External Effects in the 400-m Hurdles Race

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1 Journal of Applied Biomechanics, 2010, 2, Human Kinetics, Inc. External Effects in the 400-m Hurdles Race Mike D. Quinn A mathematical model based on a differential equation of motion is used to simulate the 400-m hurdles race for men and women. The model takes into account the hurdler s stride pattern, the hurdle clearance, and aerobic and anaerobic components of the propulsive force of the athlete, as well as the effects of wind resistance, altitude of the venue, and curvature of the track. The model is used to predict the effect on race times of different wind conditions and altitudes. The effect on race performance of the lane allocation and the efficiency of the hurdle clearance is also predicted. The most favorable wind conditions are shown to be a wind speed no greater than 2 m/s assisting the athlete in the back straight and around the second bend. The outside lane (lane 8) is shown to be considerably faster than the favored center lanes. In windless conditions, the advantage can be as much as 0.15 s for men and 0.12 s for women. It is shown that these values are greatly affected by the wind conditions. Keywords: aerodynamics, altitude, athletics, running, wind, sprinting The 400-m hurdles event is one of the most demanding track races in the Olympic program. It requires the athlete to develop speed, endurance, hurdling skill, and the concentration necessary to maintain an ideal stride pattern between the hurdles. The athlete must sprint one lap of the track and clear 10 hurdles of height m (men) or m (women), positioned 35 m apart. Since the athlete is sprinting, the hurdles should be cleared in a modified running action rather than jumped over. For an efficient hurdle clearance, the athlete s center of mass is raised by only a small amount (Lindeman, 1995). Elite athletes use a set number of strides to the first hurdle and a consistent number of strides between each hurdle. This is called the stride pattern. Performances in this event are affected by the height of the hurdle clearance, stride pattern, lane allocation, altitude of the venue, and wind conditions. A mathematical model of the 400-m hurdles race must take into account all these factors. Previously the effects of wind and altitude on performances in the 100-m sprint have been investigated in some detail (Dapena & Feltner, 1987; Dapena, 2000; Linthorne, 1994; Mureika, 2000, 2001; Ward-Smith 1985, 1999). The consensus is that a 2 m s 1 tail wind gives a s advantage over a 100-m race run in still conditions. Also an altitude of 1000 m provides an advantage of s over a 100-m race run at sea level with no wind. Wind and altitude effects have been investigated for both the Quinn is with the Department of Engineering and Mathematics, Sheffield Hallam University, Sheffield, U.K. 200-sprint (Mureika, 2003; Quinn 2003, 2004b) and the 400-m flat race (Quinn 2004a). Quinn (2004a) found that under certain circumstances a constant wind can be more favorable in the 400-m race than still conditions. Also the time difference between the running lanes in the 400-m race was affected by the wind direction. The 110-m hurdles event has been modeled by Ward-Smith (1997) and Spiegel and Mureika (2003). Ward-Smith (1997) considered both the hurdle clearance and stride pattern in his model, and estimated the effect on times of head and tail winds. The aim of this study is to produce a mathematical model that simulates the 400-m hurdles race. The model used by Quinn (2004a) to simulate the 400-m flat race is extended to predict the effect on race times of wind conditions, altitude, lane allocation, and the height of the hurdle clearance. Methods The mathematical model developed is based on Newton s second law of motion and follows from Keller s (1973) work. The 400-m model used in Quinn (2004a) is extended in two ways. Firstly, the athlete s propulsive force now includes both anaerobic and aerobic components, the latter varying with altitude. These are similar to the drive and maintenance terms used by Mureika (2001, 2003) in his sprint models but now with altitude dependence. The second extension of the Quinn (2004a) model concerns the internal resistive force, which includes both a running and a hurdling component. This is described in detail later. The equation of motion is given by the following: 171

2 172 Quinn with 1 2 d v / d t = F e + F ( 1 e ) e β t β t σ H kv α ( v v ) (1) w v = d s / d t (2) where v(t) and dv/dt are the athlete s velocity and acceleration at time t, in the direction of motion, and s(t) is the distance traveled. The athlete s propulsive force per β t unit mass has an anaerobic component F1 e 1 and an aerobic component F 1 β2t H 2 ( e ) e σ, which depends on H, the altitude of the venue. Opposing the motion of the hurdler is the resistive force kv and the wind resistance force α ( v v w ) 2. The velocity of the wind relative to the ground and tangent to the path is v w (t) and α = ρcd A / ( 2 M) (3) where ρ is the air density, M is the mass of the athlete, A is the frontal area of the athlete and C d is the drag coefficient. The resistive force kv results from frictional losses within the body (Pritchard, 1993) caused by the movement of the limbs during running and hurdling at velocity v. To take account of the extra energy expended when hurdling, we can consider the resistive force to have two components. The first is a resistive force produced by the normal running action and the second component is produced from the hurdling action. The hurdling resistive force is evident only at discrete time periods (i.e., when the 10 hurdles are cleared), but can be averaged over the entire race (Ward-Smith, 1997). This force depends on the ratio of hurdling strides to normal running strides during the race and also on the efficiency of each hurdle clearance. The hurdling action uses more energy than a normal running stride, but this can be reduced by an efficient hurdle clearance. The hurdling efficiency is related to the vertical displacement of the athlete s center of mass at each hurdle, which depends on the clearance height. The center of mass varies between different athletes, but for men it is situated at approximately 57% of the total height from the ground and at 55% for women (Page, 1978). Representative values for the height and mass of a world-class 400-m hurdler have been calculated using data from the all-time top-10 performances (Matthews, 1992, 1997, 2001). This produces heights of 1.87 m and 1.72 m, and masses of 79 kg and 61 kg for men and women, respectively. Using clearance heights of 0.32 m for men and 0.37 m for women (Ward-Smith, 1997), we obtain vertical displacements of 0.17 m and 0.18 m for men and women, respectively. Stride Pattern The 400-m hurdles race can be divided into four phases: the reaction time, the approach-run phase, the hurdle units, and the run-in phase. A hurdle unit comprises running between two sets of hurdles followed by clearing a hurdle. A consistent pattern of strides between the hurdles is essential. An efficient stride pattern is the key to a fast time in the 400-m hurdles. Analysis of the hurdle unit times and number of strides between hurdles in the 1988 Olympics is used to obtain a suitable hurdling resistive force for the model (Brüggemann & Glad, 1990). Suitable reaction times are chosen for the model by analyzing data from the 400-m hurdles finals of recent major championships (IAAF, 2008). In the model we take the reaction times to be 0.17 s for men and 0.19 s for women. In the 400-m hurdles race, the distance from the start to the first hurdle is 45 m. The approach run phase consists of running this distance and clearing the first hurdle. Times for this phase are taken as the hurdler touches down after the clearance. The number of strides in the approach run phase is N 1. Analyzing the stride pattern data from the 1988 Olympics (Brüggemann & Glad, 1990), we take N 1 = 19 for men and N 1 = 22 for women. There are 10 hurdles in the race with 35 m between each, giving nine hurdle units. Ideally the athlete attempts to maintain a constant stride pattern throughout the race. We take the number strides between each hurdle (N 2 ) to be 13 for men and 15 for women. The 40-m run-in phase is from the clearance of the 10th hurdle to the finish. The number of strides taken for this phase (N 3 ) is 18 for men and 20 for women. The total number of running strides in the race is N = N 1 + N 2 + N 3 and there are 10 hurdling strides. We define γ to be the proportion of hurdling strides to total strides in the race, so that γ = 10/(N + 10). The resistive force can be expressed as kv = ( 1 γ ) k v + γ ( η k + d) v (4) R where d is the vertical displacement of the athlete s center of mass at each hurdle and k R and η are constants to be determined. The first term on the right hand side of Equation (4), models the resistive force due to normal running strides and the second term models the hurdling action. The value of η will be greater than 1 since the hurdling resistive force will be greater than the resistive force for normal running. Bend Running On a standard running track, as described in Quinn (2003), the straight is 84.4 m and the bend is m. The athlete s speed around the bend is less than the speed attained when running in a straight line. We use Greene s (1985) method to correct the velocity for the effect of curvature, since this has been supported by empirical data. Other theoretical attempts to model the curvature effect have not been validated empirically (Alexandrov & Lucht, 1981; Mureika, 1997, 2003). The method described by Greene (1985) assumes a constant maximum velocity for the sprinter but this restriction is not essential (Ward-Smith & Radford, 2002). In the 400-m hurdles race the athlete must clear hurdles in the straights and on the bends. In the model we assume that the hurdling stride is the same for both types of clearance. In reality, hurdling on the curve can cause the athlete extra difficulty and a left lead leg is usually recommended. Athletes leading with the left leg can R

3 External Effects in 400-m Hurdles 173 keep closer to the inside of the lane and so run a shorter distance. The athlete with a right lead leg must move to the outside of the lane to clear the hurdle efficiently and legally. A right lead leg is more likely to result in the trail leg dropping below and to the side of the hurdle which can lead to possible disqualification (Lindeman, 1995). The number of hurdles on the straights and bends depends on the lane assignment. For a standard track there are five hurdles on the straights and five on the bends for lanes 1 5, but four hurdles on the straights and six on the bends for lanes 6 8 (IAAF, 2009). Wind Effects The drag coefficient of an athlete is very difficult to estimate since it depends on the shape and surface properties of the body and clothing. The value will vary during the race, but here we have assumed a constant value of C d = 0.715, based on the empirical work of Walpert & Kyle (1989). The frontal area of the athlete will also vary slightly at each hurdle clearance. In the sprint hurdles there is an exaggerated body lean that produces a smaller frontal area. In the 400-m hurdles the lower height of the barriers means the body lean is much less and the frontal area is only slightly reduced at each clearance. Here we assume a constant frontal area A of 0.51 m 2 for men and 0.43 m 2 for women using the method described in Quinn (2003). This is based on using the typical height and mass values for world-class 400-m hurdlers described previously. Taking the air density ρ = kg m 3 (Quinn, 2003), and using Equation (3) we obtain α =.0028 for men and α =.0031 for women. The value of the wind velocity v w faced by the hurdler depends on the actual wind velocity and the wind direction. A positive value for v w indicates a tail wind and a negative value indicates a head wind. However, as the athlete runs around the bend the wind conditions will change continuously. The wind velocity v w depends on both the distance run by the athlete and the radius of the running lane. The distance run around the first bend is different for each lane due to the staggered start. For example, on a standard track, the lane 1 athlete runs m around the first bend, while the lane 8 athlete runs only 89.1 m. All the athletes run the same distance along the back straight (84.4 m for the standard track), but the distance run around the second bend again differs for each lane. Finally, all the athletes run the length of the finishing straight. To take this into account the track is divided into four sections: the first bend, the back straight, the second bend and the finishing straight. If the actual wind speed (u w ) is blowing at an angle θ to the finishing straight (Figure 1), then we can calculate the wind facing the sprinter in each lane. The formulae for wind velocity in these four sections based on u w, θ, and the lane radius are given in Quinn (2004a). In the model we vary the wind speed and the direction to see the effect on the race times in each running lane. Initially we look at the effect of a uniform speed and direction throughout the duration of the race. However, in reality the presence of stadium grandstands can produce erratic wind patterns (Murrie, 1986; Dreusche, 1994; Linthorne, 2000). To simulate this we also look at examples of where either the wind speed or direction changes during the race. Using ideas from Dapena & Feltner (1987) and Quinn (2004b), we look at the effect of the wind speed or wind direction changing midway through the race. Altitude Ever since the 1968 Mexico Olympics, it has been well documented that the reduction in air density at high altitude gives an advantage to sprinters. The IAAF recognizes this by annotating with an A (for altitude assisted ), all sprint performances at altitudes of over Figure 1 Direction of the wind for the 400-m hurdles race.

4 174 Quinn 1000 m. This is the case for the 100-m, 200-m, 400-m events and the 100-m (women), 110-m (men), and 400-m hurdles races. However, high altitude also has an adverse effect on the aerobic energy system. In the 400-m hurdles event, the contribution from the aerobic energy system is thought to be about 40% (Spencer & Gastin, 2001) and therefore needs to be taken into account in the modeling process. This is not an easy task since the processes involved are complex. As altitude increases the percentage of oxygen in air remains constant at 20.94%, but the barometric pressure falls and this produces a reduction in the partial pressure of oxygen of inhaled air (McArdle et al., 2001). A lower partial pressure of oxygen affects the oxygenation of blood flowing through the lungs and results in a reduced oxygen supply to the athlete s muscles. However, it appears that other processes such as elevated ventilation rate and heart rate assist the body in minimizing the effects of a lower oxygen supply. Studies indicate that aerobic capacity is not noticeably altered up to an altitude of 1000 m (Sawka et al., 1996). There is a small decline in maximal oxygen uptake between 1000 m and 2000 m in elevation; thereafter, maximal oxygen uptake decreases by approximately 10% for each extra 1000 m of altitude (Sawka et al., 1996). This is modeled using the factor e σh in the aerobic component of Equation (1), where H is the altitude of the venue and σ is to be determined. Parameter Selection The Equations (1) and (2) are solved numerically using a fourth-order Runge Kutta method. Using data from Spencer & Gastin (2001) we can obtain a suitable balance of the aerobic and anaerobic contributions to the propulsive force. Then the model parameters are chosen to fit the 400-m hurdles data recorded in the 1988 Seoul Olympic final, where the times were recorded at the end of the approach run and at the end of each of the nine hurdle units. Using an average of the times of the three medalists, we can obtain a suitable set of values for F 1, F 2, β 1, β 2, k R, and η (Table 1). The parameter values are varied systematically by small increments. The running times at the end of the approach run and each hurdle unit are calculated from the mathematical model and compared with the track data. The parameter values that produce the lowest overall root-mean-square error are selected. A sensitivity analysis of the model confirmed that small deviations of the parameter values produced similarly small deviations in the estimates. The value of η (2.025 for men and for women) indicates that the hurdling action is approximately twice as expensive as a normal running action, which agrees with the result of Ward-Smith (1997). Results The model closely fits the performance data (Table 1), and predicts that, in windless conditions, the maximum velocity in lane 4 is reached just after the first hurdle. This agrees with the competition results as stated in Brüggemann & Glad (1990). For lane 4 the model predicts a maximum velocity of 9.47 m s 1 for men and 8.69 m s 1 for women, both attained after approximately 47 m. The least favorable wind direction for all lanes is between 30 to 30. The most favorable wind direction is between 220 and 260 (Figures 2 and 3). The model predicts that winds with speeds up to 2.3 m s 1 with direction θ = 240 can produce faster times than when there is no wind at all. For example, in lane 8 both male and female hurdlers will be approximately 0.05 s faster Table m hurdles simulation in lane 4 Distance (m) Actual time ta Men Model Time tm Error ta tm Actual Time ta Women Model Time tm Error ta tm Reaction time Finish time Note. The model parameter values used are F 1 = m s 1, F 2 = m s 1, β 1 = s 1, β 2 = s 1, k R = m, η = (men); F 1 = m s 1, F 2 = m s 1, β 1 = s 1, β 2 = s 1, k R = m, η = (women); σ = m 1.

5 External Effects in 400-m Hurdles 175 Figure 2 Effect of the direction of a constant 2 m s -1 wind on the men s 400-m hurdles race time in lanes 1, 4 and 8. Figure 3 Effect of the direction of a constant 2 m s -1 wind on the women s 400-m hurdles race time in lanes 1, 4 and 8. when the wind speed is 2 m s 1 with direction θ = 240. In lanes 1, 2 and 3, light winds (0.1 to 1 m s 1 ) blowing from the direction θ = 240 produce marginally faster times than in windless conditions. A fluctuating wind makes a significant difference to the race times. Windless conditions for the first half of the race followed by a 2 m s 1 wind (θ = 0) for the second half can produce times up to 0.15 s faster than a constant 1 m s 1 wind speed. Similar effects are seen for fluctuating wind direction. A 2 m s 1 wind in the direction θ = 270 for the first half and θ = 180 for the second half of the race is up to 0.22 s faster than a constant direction θ = 225 throughout the race (Figures 4 and 5). The model predicts that in windless conditions lane 8 is faster than lane 1 by a margin of 0.23 s for men and 0.19 s for women. The wind conditions can increase or decrease this lane difference. For example, a constant 2 m s 1 wind at an angle of 240 increases the advantage of lane 8 over lane 1: 0.30 s for men and 0.26 s for women. In contrast, a wind of the same speed at an angle of 0 reduces the advantage to 0.18 s for men and only 0.12 s for women.

6 176 Quinn Figure 4 Effect on race times of fluctuating wind speed (constant direction θ = 0) for each lane of the men s 400-m hurdles race. The three cases are (1) a 1 m s -1 wind throughout the race, (2) a 2 m s -1 for the first half of the race and no wind for the second half and (3) the reverse of case (2). Figure 5 Effect on race times of fluctuating wind direction (constant wind speed w = 2 m s -1 ) for each lane of the men s 400-m hurdles race. The three cases are (1) a constant direction θ = 225, (2) θ = 180 for the first half of the race and θ = 270 for the second half and (3) the reverse of case (2). Variation in the clearance height has a small effect on the overall race time. The model predicts that a 10% reduction in the clearance height only makes a 0.2% improvement in the race time. For men this equates to a 0.03 s (0.04 s for women) change in race time per centimeter change in clearance height. The lower air density at high altitude makes a significant difference in the races times, and for realistic altitudes it far outweighs any adverse effects of the aerobic energy system. The model predicts this is true up to an altitude of about 5000 m. At a height of 2000 m the model predicts an advantage of 0.21 s for men and 0.24 s for women (Figure 6). Discussion Using data from the last six World Championship 400-m hurdles finals (IAAF, 2008), we obtained the average reaction times of 0.17 s for men and 0.19 s for women. These figures were used in the model rather than those

7 External Effects in 400-m Hurdles 177 Figure 6 Time advantage in lane 4 for different altitudes (m) in windless conditions. from the 1988 Olympic final (Brüggemann & Glad, 1990). In all these World Championships the starter used a silent gun, which transmits the sound to speakers in the athlete s starting blocks. This technology was not used in the 1988 Olympics. Instead the traditional starting gun was used, which means that the athletes further from the starter tend to have slower reaction times. In the model we have assumed the athlete uses a modified running action to clear the hurdle. The value obtained for the parameter η (2.025 for men and for women) in Equation (4) indicates that the hurdling stride produces about twice as much resistive force as a normal running stride. The model predicts that changes in the clearance height make relatively small differences to the race time. However, for world-class athletes an improvement by a few hundredths of a second can be significant. We have assumed there are an odd number of strides between the hurdles (13 for men and 15 for women). This ensures that the athlete will hurdle with the same lead leg throughout the race. If an even number of strides is taken between hurdles then the athlete must be able to alternate the lead leg between consecutive hurdles. The hurdling stride for world-class male athletes is approximately 3.5 m in length, with the athlete taking off 2.2 m before the hurdle and landing 1.3 m afterward (Lindeman, 1995). Using the choice of N 1, N 2, and N 3 for the model, we obtain approximate male stride lengths of 2.25 m in the approach phase, 2.42 m between the hurdles, and 2.15 m in the run-in. The stride lengths for the women s model are 1.95 m in the approach, 2.10 m between hurdles, and 1.94 m in the run-in. The model simulates the ideal race, where a constant stride pattern is maintained throughout the race. In reality not all top athletes maintain the same stride pattern during the race; some choose to increase the number strides between hurdles after the first five or six hurdles have been cleared. The choice of which foot is forward in the starting blocks is crucial. In the model, the male hurdler takes 19 strides in the approach phase, which means the foot of the lead leg is placed in the front starting block. There are 22 strides in the approach phase of the women s model so the lead leg pushes off from the rear block at the start of the race. The results indicate that a slight tail wind in the back straight and round the final bend is advantageous to the 400-m hurdler. Although this means there is a head wind in the final straight, the time gained earlier in the race compensates for this. This agrees with the predicted results from the 400-m model (Quinn, 2004a). The optimal wind conditions for each lane of a standard track are shown in Table 2. Winds of speed greater than 2 m s 1 have not been used in the model simulations since they may disrupt the athlete s stride pattern. If there are strong winds during a race the athlete may need to adjust the stride pattern and this will cause them to slow down (Lindeman, 1995). An analysis of the top-50 best-ever 400-m hurdles times shows very few altitude-assisted performances. For men there are only two: 35th best (at an altitude of 2250 m) and 44th best (altitude 1750 m). In the women s event there are three altitude-assisted performances, the best being in 30th position (altitude 1750 m). This may suggest that altitude is less advantageous than predicted Table 2 Predicted time in each lane under optimal wind conditions Lane Wind Speed (m s 1 ) Men Women Note. The wind direction is θ = 240 for all lanes.

8 178 Quinn by the model. However, it is more likely the result of no Olympics or World Championships being held at altitude since In the Mexico Olympics, the winning time in the men s 400-m hurdles was a world record by 0.5 s and 40 years later it is still in the top-50 best-ever times. Women did not race this event at world level until 1983 and so have never competed at altitude in a major championship. Most top 400-m hurdlers prefer one of the middle lanes (3, 4, 5, or 6) and these are used for the seeded runners in championship finals. The reasons for this preference appear to be either psychological or tactical, where the hurdler can use athletes on the outside lanes to judge their own performance. However, the model predicts that the outside lane is the fastest and significantly better than the seeded lane 3. This is due to the tighter curvature of the track in the inside lanes, which reduces the athlete s maximum speed. On a standard track in windless conditions, the time advantage of lane 8 over lane 3 is 0.15 s for men and 0.12 s for women. As shown earlier, the wind conditions can increase or reduce this difference. The model can predict the effect of the most favorable conditions on the current world records. A simulated race in lane 8, at an altitude of 2250 m (Mexico City) with a 2 m s 1 wind speed with direction θ = 240 would produce improvements of 0.39 s for men and 0.41 s for women. The current study has shown that the mathematical model used for the 400-m flat race (Quinn, 2004a) can be successfully extended to simulate the 400-hurdles event. The results for wind and altitude seem to be in line with much of the previous results for other events. The model provides an interesting insight into a range of factors affecting world-class hurdlers. The effect of wind conditions in each running lane could be useful knowledge to the hurdler when preparing for a race. It could lead to an alteration in the stride pattern due to adverse weather conditions. The simulation has considered winds with constant direction and speed, and some simple fluctuating winds. This is a simplification of what might occur in reality when swirling winds are possible. The model predicts significant advantages to performances at altitude. However, this part of the model might need refinement when there is more information on the contribution from the aerobic energy system in the 400-m hurdles and how it is adversely affected by altitude. The model also highlights the importance of improving the efficiency of the hurdle clearance and emphasizes the advantage of running in the outside lanes. References Alexandrov, I., & Lucht, P. (1981). Physics of sprinting. American Journal of Physics, 49, Brüggemann, G.P. & Glad, B. (1990). Time analysis of the sprint events. Final Report of the Scientific research Project of the XXXIV Olympiad-Seoul 1988, Dapena, J. (2000). In E.G. 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