The evolution of speed, size and shape in modern athletics

Broadly speaking, larger animals move faster on earth than smaller animals(Bejan, 2000; Hoppeler and Weibel, 2005; Weibel, 2000). This aspect of animal scaling is often overlooked because large animals appear sluggish. Geese flap their wings infrequently. Whales swing their tails as if not in a hurry. The scaling relations of animal locomotion have been measured and studied empirically as three separate locomotion mechanisms: flying(Bartholomew and Casey, 1978; Greenewalt, 1975; Lighthill, 1974; Marden et al., 1997; May, 1995; Tennekes, 1997; Wakeling, 1997), running(Heglund et al., 1974; Iriarte-Diaz, 2002; Marsh, 1988; Pennycuick, 1975) and swimming(Arnott et al., 1998; Brett, 1965; Childress and Dudley, 2004; Drucker and Jensen, 1996; Kiceniuk and Jones, 1977; Peake and Farrell, 2004; Rohr and Fish, 2004; Videler, 1993).

More recently, a broader view of the commonality of animal locomotion has been emerging (Ahlborn, 2004; Bejan, 2000; Bejan, 2005; Bejan and Marden, 2006; Bejan and Lorente, 2008; Hoppeler and Weibel, 2005; Marden and Allen, 2002; Muller and van Leeuwen, 2004; Taylor et al., 2003; Weibel, 2000). For example,according to constructal theory, animal locomotion is a rhythm of body motion constructed such that the animal achieves a balance between two expenditures of useful energy: lifting weight on the vertical, and overcoming drag while progressing on the horizontal (this analysis is reviewed in the Appendix). The sum of the two efforts is minimal when the two efforts are of the same size. In this regime the body-mass scaling relations (slope and intercept) of animal locomotion are predicted and are in agreement with the measurements of swimmers, runners and fliers over the body mass (M) range 10^–6–10³ kg:

• travel speeds (V) proportional to M^1/6, where M is the body mass, ρ is the body density and gis gravitational acceleration, for example, for running on soft ground and swimming:

  \[\ V{\sim}\mathbf{g}^{1{/}2}{\rho}^{-1{/}6}M^{1{/}6},\]

  (1)
• body frequencies (t^–1) (flapping, stride, fish tailing) proportional to M^–1/6:

  \[\ t^{-1}{\sim}\mathbf{g}^{1{/}2}{\rho}^{1{/}6}M^{-1{/}6},\]

  (2)
• body force (F) scale equal to M:

  \[\ F{\sim}M\mathbf{g},\]

  (3)
• food requirement (useful energy, W) per distance traveled(L_x), which is proportional to M:

  \[\ \frac{W}{L_{\mathrm{x}}}{\sim}M\mathbf{g}.\]

  (4)

The corresponding scaling laws for flying and running with air drag are similar to Eqns 1, 2, 3, 4. These relations are accurate within a dimensionless factor of order 1, as they were derived based on scale analysis. In spite of this built-in approximation, they agree well with the large body of experimental data available(Bejan and Marden, 2006).

We used this constructal framework to examine the evolution of speeds in modern athletics: the evolution of the sport (winning speeds and body metrics), not the evolution of the athletes. We focused on the two most documented probes for men, the 100 m-freestyle in swimming and the 100 m-dash in track. These are sprint probes, not endurance events. Sprint probes require intense expenditure of work during a relatively short period of time.

In Fig. 1A and Table 1 we see the evolution of the world speed record (V) for male 100 m-freestyle swimming since 1912. Because of the theoretical scaling (i), we also researched the evolution of the body masses of the record-breaking athletes(Fig. 1B). Both V and M have been increasing in time (t). By eliminating t between Fig. 1A,B,we found Fig. 1C, which shows the evolution of V vs M.

Noteworthy are the factors 0.72 and 4.85 in Eqns 5a and 6, respectively. These factors of order 1 agree with the theoretical scaling(Eqn 1), which after substituting the values for g and ρ yields for swimming and running V∼1×M^1/6. Here `1' is the intercept of the line plotted as log V vs log M. Also noteworthy is that the factor for running (4.85) is greater than the factor for swimming (0.72). This also agrees with the manner in which the empirical factor (not shown in Eqn 1 but reported in the Appendix) differentiates between the power-law correlations of animal speed data for runners and swimmers (Bejan and Marden, 2006).

The proportionality between speeds and body length raised to the power 1/2(Eqns 7 and 8) suggests a simpler way to derive the speed–mass scaling rule (Eqn 1), much simpler than the analysis shown in the Appendix. During each cycle of locomotion the body falls from a height of order L_b. The time scale of the fall is of order t∼(L_b/g)^1/2. The body falls forward to a distance of order L_b; therefore,the horizontal velocity scale is V∼L_b/t∼(gL_b)^1/2. Combining this V scale with L_b∼(M/ρ)^1/3 we arrive at Eqn 1.

The S effect differentiates between running and swimming. Dividing Eqns 11 and 10 we anticipate V_run/V_swim∼S^1/2,which is a number of the same order as the ratio between Eqns 6 and 5a. The two-scale model also suggests that from among athletes with the same mass, the ones with larger S values are more likely to run fast. In swimming, the Seffect is the opposite but weaker: swimmers would be slightly faster if more robust (smaller S).

The same S data are plotted against M in Fig. 6A,B. The data are too sparse to yield statistically significant correlations; however, qualitatively they suggest a slight increase in S vs M for running and a slight decrease in S vs M for swimming.

The scaling trends revealed by the speed data suggest that speed records will continue to be dominated by heavier and taller athletes. This trend is due to the scaling rules of animal locomotion, not to the contemporary increase in the average body size of humans. The mean height of humans has increased by roughly 5 cm from 1900 to 2002(Plastic Soldier Review,2002). During the same century, the mean height of champion swimmers and runners has increased by 11.4 cm and 16.2 cm, respectively(Fig. 3C, Fig. 4C).

The insight gained in this paper allows us to speculate what the running speeds might have been in ancient Greece and the Roman Empire. There is no record of what the winning speeds were then, because the competition was for winning the race, not for breaking a time record. Chronometry did not exist. In antiquity body masses were roughly 70% of what they are today(Plastic Soldier Review, 2002; Hpathy 2009; National Health and Nutrition Examination Survey, 1999). According to Eqn 6, this means that speeds were lower by a factor of roughly(0.7)^1/6=0.94. In other words, if the 100 m dash in military training today is won in 13 s, 2000 years ago it would have been won in∼14 s.

This insight also teaches us why certain training techniques are successful in high-performance sports. For example, in modern speed swimming, the doctrine holds that the swimmer must raise his body to the highest level possible above the water. Two explanations are given for this swimming doctrine: air drag is much smaller than water friction, and the water wave generated by the body propels the body better(Collela, 2009). The doctrine is correct but for a different reason, which is evident in Eqn 7. When the body is high above the water it falls faster (and forward) when it reaches the water line. For the same reason, the speeds of all water waves exhibit the same scale as in Eqn 7, in which L_b is the length scale of the wave(Prandtl, 1969). The crest of the wave falls with a speed of order(gL_b)^1/2, which becomes visible as the forward speed of the traveling wave.

In the future, the fastest athletes can be expected to be heavier and taller. If the winners' podium is to include athletes of all sizes, then speed competitions might have to be divided into weight categories. This is not at all unrealistic in view of the body force scaling(Eqn 3), which was recognized from the beginning in the structuring of modern athletics. Larger athletes lift, push and punch harder than smaller athletes, and this led to the establishment of weight classes for weight lifting, wrestling and boxing. Larger athletes also run and swim faster.

Here is a brief summary of the scale analysis of animal locomotion, which leads to Eqns 1, 2, 3, 4. It was first done for flying(Bejan, 2000) and then generalized to all locomotion: running, flying and swimming(Bejan and Marden, 2006).

In conclusion, the scaling relations (Eqns 1, 2, 3, 4) have been derived here in Eqns A7, A8, A9, A10. The modifying factor(ρ_m/ρ)^1/3 depends on the medium. In flying, theρ _m (air) is roughly equal to ρ/10³, and the factor (ρ_m/ρ)^1/3 is close to 1/10. In swimming,the ρ_m (water) is the same as the body density, and the factor(ρ_m/ρ)^1/3 is 1. In running, the modifying factor is between 1/10 and 1, and depends on the nature of the running surface and the speed. For example, running through snow, mud and sand is represented by a(ρ_m/ρ)^1/3 value close to 1. Running at high speed on a dry surface is represented more closely by a(ρ_m/ρ)^1/3 factor similar to the one that represents flying.

The data collected in fig. 2of Bejan and Marden (Bejan and Marden,2006) also confirm that the medium factor(ρ_m/ρ)^1/3 does not have an effect on the frequencies (t^–1) and forces (F) of flyers,runners and swimmers. This is in accordance with Eqns A8 and A9, in which(ρ_m/ρ)^1/3 is not present.

LIST OF ABBREVIATIONS

 
• C_D

  drag coefficient

 
• F

  force, N

 
• F_drag

  drag force, N

 
• g

  gravitational acceleration, m s^–2

 
• H

  body height, m

 
• L

  body width, m

 
• L_b

  single body length scale, m

 
• L_x

  distance traveled, m

 
• M

  body mass, kg

 
• P

  probability of true null hypothesis

 
• R²

  coefficient of determination

 
• S

  body slenderness, H/L

 
• t

  period, s

 
• t

  time, years

 
• t^–1

  frequency, s^–1

 
• V

  speed, m s^–1

 
• W

  ork, useful energy, J

 
• W₁

  work done vertically, J

 
• W₂

  work done horizontally, J

 
• ρ

  body density, kg m^–3

 
• ρm

  density of the medium, kg m^–3