A mathematical model is used to determine the effect of wind resistance and altitude on 200-m race times. The model is used to simulate the effect of different wind speeds, wind directions, and lane allocation in the 200-m sprint. For the 200-m event there is a single wind gauge recording only the wind’s component velocity in the straight, and not the wind direction. For record purposes the wind reading should not exceed 2 m/s. It is evident that for the same official wind reading, an athlete may face vastly different conditions. The model estimates that for the same wind reading, the prevailing conditions can produce a time difference of as much as 0.5 s. Results indicate that many legal performances in the 200-m are currently ruled out for record purposes. Some performances which are officially wind-assisted have in fact been hindered by an overall head wind when it is averaged throughout the race. Conversely, some performances which are run into a head wind in the straight have benefited from an overall tail wind. We estimate that, on average, for a 2-m/s wind blowing down the straight, the 200-m runner benefits from an overall tail wind of only 0.95 m/s. The lower air density at an altitude of 1,500 meters produces an advantage of 0.11 s in the 200-m, which equates to a 2-m/s tail wind at sea level blowing directly down the straight. Correction estimates are provided for the combined effect of both wind and altitude in 200-m races. A new all-time world top five ranking list for men and women is produced for the 200-m event, corrected for wind and altitude effects.

Key Words: athletics, mathematical model, running, air resistance

Introduction

Over recent years there has been much interest in modeling the effects of wind and altitude on sprinting. A sprinter is hindered by an aerodynamic drag force which depends on both the air density and the prevailing wind. A head wind increases the drag force, which will reduce the athlete’s speed. Conversely, the drag force will be reduced if there is either a tail wind or a lower air density (as occurs at high altitude). A sprinter’s performance can be significantly affected by these factors.

Several studies have used mathematical models to quantify the effects of wind and altitude. Many of the models derive from the pioneering work of Hill (1928) and later Keller (1973). Subsequently, Ward-Smith (1984,1985a, 1985b, 1999), Kyle and Caiozzo (1986), Dapena and Feltner (1987), Pritchard (1993), and Behncke (1994) have modeled the effect of wind on running. In addition, Linthorne (1994) conducted a comprehensive statistical analysis of the effect of wind on 100-m sprint times. All previous research has concentrated on modeling the 100-event in which the sprinter runs in a straight line, and there has been little attention paid to the equivalent effects in the 200-m sprint.

The effects in the 200-m event are much more complicated than in the 100-m event. The first half of the race is run around a bend where there is no information about wind speed. The only officially recorded information is the wind’s component velocity in the straight, but not the direction of the wind. The conditions faced by the 200-m sprinter depend on wind speed, wind direction, and lane allocation. Moreover, the radius of each lane depends on the track specification, which can vary between venues. Furthermore, the air density will vary with altitude. Clearly we need a better understanding of these effects in the 200-m sprint.

The purpose of this study was to produce a mathematical model to simulate the 200-m event and use it to analyze the effects of wind and altitude. The first objective was to use the model to simulate the 100-m sprint and then compare the results for wind and altitude effects with previous studies. Four research questions are posed: ( 1) What is the effect of different wind speeds and directions on the performance in the 200-m event? ( 2) What is the effect of the lane allocation in the 200-m event under identical wind readings? ( 3) What is the combined effect of wind and altitude in the 200-m sprint? ( 4) How does the combined effect of wind and altitude affect the current all-time world rankings in the sprints?

Methods

The Keller model is based on Newton’s law; the equation of motion of the runner is given by:

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where v(t) is the runner’s velocity at time t, in the direction of motion, and f(t) is the runner’s propulsive force per unit mass. The resistive force is linear in the velocity v, and τ is the time constant. We assume that f(t) ≤ F, where F is the maximum force per unit mass the runner can exert. In this study the Keller model was extended to include the reaction time of the sprinter and an air resistance term, which depends on the relative velocity between the sprinter and the air. The equation of motion for the extended model is given by:

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where v[subw] is the velocity of the wind relative to the ground and

α = ρC[subd]A/2M (3)

A positive value for v[subw] corresponds to a tail wind, while a negative value corresponds to a head wind. The value of a depends on p the air density, M the mass of the athlete, A the frontal area of the athlete, and C[subd] the drag coefficient. C[subd] depends on the shape and surface properties of the athlete’s body and clothing, and is determined empirically. The correct value to take for C[subd] is open to debate and previous authors have used values between 0.7 and 1.1. In this study we take C[subd] = 1.0, which is consistent with the value used by Dapena and Feltner (1987) and Pritchard (1993). The frontal area A is the area of the projection of the body onto a plane orthogonal to the direction of v. We can assume the frontal area A is proportional to the total surface area of the athlete A[subb]:

```
A = KA[subb] (4)
```

where K is the constant of proportionality, which is determined empirically. Here we take K = 0.24 (Dapena & Feltner, 1987; Davies, 1980). The total surface area is related to the standing height h and mass M of the athlete by the equation

A[subb] = 0.2l7h[sup0.725] M[sup0.425] (5)

(Dapena & Feltner, 1987; Vaughan, 1983). Typical M and h values are obtained by averaging the heights and weights of the top 20 all-time sprinters (Matthews, 1992, 1997, 2001). The values obtained for h and M, respectively, are 1.80 m and 74 kg for men, and 1.67 m and 57 kg for women. Using these values, the estimates for the frontal area A are 0.48 m² for men and 0.40 m² for women, which compare well with the values of 0.50 m² for men and 0.42 m² for women obtained by Linthorne (1994). For a sprint race at sea level at a temperature of 25 °C, an appropriate figure for summer competition, the air density is estimated to be p = 1.184 kg/m³ (Dapena & Feltner, 1987).

In the extended Keller model we assume that after a reaction time of 8 seconds, the runner exerts maximum force f(t) = F. Consider at first a head wind of constant velocity v[subw] < 0 against the direction of motion. For F, T, and a in the appropriate range, the velocity of the runner for t ≥ δ is given by:

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where

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In the case of a constant tail wind of velocity v[subw] > 0 directly behind the runner, there will be a short period at the beginning of the race when the runner’s velocity is less than v[subw]. Over this period δ ≤ t ≤ t[sub1], the value of a will be slightly different, since the area A is now the back of a runner who is leaning forward in the acceleration phase. There is little research to indicate the correct value of A to take; however, the overall effect on the time is extremely small except when there is an unusually strong tail wind, for example in excess of 8 m/s. For t > t[sub1] the runner’s velocity exceeds the tail wind and is given by Equation 6 with 5 replaced by t[sub1] A least-squares fit of the model to the current world records (Table 1) produced constant values of F = 12.08 m/s, T = 0.955 s for men, and F = 12.19 m/s, τ = 0.848 s for women.

In the calculations the reaction time δ has been set at the minimum allowable value of 0.1 s (IAAF Handbook 2001–2002). In reality, the reaction time for a world class sprinter is rarely below 0.13 s and averages about 0.15 s. This is illustrated by results from the 100-m at the 2001 World Athletics Championships in Edmonton. Data from the quarter final stage onward is used in order to eliminate any inexperienced runners from Round 1. In the men’s event the reaction times of 61 performances ranged from 0.123 to 0.186 s, with a mean of 0.154 s. Only two athletes had reaction times faster than 0.130 s. The reaction times of the women sprinters were slightly quicker, ranging from 0.119 to 0.189 s, with a mean of 0.146 s.

On a standard outdoor track, the 200-m event involves bend running for more than 100 m. The wind conditions for the athlete vary around the bend until entering the straight. This means that Equation 6, which assumes a constant wind velocity, is unrealistic for 200-m races. The precise wind conditions facing the runner depend on the track specification, the lane allocation, and the strength and direction of the wind.

The specification of a running track must satisfy IAAF recommendations which require a flat surface with two parallel straights and two semicircular bends of equal radius. In particular, a track must have an inside curb radius of between 32 and 42 m for a 400-m circuit. The measurement of Lane 1 of the track is taken to be 0.3 m from the outer edge of the inside curb. The measurement in the other lanes is taken to be 0.2 m from the outer edge of the inside lane markings. Lane width should be a minimum of 1.22 m and a maximum of 1.25 m, measured radially from the inside edge of the lane to the inside edge of the adjacent lane. The minimum allowable radius of 32 m produces a track with long straights (98.5 m), but most athletes find the bends too tight. Conversely, a radius of 42 m produces wider bends but with very short straights (67.1 m).

The standard track has a radius of 36.5 m and a lane width of 1.22 m. Taking these dimensions into account gives a track with straights of 84.4 m, which are extended backward to facilitate races such as 100-m and 110-m hurdles. It means that 115.6 m of the 200-m event is run around a bend, the curvature of which varies from lane to lane. This makes the exact effect of a tail wind or head wind difficult to calculate. However, at any particular moment we can calculate the effective wind faced by the runner and estimate an average for the entire race. This can be used in Equation 6 to model the effect of different wind conditions on 200-m times.

Since 1936 the IAAF has agreed that for official recognition of records in the 100-m, the tail wind must not exceed 2 m/s. Wind velocity is measured for 10 s by a wind gauge and must follow specific regulations (IAAF Handbook 2001—2002). The wind gauge is placed at a height of 1.22 m, a distance of 50 m from the finish line, and not more than 2 m away from the track. The wind velocity is measured in m/s, rounded up to the next 0.1 m/s in the positive direction. In the 1950s the IAAF extended the wind assistance rule for official recognition of records to the 200-m event but decided that a single wind gauge was sufficient. The requirement was that the wind’s component velocity in the direction of the straight should not exceed 2 m/s, the same limit as for the 100-m event. The wind reading is taken only for a period of 10 s beginning when the first runner enters the straight. Since less than half the race is run along the straight, this can produce some surprising results. It means that even when there is a tail wind registered, for a major part of the race the wind may be hindering the runner.

One problem is that the actual direction of the wind is not recorded, only the magnitude of the component of the wind’s velocity in the direction of the straight. This means that a 2-m/s wind reading can be produced from winds of different strengths coming from different directions. The angle the wind direction makes with the straight is ± 9 (Figure 1). We can vary the angle 6 and the wind strength but maintain the same wind reading in the straight.

The air density at high altitude is less than at sea level. This makes a noticeable difference for sprint events. Although the IAAF does not restrict the altitude of a competition venue for the acceptance of records, it is generally accepted that performances at altitudes of over 1,000 m are “altitude assisted.” According to Moran, Morgan, and Pauley (1996), the air density, p[subH], at an altitude of H meters above sea level, is related to the density at sea level, p[sub0], by the equation

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where T is the air temperature in degrees Celsius, g is gravitational acceleration, and the gas constant R = 287 J/kg. This gives, for example, an air density of p = 0.915 kg/m³ at 25 °C in Mexico City (altitude 2,250 m). There is no evidence that C[subd] is affected by an increase in altitude, and any variations in the gravitational acceleration due to the location are insignificant (Behncke, 1994). Therefore, in windless conditions it appears that any advantage to sprinting at altitude results from reduced air density alone.

Results

The extended Keller model predicts 100-m wind and altitude effects in agreement with the research of Linthorne (1994) and Ward-Smith (1999). The largest legal wind advantage is approximately 0.1 s (0.101 s for men and 0.110 s for women) (Table 2). In contrast, a 2-m/s head wind can cost a male sprinter 0.121 s and a female sprinter 0.134 s. With no wind, an altitude of 1,000 m provides an advantage of 0.032 s in the 100-m.

In the 200-m there is a large variation in the time advantage depending on wind speed and wind direction. Given an official wind reading of +2 m/s, we find that the prevailing wind conditions throughout the 200-m race vary greatly with wind direction (Table 3). If the wind direction is ϑ = -70° (Figure 1), the actual wind strength is 5.85 m/s and the overall effect on the runner is to face a head wind of 1.21 m/s, averaged throughout the race. This slows the male sprinter by 0.16 s and the female sprinter by 0.17 s. In contrast, if ϑ = +53.2°, the wind strength is +3.34 m/s and the runner has the benefit of an overall tail wind of 2 m/s averaged throughout the race. This gives the sprinter an advantage of 0.22 s for men and 0.25 s for women. In both examples above, the average value of the eight running lanes has been taken.

There is a significant difference in wind conditions encountered by runners in the eight lanes. Since the wind direction is not recorded, the best strategy is to assume a median value of ϑ = 0, which is a wind blowing directly along the straight. If the wind reading is +2 m/s, then the time advantage in Lane 1 is 0.099 s for men and 0.110 s for women, while in Lane 8 it is 0.126 s for men and 0.139 s for women (Table 4). A wind reading of +4.22 m/s, again blowing directly down the straight, produces an average assistance throughout the 200-m race of 2 m/s, which equates to the legal limit in the 100-m sprint. For a wind of this strength, the time advantage in Lane 1 is 0.202 s for men and 0.253 s for women, while in Lane 8 it is 0.222 s for men and 0.277s for women (Table 4). The time lost or gained due to wind conditions for the 200-m can be compared to the results for the 100-m (Table 2). The results assume ϑ = 0 and take an average of the eight lanes.

In the 200-m event there is a significant combined effect of wind and altitude. For example, a zero wind at an altitude of 2,000 m produces a time advantage of 0.14 s, and a 2-m/s tail wind at the same altitude gives a male athlete an advantage of 0.25 s, compared to 0.26 s for women (Table 5).

The combined correction for both wind and altitude has a significant effect on the world 200-m rankings. A new top five 200-m list for men and women is produced which has been corrected for the effects of both factors (Table 6). By removing the advantages given by favorable wind and altitude, we can find the best overall performance.

Discussion

By a careful choice of the parameters, the extended Keller model has produced results for the 100-m which are in close agreement with the statistical study conducted by Linthorne (1994). The model predicts an advantage of 0.101 s for a male sprinter with a 2-m/s tail wind, compared to a figure of 0.10 s (Linthorne, 1994). For a 2-m/s head wind, the time lost by a male runner is estimated to be 0.121 s, compared to Linthorne’s figure of 0.123 s. The results for stronger winds are also very similar. For a 5-m/s head wind, the time lost is estimated to be 0.340 s for men (Table 2) compared to Linthorne’s figure of 0.350 s.

The effects of altitude predicted by the model also agree with Linthorne’s (1994) results. At an altitude of 1,380 m at Provo, Utah, we calculate that the advantage in the men’s 100-m is 0.04 s. This result is consistent with the statistical analysis of the 1989 NCAA Championship results held in Provo, cited by Linthorne (1994). The results for women athletes are not quite as good but still show agreement. For example, the model predicts an advantage of 0.110 s for a female sprinter with a 2-m/s tail wind compared to a figure of 0.12 s in Linthorne (1994). The predictions for both men and women in the 100-m are also in agreement with the results of Ward-Smith (1999).

Wind speed and wind direction have a major effect on 200-m times (Table 3). The same wind reading but different wind direction can produce a time difference as much as half a second. For a given wind reading in the straight, we can estimate the average wind conditions faced by each of the eight runners over the entire race. If the wind reading is 2 m/s and the wind direction ϑ lies between — 90° and −50.3°, the runner actually faces a head wind when averaged throughout the race. Since this comprises 22% of all possible wind directions, there will be many performances ruled as wind-assisted which had no benefit from the wind at all. If 0 lies between −50.3° and 53.2°, the sprinter experiences an overall tail wind below 2m/s. Only when ϑ lies between 53.2° and 90° is the advantage to the runner, averaged over the whole race, greater than 2 m/s. In fact for 80% of all possible wind directions, the runner is either facing a head wind or getting a tail wind below the legal limit for the 100-m.

Comparing the correction estimates for wind direction ϑ = −70° and ϑ = +70° (Table 3), we have, on average, a difference of 0.49 s for men and 0.53 s for women. This is a huge variation, considering that officially the wind conditions are identical. Similar anomalies occur when there is a head wind in the straight. For example, a 2-m/s head wind recorded in the straight which originates from a wind blowing at an angle of—75° produces a tail wind of +2 m/s averaged throughout the race. This makes comparisons of 200-m performances extremely difficult, even when the official wind readings are the same.

To compare the wind conditions encountered by runners in each of the eight lanes, we can assume that the wind is blowing directly along the straight. When there is a 2-m/s tail wind recorded in the straight, runners in each of the eight lanes have an overall tail wind which is well below the legal limit allowed for the 100-m race. We estimate that, on average, the runners benefit from a 0.95-m/s tail wind which is less than half the allowable limit for the 100-m. This compares with the figure of 0.91 m/s calculated by Heidenstrom (1992) for a slightly different track specification. In terms of time, the men have a 0.112-s advantage and the women have a 0.123-s advantage (Table 4). This is only slightly more than in the 100-m (Table 2) and is considerably less than the corrected values given by Behncke (1994), although it is not clear which track specification he used.

It is evident that much more information is required about the wind conditions in the 200-m. Even in 100-m races, the official wind reading does not accurately represent the conditions affecting each sprinter (Linthorne, 2000). A minimum requirement should be two wind gauges, one placed on the bend and one in the straight. However, this alone would not be enough to take into account wind direction and the differences between lanes, let alone any swirling winds present in the stadium. Ideally, a series of wind gauges placed along the length of the 200-m on both sides of the track would improve the accuracy of the wind reading.

The combined effect of wind and altitude is clearly evident in the 200-m sprint (Table 5). An altitude of around 1,500 m produces a time advantage which is comparable with the maximum legal wind advantage at sea level. This assumes a wind along the straight and an average of all eight lanes. In the men’s top 20 alltime performances, 6 were run at an altitude of over 1,000 m. Correcting performances for the effects of both wind and altitude changes, the all-time top 5 lists for men and women are shown in Table 6. In both lists the top performers are the same, but further down the lists there are significant changes. In particular, we can reassess the 1979 performance of Pietro Mennea at high altitude in Mexico City, when he clocked 19.72 s for the 200-m.

This amazing world record, which stood for almost 17 years, is still the third best-ever performance and was much faster than Mennea ever ran before or since. Correcting the time for both altitude and the tail wind of 1.8 m/s produces a revised time of 19.97 s, which drops it down to only 18th best. This compares to Mennea’s best low altitude time of 19.96 s, run in windless conditions. This illustrates the significant combined effect of altitude and a tail wind.

This study has several limitations. Although the basic Keller model has been extended, it does not accurately predict the observed velocity profile of world class sprinters (Ferro, Rivera, Pagola, et al., 2001). In addition, the unreliability of official wind readings makes the real effect of the wind in a 200-m race very difficult to predict. It is clear that more information is needed concerning the wind speed and direction throughout the entire 200-m race. Despite these limitations, this simulation can provide valuable insight into the effects of wind and altitude in the 200-m sprint.

Table 1 Men’s and Women’s Sprint World Records

Current world records Time in seconds

Distance Men Women

50 m (indoors) 5.56 5.96

60 m (indoors) 6.39 6.92

100 m 9.79 10.49

200m 19.32 21.34

Source: http: // www.iaaf.org

Table 2 Effect of Wind (m/s) on the Time in the 100-m and 200-m

Sprint

Head wind:

time

lost (s) −5.0 −4.0 −3.0 −2.0 −1.0

Men’s 100 0.340 0.262 0.189 0.121 0.058

Men’s 200 0.318 0.251 0.184 0.121 0.059

Women’s 0.378 0.291 0.209 0.134 0.064

100

Women’s 0.357 0.280 0.205 0.135 0.065

200

Tail wind:

time

gained (s) +1.0 +2.0 +3.0 +4.0 +5.0

Men’s 100 0.053 0.101 0.145 0.183 0.217

Men’s 200 0.056 0.112 0.164 0.214 0.262

Women’s 100 0.058 0.110 0.156 0.196 0.231

Women’s 200 0.062 0.123 0.180 0.235 0.287

Note: In the 200-m sprint, wind direction is assumed to be

along the straight (ϑ = 0) and an average of the 8

lanes is taken.

Legend for Chart

A-Wind direc. θ°

B-Actual Wind speed (m/s)

C-Lane 1

a-Men

b-Women

D-Lane 8

E-Avg of all 8 lanes

Table 3 Correction Estimates (s) in the 200-m for a +2 m/s

Official Wind Reading

A B C D

a b a b

E

a b

−70 5.85 −0.14 −0.16 −0.14 −0.17

−0.16 −0.17

−50.3 3.13 0 0 0.01 0.01

0 0

−30 2.31 0.05 0.06 0.07 0.08

0.06 0.07

0 2 0.10 0.11 0.13 0.14

0.11 0.12

+30 2.31 0.15 0.16 0.18 0.20

0.16 0.18

+53.2 3.34 0.20 0.22 0.24 0.27

0.22 0.25

+70 5.85 0.30 0.33 0.35 0.38

0.33 0.36

Table 4 Time Advantage (s) for Each Lane in the 200-m Sprint With

Two Wind Speed Readings

θ = 0 Wind reading = +2 m/s Wind reading = +4.22 m/s

Lane Men Women Men Women

1 0.099 0.110 0.202 0.222

2 0.100 0.111 0.203 0.223

3 0.105 0.116 0.211 0.232

4 0.109 0.121 0.219 0.241

5 0.113 0.125 0.228 0.250

6 0.116 0.128 0.234 0.256

7 0.123 0.136 0.247 0.271

8 0.126 0.139 0.253 0.277

Avg. 0.112 0.123 0.225 0.247

Note: Wind direction is assumed to be along the straight

(θ = 0).

Table 5 200-m Time Advantage (s) for Varying Tail Winds and

Altitude

Wind Speed (m/s)

Altitude +0.0 +1.0 +2.0

(m) Men Women Men Women Men Women

0 0 0 0.06 0.06 0.11 0.12

500 0.04 0.04 0.09 0.10 0.15 0.16

1,000 0.07 0.07 0.13 0.13 0.18 0.20

1,500 0.11 0.11 0.16 0.17 0.22 0.23

2,000 0.14 0.14 0.19 0.20 0.25 0.26

2,500 0.17 0.17 0.22 0.23 0.28 0.29

Note: Wind direction is assumed to be along the straight

(ϑ = 0) and the time given is the average value of all

8 lanes.

Legend for Chart

A-World ranking

B-Athlete

C-Raw time

D-Year

E-Wind (m/s) reading

F-Venue/Altitude

G-Corrected time

H-New ranking

Table 6 Men’s and Women’s All-Time 200-m Top 5 Rankings Corrected

for Wind Speed and Altitude

A B C D E F

G H

Men

1 M.Johnson 19.32 1996 +0.4 Atlanta/315m

19.37 1

2 F.Fredericks 19.68 1996 +0.4 Atlanta/315m

19.73 =2

3 P.Mennea 19.72 1979 +1.8 Mexico City/2,250m

19.97 18

4 M. Marsh 19.73 1992 −0.2 Barcelona/95 m

19.73 =2

5 C.Lewis 19.75 1983 +1.5 Indianapolis/220 m

19.85 =6

7 A. Boldon 19.77 1997 +0.7 Stuttgart/250 m

19.83 =4

10 J. Capel 19.85 2000 −0.3 Sacramento/10 m

19.83 =4

Women

1 F. Griffith-

Joyner 21.34 1988 +1.3 Seoul/84m

21.43 1

2 M.Jones 21.62 1997 −0.6 Johannesburg/1,750m

21.70 3

3 M. Ottey 21.64 1991 +0.8 Bruxelles/35 m

21.69 2

=4 M.Koch 21.71 1979 +0.7 K-Marx-Stadt/410m

21.78 =6

=4 H.Drechsler 21.71 1984 +1.2 Jena/900m

21.85 13

=6 G. Torrence 21.72 1992 −0.1 Barcelona/95 m

21.72 4

10 J. Cuthbert 21.75 1992 −0.1 Barcelona/95 m

21.75 5

Note: Wind direction is assumed to be along the straight

(ϑ = 0) and the time correction used is the average

value of all 8 lanes.

© 2003 by Human Kinetics Publishers, Inc.

DIAGRAM: Figure 1—Wind direction for the 200-m sprint.

References

Behncke, H. (1994). Small effects in running. Journal of Applied Biomechanics, 10, 270–290.

Dapena, J., & Feltner, M.E. (1987). Effects of wind and altitude on the times of 100-meter sprint races. International Journal of Sport Biomechanics, 3, 6–39.

Davies, C.T.M. (1980). Effects of wind assistance and resistance on the forward motion of a runner. Journal of Applied Physiology, 48, 702–709.

Ferro, A., Rivera, A., Pagola, I., Ferreruela, M., Martin, A., & Rocandio,V. (2001). Biomechanical research project at the 7th World Championships in Athletics, Seville 1999. New Studies in Athletics, 16, 25–60.

Heidenstrom, P.N. (1992). A new attitude to altitude. Athletics: The International Track and Field Annual—1993, pp. 121–124.

Hill, A.V. (1928). The air-resistance to a runner. Proceedings of the Royal Society of London, Series B, 102, 380–385.

IAAF Handbook 2001–2002. (2001). Monaco: International Amateur Athletic Federation.

Keller, J.B. (1973). A theory of competitive running. Physics Today, 26, 42–47.

Kyle, C.R., & Caiozzo, V.J. (1986). The effect of athletic clothing aerodynamics upon running speed. Medicine and Science in Spans and Exercise, 18, 509–515.

Linthorne, N.P. (1994). The effect of wind on 100-m sprint times. Journal of Applied Biomechanics, 10, 110–131.

Linthorne, N.P. (2000). Accuracy of wind measurements in athletics. In A.J. Subie & S.J. Haake (Eds.), Proceedings of the 3rd International Conference on The Engineering of Span (pp. 451–458). Oxford: Blackwell Scientific.

Matthews, P. (1992). Biographies of leading athletes. Athletics: The International Track and Field Annual—1993, pp. 136–249.

Matthews, P. (1997). Biographies of leading athletes. Athletics: The International Track and Field Annual—1997, pp. 127–243.

Matthews, P. (2001). Biographies of leading athletes. Athletics: The International Track and Field Annual—2001, pp. 129–237.

Moran, J.M., Morgan, M.D., & Pauley, P.M. (1996). Meteorology: The atmosphere and science of weather. Englewood Cliffs, NJ: Prentice Hall.

Pritchard, W.G. (1993). Mathematical models of running. SIAM Review, 35, 359–379.

Vaughan, C.I. (1983). Simulation of a sprinter. Part 1. Development of a model. International Journal of Biomedical Computing, 14, 65–74.

Ward-Smith, A.J. (1984). Air resistance and its influence on the biomechanics and energetics of sprinting at sea level and at altitude. Journal of Biomechanics, 17, 339–347.

Ward-Smith, A.J. (1985a). A mathematical theory of running based on the first law of thermodynamics, and its application to the performance of world-class athletes. Journal of Biomechanics, 18, 337–349.

Ward-Smith, A.J. (1985b). A mathematical analysis of the influence of adverse and favourable winds on sprinting. Journal of Biomechanics, 18, 351–357.

Ward-Smith, A.J. (1999). New insights into the effect of wind assistance on sprinting performance. Journal of Spans Sciences, 17, 325–334.