impact of UB's change in mass between Berlin 2009 and Rio 2016. In section 3, we collect the data of his sprints from Beijing 2008 to Rio 2016 to get an insight into the main physical variables, such as the force and power he exerted in these events. In section 4, we fully exploit the predictive power of the model, and apply it by simulating the events that took place in the time span previously mentioned, in three different hypothetical setups: whether all events were run with the same tailwind conditions, whether UB had the same mass each time, and whether all events were run with the same mass and tailwind conditions. The latter setup allows us to isolate the age of the runner. Finally, in section 5 we pose some interesting concluding remarks.

The mechanical model of the 100 m sprint by Hernández-Gómez et al (2013) follows two basic assumptions:

• That the sprinter is able to exert a constant force F₀ during the race; this assumption is based on the essentially constant speed that UB developed in the 100 m and 200 m races in Berlin 2009, and has been successfully validated in (Hernández-Gómez et al 2013);
• That the air drag is proportional to both the speed and the square of the speed of the runner.

This yields to the following motion equation:

$$\begin{eqnarray}&&m\dot{v}={F}_{0}-\gamma v-\sigma {v}^{2},\end{eqnarray} \tag{ 1 }$$

where v is the speed of the sprinter, m is the sprinter's mass, and γ and σ are the air resistance and the hydrodynamical drag proportionality constants respectively. With the aid of extremely accurate measurements of UB's speed as a function of his position in Berlin 2009, Hernández-Gómez et al (2013) were able to fit the model parameters so as to obtain F₀, γ and σ. Key features, such as his drag coefficient C_d = 1.2, were also obtained. With such a model, the possible run times with different tailwinds can be predicted by calculating the value of $\sigma ^{\prime}$ for each tailwind speed.

Some of the physical variables change at the different sprint events in which Bolt participated, and others do not:

• (i)  
  The air resistance term arises from air viscosity, which is independent of air pressure. The fitted γ value turns out to be $\gamma =59.7\ \mathrm{kg}\ {{\rm{s}}}^{-1}$ (Hernández-Gómez et al 2013).
• (ii)  
  As the exerted constant force F₀ is particular to each runner and depends on the specific conditions of the runner during the sprint, in this study we set F₀ as a parameter to be fitted.
• (iii)  
  The estimation of the value of σ for other sprint events can be done as follows: first, we take UB's drag coefficient C_d = 1.2 as well as his cross section area⁴ $A=0.8\ {{\rm{m}}}^{2}$ (Hernández-Gómez et al 2013). Then, the air density at the sprint location is calculated considering the altitude above sea level as well as the temperature and humidity during the sprint; the latter values are taken from the official data sheets of the events. Then,

  $$\begin{eqnarray*}&&{\sigma }_{{\rm{s}}}=\displaystyle \frac{1}{2}\rho {C}_{{\rm{d}}}A.\end{eqnarray*}$$
• (iv)  
  The above procedure calculates ${\sigma }_{{\rm{s}}}$ in still air. To estimate its value in the wind conditions of the particular sprint (wind speed v_w), it is necessary to first estimate the terminal speed,

  $$\begin{eqnarray*}&&{v}_{{\rm{T}}}=\displaystyle \frac{-\gamma +\sqrt{{\gamma }^{2}+4{F}_{0}{\sigma }_{{\rm{s}}}}}{2{\sigma }_{{\rm{s}}}},\end{eqnarray*}$$

  so a first guess on the value of the parameter F₀ is required—for instance, the value it took in Berlin 2009. Then, the corrected hydrodynamical coefficient is given by (Hernández-Gómez et al 2013),

  $$\begin{eqnarray*}&&\sigma ={\sigma }_{{\rm{s}}}\left(1+\displaystyle \frac{2{v}_{{\rm{w}}}}{{v}_{{\rm{T}}}}\right).\end{eqnarray*}$$
• (v)  
  With the set value of the parameters F₀, γ and the corrected value of σ, we compute the values of A, B and k,

  $$\begin{eqnarray*}&&A=\displaystyle \frac{\gamma +\sqrt{{\gamma }^{2}+4{F}_{0}\sigma }}{2\sigma },\end{eqnarray*}$$

  $$\begin{eqnarray*}&&B=\displaystyle \frac{-\gamma +\sqrt{{\gamma }^{2}+4{F}_{0}\sigma }}{2\sigma },\end{eqnarray*}$$

  $$\begin{eqnarray*}&&k=\displaystyle \frac{\sqrt{{\gamma }^{2}+4{F}_{0}\sigma }}{m},\end{eqnarray*}$$

  where m can change from event to event. The explicit forms for the position x(t), velocity v(t) and acceleration a(t) of the runner each sprint are (Hernández-Gómez et al 2013):

  $$\begin{eqnarray}&&x(t)=\displaystyle \frac{A}{k}\mathrm{log}\left(\displaystyle \frac{A+{{B}{\rm{e}}}^{-{kt}}}{A+B}\right)+\displaystyle \frac{B}{k}\mathrm{log}\left(\displaystyle \frac{{{A}{\rm{e}}}^{{kt}}+B}{A+B}\right),\end{eqnarray} \tag{ 2 }$$

  $$\begin{eqnarray*}&&v(t)=\displaystyle \frac{{AB}(1-{{\rm{e}}}^{-{kt}})}{A+{{B}{\rm{e}}}^{-{kt}}},\end{eqnarray*}$$

  $$\begin{eqnarray*}&&a(t)={ABk}(A+B)\displaystyle \frac{{{\rm{e}}}^{-{kt}}}{{(A+{{B}{\rm{e}}}^{-{kt}})}^{2}}.\end{eqnarray*}$$
• (vi)  
  In order to fit the value of the parameter F₀, equation (2) must be constrained as follows: if t_s is the time of the sprint and t_r is the runner's reaction time, it ought to follow that

  $$\begin{eqnarray*}&&x({t}_{{\rm{s}}}-{t}_{{\rm{r}}})=100\,{\rm{m}}.\end{eqnarray*}$$

  This procedure is repeated recursively from steps (ii) to (vi), varying the values of F₀ until the above constriction is satisfied.
• (vii)  
  When finishing this process, we end up with the values of the physical parameters F₀, γ and σ, which will allow us to quantitatively observe the difference in Bolt's performance over different competitions.

2.1. Application example: Usain Bolt at 2016 Rio OG

As mentioned in point (v), it is important to take UB's current mass, which at Rio was m = 94 kg. This is 9.30% higher than the mass of 86 kg he had when he broke the world record at Berlin 2009 (Helene and Yamashita 2010, Hernández-Gómez et al 2013), so this increment ought to have played an important role in the dynamics of the sprint in Rio. Taking into account m = 94 kg as well as the Rio conditions (altitude of 11 masl, humidity of 53% and temperature of 23 °C), ${t}_{{\rm{s}}}=9.81\ {\rm{s}}$ and ${t}_{{\rm{r}}}=0.155\ {\rm{s}}$ (see table 3) (IAAF Rio 2016 Olympic Games 2016), we obtain the physical parameters of the model for his 100 m sprint at the 2016 Rio OG. These values are shown in table 1, where we also present the corresponding values for Berlin 2009 (Hernández-Gómez et al 2013) for comparison purposes.

Table 1.  Values of the physical parameters F₀, γ and σ for both Berlin 2009 and Rio 2016.

Constant	Value at Berlin 2009	Value at Rio 2016	Percentage difference (%)
F0 (N)	815.8	800.7	−1.85
γ (kg s−1)	59.7	59.7	NA
σ (kg m−1)	0.60	0.57	−5.00

As can be observed in table 1, seven years after Berlin (2009), the exerted force has reduced by 1.85%, while σ has decreased 5.00%. Although the changes are small, they are not negligible, as shown in figure 1, where his overall performance can be appreciated. Figure 1(a) compares the position of the sprinter as a function of time for both competitions. In both cases, after about the first two or three seconds, his speed essentially remains constant.

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From figure 1(b), it is clear that his speed at Rio 2016 does not surpass that of Berlin 2009 at any moment. This is due, in part, to the fact that in Berlin, he acquired a terminal speed of v_T = 12.16 m s^–1, while for Rio it was v_T = 12.03 m s^–1, which is slightly slower. Also, there is a noticeable decrease in his initial acceleration, shown in figure 1(c), from $a(0)=9.49\ {\rm{m}}\,\,{{\rm{s}}}^{-2}$ (which was $0.97g$) in Berlin to $a(0)=8.52\ {\rm{m}}{{\rm{s}}}^{-2}$ (which is only $0.87g$) in Rio. This 10.22% reduction in initial acceleration, notwithstanding the fact that the force only decreases 1.85%, is most likely due to his greater mass in Rio compared to Berlin. The 10.22% reduction in initial acceleration is consistent with the ∼9.30% by which his mass increased.

The instantaneous power he develops is given by (Hernández-Gómez et al 2013)

$$\begin{eqnarray}&&P(t)={Fv}=m\dot{v}v=m{({AB})}^{2}k(A+B)\displaystyle \frac{(1-{{\rm{e}}}^{-{kt}}){{\rm{e}}}^{-{kt}}}{{(A+{{B}{\rm{e}}}^{-{kt}})}^{3}}.\end{eqnarray} \tag{ 3 }$$

In figure 1(d) we plot the power of the sprint for both Berlin 2009 and Rio 2016, where distinguishable differences between both runs are observed. While at Berlin 2009, the maximum power of ${P}_{\max }=2619.49$ W (3.51 HP) was reached promptly in a time of ${t}_{P\max }=0.89\,{\rm{s}}$, at Rio 2016 the maximum power reduces to ${P}_{\max }=2534.73$ W (3.40 HP) and it was reached in a time of ${t}_{{P}{\rm{m}}{\rm{a}}{\rm{x}}}=0.99\,{\rm{s}}$. Although the reduction is of just 3.24% with respect to Berlin 2009, the sprinter's interplay with the air drag is better observed when the full energy is calculated. While at Berlin 2009, the mechanical work exerted is ${W}_{{\rm{B}}}={F}_{0}d=81.58$ kJ, where d = 100 m, at Rio 2016 this value reduces to ${W}_{{\rm{B}}}=80.07$ (−1.85%). Nevertheless, the effective work is (Hernández-Gómez et al 2013):

$$\begin{eqnarray}&&{W}_{{\rm{e}}\mathrm{ff}}={\int }_{0}^{{t}_{{\rm{s}}}-{t}_{{\rm{r}}}}P(t){\rm{d}}{t}={\int }_{0}^{{t}_{{\rm{s}}}-{t}_{{\rm{r}}}}\displaystyle \frac{1}{2}{{m}{\rm{d}}{v}}^{2}=\displaystyle \frac{1}{2}{{mv}}^{2}({t}_{{\rm{s}}}-{t}_{{\rm{r}}}),\end{eqnarray} \tag{ 4 }$$

that is, the area under the curve of figure 1(d), which at Berlin 2009 was ${W}_{{\rm{e}}\mathrm{ff}}=6.36$ kJ while at Rio 2016 it was ${W}_{{\rm{e}}{\rm{ff}}}=6.80$ kJ. Although the effective work increased by 6.92% with respect to Berlin 2009, this is simply explained by the fact that at Rio 2016, the achieved speed was lower than at Berlin 2009, so the drag terms, which depend upon v and v², dissipate less energy. Actually, while at Berlin 2009 the amount of energy used to achieve the motion was 7.79% (of the total work exerted in such an event), while at Rio 2016 it was 8.49%.

2.1.1. Impact of weight between Berlin 2009 and Rio 2016

In order to gain a physical insight into the impact of UB's weight gain on his sprinting performance, we predict his running time as if he had had a mass of m = 86 kg (mass at Berlin 2009 (Helene and Yamashita 2010, Hernández-Gómez et al 2013)) at Rio 2016. Assuming that despite the change in mass, he is able to exert essentially the same force as that calculated in the previous section (F₀ = 800.7 N), taking the same value of $\sigma =0.57$, because it does not depend on the mass of the runner, considering that his reaction time ${t}_{{\rm{r}}}=0.155\,{\rm{s}}$ is essentially the same, independent of his mass, and considering the same sprint conditions as in the previous section, we predict that he would have run with a time of ${t}_{{\rm{s}}}=9.70\,{\rm{s}}$. Furthermore, if Rio 2016 had had the same tailwind conditions as in Berlin 2009 (v_w = 0.9), his run time would have been ${t}_{{\rm{s}}}=9.62\,{\rm{s}}$, which is slightly slower than the Berlin 2009 world record (9.58 s), by just 0.04 s.

The overall performance of this hypothetical sprint is depicted in figure 2, where again we show Rio 2016 against the Berlin 2009 world-record-breaking performance. The great resemblance between our hypothetical sprint at Rio 2016 and the real one at Berlin 2009 is mainly due to the very similar circumstances in altitude, temperature and humidity found during both races, despite the lower force exerted at Rio 2016. The latter fact allows us to emphasise the importance of the role of mass on 100 m sprinters. To make a fair comparison, we calculated Bolt's sprint time with his current mass of 94 kg, but with the same tailwind as in Berlin 2009 (0.9 m s^–1), which turns out to be ${t}_{{\rm{s}}}=9.74\,{\rm{s}}$. These results are condensed in table 2.

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Table 2.  The hypothetical run times at Rio 2016 with the mass and tailwind conditions of both Rio 2016 and Berlin 2009. The real time is denoted in italics.

${{v}}_{{\rm{w}}}$ (m s–1)\${m}$ (kg)	$86$	$94$
$0.2$	9.70	9.81
$0.9$	9.62	9.74

As can clearly be observed in table 2, the mass of the runner greatly influences the run times for the 100 m sprint—maybe even more than the tailwind speeds. These results suggest that, at least for the 100 m sprint, runners should be categorised by their mass, just as in other sporting disciplines, such as boxing, so as to establish a fairer competition.

To trace UB's performance throughout the years from Beijing 2008 to Rio 2016 in a more precise way, a record of his mass in the intermediate years is required. We tried to contact Bolt's team to obtain an accurate record, but at the time of writing the paper, we did not obtain any response⁵ . Surprisingly, the increment in his mass was not linear between Beijing 2008 and Rio 2016, as might have been expected, though it was sudden between 2009 and 2012; as for the London 2012 OG it was m = 93 kg (BBC Sport 2012). The progression of Bolt's mass, as well as other relevant data from each competition in which he obtained first place in the 100 m sprint, are shown in table 3.

Table 3.  Key facts for Usain Bolt's 100 m sprints at Berlin 2009, London 2012, Moscow 2013, Beijing 2015 and Rio 2016.

Event	Bolt's mass	Official time	Reaction time	Wind speed	Temperature	Humidity	Altitude
	m (kg)	${{t}}_{{\rm{s}}}$ (s)	${{t}}_{{\rm{r}}}$ (s)	${{v}}_{{\rm{w}}}$ (m s–1)	(°C)	(%)	(masl)
Beijing 2008a	86b	9.69	0.165	0.0	21	71	43
Berlin 2009c	86d	9.58	0.142	+0.9	26	39	34
London 2012e	93f	9.63	0.165	+1.5	17	73	35
Moscow 2013g	93h	9.77	0.163	−0.3	21	62	156
Beijing 2015i	94j	9.79	0.159	−0.5	22	78	43
Rio 2016k	94	9.81	0.155	+0.2	23	53	11

^aIAAF Beijing 2008: The XXIX Olympic Games (2008), World Weather Online (2017). ^bHelene and Yamashita (2010). ^cIAAF Berlin 2009 World Athletics Championships (2009). ^dHelene and Yamashita (2010), Hernández-Gómez et al (2013). ^eIAAF London 2012 Olympics Games (2012). ^fBBC Sport (2012). ^gIAAF Moscow 2013 World Athletics Championships (2013). ^hWe were not able to retrieve UB's mass at Moscow 2013; nevertheless, as at Glasgow 2014, it was reported as m = 93 kg (The Glasgow XX Commonwealth Games 2013), and we have supposed that his mass was constant from London 2012 up to 2014. ⁱIAAF Beijing 2015 World Athletics Championships (2015). ^jHerald Sun (2015). ^kIAAF Rio 2016 Olympic Games (2016).

Some preliminary conclusions can be drawn from table 3; for instance, from Berlin 2009 to London 2012, it seems that his reaction time might have been influenced by his 2011 disqualification in Korea, and that his second-best time was aided by a +1.5 m s^–1 tailwind; nevertheless, his third-best time was obtained without a tailwind. To acquire a better understanding of the evolution of Bolt's performance throughout years, the method depicted in section 2 is now also considered for Beijing 2008, London 2012, Moscow 2013 and Beijing 2015.

In table 4, we observe the overall evolution of the sprinter's performance from Beijing 2008 to Rio 2016. The exerted force (table 4, column 2) clearly decreases with his increasing mass. Furthermore, at Berlin 2009, the F₀ was 96.79% of his weight for the event; this percentage is only 95.47%, 88.38%, 88.75%, 87.99% and 86.92% for Beijing 2008, London 2012, Moscow 2013, Beijing 2015 and Rio 2016 respectively. This fact reveals the direct impact that the mass has on the ability of a sprinter to exert force. According to the model, in table 4, column 3, we feature the terminal speed that Bolt could achieve. The particularly high value of achievable terminal speed at London 2012 was due to the higher tailwind during the sprint. Nevertheless, due to the small exerted force (table 4, column 2) at this particular event, the maximum speed achieved at London 2012 was only 12.03 m s^–1, which is a long way below the achievable terminal speed of v_T = 12.35 m s^–1 at this sprint.

Table 4.  The overall performance of UB from Beijing 2008 to Rio 2016.

Event	Exerted force	Terminal speed	Initial acceleration	Maximum power	Total work	Effective work
	${{F}}_{0}$ (N)	${{v}}_{{\rm{T}}}$ (m s–1)	${{a}}_{0}$ (m s–2)	${{P}}_{\max }$ (W)	${W}$ (kJ)	${{W}}_{\mathrm{eff}}$ (kJ)
Beijing 2008	804.7	12.04	9.36	2554.35	80.465	6.235
Berlin 2009	815.8	12.16	9.50	2619.50	81.580	6.360
London 2012	805.7	12.35	8.66	2594.96	80.570	7.078
Moscow 2013	808.9	12.06	8.70	2575.38	80.890	6.755
Beijing 2015	810.6	12.03	8.62	2579.88	81.057	6.798
Rio 2016	800.7	12.03	8.52	2534.73	80.070	6.795

The influence of mass on the performance of the 100 m sprinter is also revealed in table 4, column 4, in which we confirm the drastic decrease in initial acceleration. Clearly, his initial acceleration reduced from $0.97g$ at Berlin 2009 to $0.95g$, $0.88g$, $0.89g$, $0.88g$ and $0.87g$ at Beijing 2008, London 2012, Moscow 2013, Beijing 2015 and Rio 2016 respectively.

In the last three columns of table 4 we can observe the aspects of Bolt's performance related to energy. For instance, table 4, column 5 shows the maximum instantaneous power reached at ${t}_{{P}_{\max }}=0.90\,{\rm{s}}$, ${t}_{{P}_{\max }}=0.89\,{\rm{s}}$, ${t}_{{P}_{\max }}=1.01\,{\rm{s}}$, ${t}_{{P}_{\max }}=0.97\,{\rm{s}}$, ${t}_{{P}_{\max }}=0.97\,{\rm{s}}$ and ${t}_{{P}_{\max }}=0.99\,{\rm{s}}$ at Beijing 2008, Berlin 2009, London 2012, Moscow 2013, Beijing 2015 and Rio 2016 respectively. The particularly high time taken to achieve maximum instantaneous power at London 2012 (1.01 s) is due to the particularly high tailwind at this competition, which in turn reduced the relative speed between the sprinter and the medium, so the drag terms are smaller. This later effect can be more clearly observed in table 4, columns 6 and 7, in which we show the total work done, as well as the effective work (the energy effectively transformed to motion) respectively. It results that the effective work is only 7.75%, 7.80%, 8.78%, 8.35%, 8.39% and 8.49% of the total work for Beijing 2008, Berlin 2009, London 2012, Moscow 2013, Beijing 2015 and Rio 2016 respectively. In the particular case of London 2012, we see that despite the small differences in run times with Berlin 2009 (9.63 s and 9.58 s respectively), the total work exerted to achieve 9.63 s at London was much lower than what was required at Berlin 2009. Likewise, the effective work was superior at London 2012, due to the smaller amount of dissipation from the drag terms induced by the particularly high tailwind.

As mentioned in the introduction, many factors can affect the effectiveness of a high-performance athlete. From the above analyses, the impact of tailwind speed and the mass of the runner on the run time of the 100 m sprinter has clearly been revealed. Nevertheless, the question of the impact of the increment of age on the time is still open. To tackle the age issue, we analyse Bolt's performance from Beijing 2008 to Rio 2016 in three hypothetical cases: as if all the events had occurred with the same tailwind, as if Bolt had had the same mass in each event, and as if all the events had had the same mass and tailwind conditions. The latter hypothetical case removes the influence of tailwind and mass, so it ought to show the influence of age on his performance. These cases have been calculated with the methodology described in section 2.

4.1. 2008–2016: without tailwind

The key physical facts of the events from Beijing 2008 to Rio 2016 were recalculated with standardised wind conditions, as if they had been run with no tailwind (${v}_{{\rm{w}}}^{({\rm{S}})}=0$). Clearly, the Beijing 2008 sprint is of particular importance as it was run without a tailwind (see table 3), so it does not change in this hypothetical case. Although several results regarding UB's physical parameters in each event are illuminating, we only show the impact of the hypothesis in the run time, which is shown in figure 3.

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We also condense the run time results in table 5 for the no-tailwind case. The obtained results confirm the conclusion of Hernández-Gómez et al (2013), in which a tailwind of ±1.0 m s^–1 changes the total sprint time by approximately $\mp 0.1\,{\rm{s}}$.

Table 5.  The hypothetical run times of UB in the 100 m sprint at Beijing 2008, Berlin 2009, London 2012, Moscow 2013, Beijing 2015 and Rio 2016, calculated with the same tailwind conditions (${v}_{{\rm{w}}}^{({\rm{S}})}=0$ m s^–1).

Event	Original wind	Standardised wind	Original running	Standardised running	${{t}}_{{\rm{O}}}-{{t}}_{{\rm{S}}}$
	speed ${{v}}_{{\rm{w}}}^{({\rm{O}})}$ (m s–1)	speed ${{v}}_{{\rm{w}}}^{({\rm{S}})}$ (m s–1)	time ${{t}}_{{\rm{O}}}$ (s)	time ${{t}}_{{\rm{S}}}$ (s)	(s)
Beijing 2008	0.00	0.00	9.69	9.69	0.00
Berlin 2009	0.90	0.00	9.58	9.67	−0.09
London 2012	1.50	0.00	9.63	9.79	0.16
Moscow 2013	−0.30	0.00	9.77	9.74	+0.03
Beijing 2015	−0.50	0.00	9.79	9.74	+0.05
Rio 2016	0.20	0.00	9.81	9.83	−0.02

4.2. 2008–2016: with the same mass

Bolt's performances in the events from Beijing 2008 to Rio 2016 were recalculated with a standardised mass of ${m}^{({\rm{S}})}=86\,\mathrm{kg}$, as if he had not changed mass from Berlin 2009 to Rio 2016 (including Beijing 2008). We assumed that the force he was able to exert did not change with his mass increments. Normally, a well-trained athlete can exert a greater force when he gains a small fraction (∼5%) of his weight, so this assumption is only addressed for comparison purposes. Clearly, the Beijing 2008 sprint is of particular importance as it was run with the same mass that he had at Berlin 2009, so it does not change in the hypothetical case. Although several results regarding the physical parameters in each event are illuminating, we only show the impact of the hypothesis on the run time, which is shown in figure 4.

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The results of this hypothetical case have been summarised in table 6. The average percentage increase in his mass was 8.02%, whilst the average difference in ${t}_{O}-{t}_{{\rm{S}}}$ was 0.11 s; this suggests that an 8% increase in mass results in an increment of approximately 0.11 s in the run time. The times in table 6 consider the sprinter's reaction time in each event from 2008 to 2016, where we can observe that a new world record would have been established in London 2012 (9.53 s) under these conditions. It is possible that if Bolt had not been disqualified at Korea 2011, he would have had a quicker reaction time—perhaps closer to 0.142 (the reaction time of Berlin 2009)—at London 2012; in such a hypothetical case, if in addition he had not gained any weight, in the best case, the 100 m world record would have been 9.51 s, favoured by the +1.5 m s^–1 tailwind at the event.

Table 6.  The hypothetical run times of UB's 100 m sprints at Beijing 2008, Berlin 2009, London 2012, Moscow 2013, Beijing 2015 and Rio 2016, calculated as if they had been run with the same mass that he had in Berlin (${m}^{({\rm{S}})}=86$ kg).

Event	Original mass	Standardised	${{m}}^{({\rm{S}})}/{{m}}^{({\rm{O}})}$	Original	Standardised	${{t}}_{{\rm{S}}}/{{t}}_{{\rm{O}}}$	${{t}}_{{\rm{O}}}-{{t}}_{{\rm{S}}}$
	${{m}}^{({\rm{O}})}$ (kg)	mass ${{m}}^{({\rm{S}})}$ (kg)	(1)	time ${{t}}_{{\rm{O}}}$ (s)	time ${{t}}_{{\rm{S}}}$ (s)	(1)	(s)
Beijing 2008	86	86	1.000	9.69	9.69	1.000	0.000
Berlin 2009	86	86	1.000	9.58	9.58	1.000	0.000
London 2012	93	86	0.925	9.63	9.53	0.989	0.102
Moscow 2013	93	86	0.925	9.77	9.67	0.990	0.099
Beijing 2015	94	86	0.915	9.79	9.68	0.989	0.112
Rio 2016	94	86	0.915	9.81	9.70	0.988	0.114

4.3. 2008–2016: without tailwind and with the same mass

Finally, Bolt's hypothetical performance in the events during the time span 2008–2016 was calculated, considering the same standardised wind conditions ${v}_{{\rm{w}}}^{({\rm{S}})}=0$ m s^–1 as well as with the same standardised mass ${m}^{({\rm{S}})}=86\,\mathrm{kg}$. Clearly, the Beijing 2008 sprint is of particular importance as it was run in the same conditions as the standardised case, so it does not change in the hypothetical case. We again assume that the exerted constant force has not changed from what was calculated in the real case. The results for the run times are shown in figure 5.

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From table 7 we can observe that the mass and tailwind conditions linearly affect the run time result. Moreover, if Bolt had not changed his mass from Berlin 2009, and if all sprint events had been run under the same tailwind conditions (${v}_{{\rm{w}}}^{({\rm{S}})}=0$), the current world record would have not been established in Berlin 2009 but in Beijing 2015, and it would have been 9.63 s. We must recall that these results take into account Bolt's real reaction times (see table 3), which cannot be estimated hypothetically, as the psychological aspect plays an important role on reaction times, besides the essential physiological and physical determining reasons. This way, in real terms, we can affirm that UB's best year in terms of performance was 2015.

Table 7.  The hypothetical run times of UB's 100 m sprints at Beijing 2008, Berlin 2009, London 2012, Moscow 2013, Beijing 2015 and Rio 2016, with the same tailwind conditions (${v}_{{\rm{w}}}^{({\rm{S}})}=0$ m s^–1) as if his mass had not changed from that at Berlin 2009 (${m}^{({\rm{S}})}=86$ kg).

Event	Original mass	Original wind	Original	Standardised	${{t}}_{{\rm{O}}}-{{t}}_{{\rm{S}}}$
	${{m}}^{({\rm{O}})}$ (kg)	speed ${{v}}_{{\rm{O}}}$ (m s–1)	time ${{t}}_{{\rm{O}}}$ (s)	time ${{t}}_{{\rm{S}}}$ (s)	(s)
Beijing 2008	86	0.00	9.69	9.69	0.00
Berlin 2009	86	+0.90	9.58	9.67	−0.09
London 2012	93	+1.50	9.63	9.69	−0.06
Moscow 2013	93	−0.30	9.77	9.64	+0.13
Beijing 2015	94	−0.50	9.79	9.63	+0.16
Rio 2016	94	+0.20	9.81	9.72	−0.09

With respect to the question of the impact of age on Bolt's overall performance in the time span considered here, it is difficult to draw definite conclusions from the standardised results (see figure 5). On the one hand, the best and worst run times, under the same mass and wind speed conditions, would have happened in the consecutive years 2015 and 2016. It is important to observe that the differences between the standardised run times are not as drastic as they are in the real ones, which shows UB to be a more stable athlete with respect to his performance as a 100 m sprinter. Nevertheless, the least square fit to the standardised data reveals that from 2008 to 2016 his run times have remained essentially constant (see figure 5). This result would suggest an answer to the question posed in the introduction of this paper, namely that in the particular case of UB, his performance does not seem to have been affected by his age, from his world-record-breaking 100 m sprint in Berlin 2009 up to the 2016 Rio OG.

Although we know that we have very little data to be able to interpret confidently the least square fits in figures 3, 4 and 5, we just interpret the linear fits as main tendencies. Maybe with the data of future events, the tendency will be clearer, allowing us to perform a more conclusive fit.

In this paper, we addressed the main query of why UB has not been able to break his own Berlin 2009 world record in the 100 m sprint. Although we do not possess highly validated measurements of Bolt's mass for the different events studied herein, in light of having the best available data for such masses, we conclude that his mass increment has been a determining factor, aided in some cases by the tailwind at particular events. With respect to the run times in standardised wind and mass conditions, the question naturally arises of whether the differences in standardised run times between different events may be considered as fluctuations from the almost constant running time shown by the data fit featured in figure 5, or not; these fluctuations would have been influenced by diverse and mainly nonphysical factors. Although we do not have enough results to state such a conclusion, this issue will be illuminated by Bolt's performance in the 100 m sprint at the 2017 IAAF World Athletics Championships, which will be held in London from August 4th to 13th, 2017. If his time in London 2017 is slower than at Rio 2016, we will then be able to confirm that in his case, they are effectively fluctuations. But if his London 2017 time is quicker than his time at Rio 2016, that would mean that Beijing 2015 had been his best year, and that from then on, age had started to play an important role on his overall performance throughout the years.

First of all, for future investigations it would be extremely useful for researchers to have access to accurate mass records of runners before each event. We think that a useful idea would be for the IAAF to establish a weight measure of runners at the beginning and at the end of each sprint event, just as is done for Formula One drivers. In such a case, this study could be repeated for other sprinters with a well-known mass record across the years. It would also be interesting for future studies to determine the influence of mass on high-performance short sprinters in controlled experimental environments. A suit that artificially adds different weights to the athlete, equally distributed along their body in order to emulate the fattening process, while minimising changes in aerodynamics and/or drag coefficient, might help to realise such a study. Such experimental investigations, together with the conclusions of this paper, are likely to have an influence on the performance of the so-called 'queen' of track competitions in athletics, the 100 m sprint.

The authors acknowledge the partial support of projects 20171536, 20170721 and 20171027, as well as an EDI grant, all provided by SIP/IPN.