^{1}

^{1}

^{1}

^{*}

Based on the v-t (speed-time) curves in physics, this study analyzes 4 × 100 m relay baton transition period and describes 4 different relay baton transition period v-t state characteristics curves. We can calculate from the curves that the speed of the first three athletes at the moment of transition is above the 96% of the average speed of 100-meter. The research reveals that the ideal v-t curve is a smooth semi-circular curve (without the launching phase). The stability and continuity during the baton transition period are the key factors in 4 × 100 m relay race.

During Beijing Olympic Games in 2008, Chinese men’s 4 × 100 m relay team reached the finals for the first time. However, the ranking of them was cancelled for the mistake that happened during the anchor. The crack U.S. team and defending champion England also had some problem in the phase, and for this reason they were all eliminated in the preliminaries (Wang et al., 2009). The cases clearly show that the baton transition period is the key factor in 4 × 100 m relay race (Baeg & Cho, 2005). In the 4 × 100 m relay race, the average speed is higher than the hundred meters because of the saves of launching time. That is the fascination of 1 + 1 + 1 + 1 > 4 in the relay race in which the secret is at the baton transition period (Berthelot et al., 2008). Based on the physics formula of v = v_{0} + at (Zhang, 1999, 2000; Zhu, 2004) …... the present study employed the v-t curves to show the relationship between speed and time at the different baton transition periods (Du & Lin, 2005).

The purpose of the study is to analyze 4 × 100 m relay baton transition period through using the v-t (speed-time) curves in physics.

Using the CNKI and to search and categorize the related information which will provide the theoretical evidence for the research.

The v-t curves in physics were employed to analyze 4 × 100 m relay baton transition period, and the researcher advanced 4 different relay baton transition period v-t state characteristics curves.

Based on the actual case of Track and Field World Championships, the present study calculated the specific values of the speed in each separated period. Then, the researcher compared and analyzed them. At last, the graphics and reference data were found in the qualitative and quantitative research.

Firstly, based on the formula (1) v = v_{0} + at and to make the v-s curves.

The speed of each team member is represented by v = v_{0} + at, v_{0} is for initial velocity, a is for acceleration, it is for time, and v is for terminal velocity that is the final velocity after a period of acceleration time. In a relay race, the entire runner should start from zero speed, and accelerate the process of running to reach the maximum speed which should be kept by the runner till baton transition or sprint finish (Chollet et al., 1996, 1997, 1999).

The baton transition (Messerli, 1952) is the most critical link in relay race. In order to highlight this main contradiction the researcher suppose that the 4 team members’ hundred meters rate and the process of running is almost same. The v-t curves also for the schematic diagram (Tourny et al.,1992), and then the present study exaggerates the baton transition period, and supposes the last three team members’ speed as hundred meter rate (Salo & Bezodis, 2004). As t_{1}, t_{2}, t_{3} for the baton transition period speed of the 1^{st}, 2^{nd}, and 3^{rd} team members, respectively. t_{c} is for the total time of the full competition (c represents the runner’s score).

In figures 1-4, the v-s curve represents the 4 × 100 m relay under different kinds of baton transition. In the mathematic, the area under the v-s curve represents distance, and expressed by x. the full game can be ex-

pressed by a formula of

m. Integral limit t_{c }represents the total time of the game which is the more quickly the better. The 4 × 100 m relay is reflected in the v-t curves. In other words, we pursuit of the smallest t_{c }under the circumstance that the areas are equal in figures 1-4. It requires that v-t curve is a smooth motion high line (except the scratch line), in this way we can only guarantee the areas under the curves are equal, and the time t_{c} is the most short.

The first runner’s initial velocity is represented by v_{01}, and v_{01} = 0. the runner is required to have fast response in scratch line, great acceleration, and should reach the v_{w} (speed way) in the short time (see the first part of skew lines in figures 1-4), and then to keep the speed v_{w} till the baton shall be passed within the take-over zone.

The second runner must start from zero speed (this part should be expressed by dotted line for not carrying baton, and it cannot be included in the t_{c}). The second runner should start running before the first runner enter take-over zone (Barbosa et al., 2008). When the speed nearly equals to v_{w} the second runner should take over the baton quickly. We suppose that at the time the baton is passed the second runner’s speed is transfer speed (v_{T}), and v_{T} as his initial speed v_{02}. Obviously, v_{02} ≠ 0. thus, the second runner’s speed can be expressed as the formula v = v_{02} + at = v_{T} + at. In the same way, the third and fourth runners’ initial speed all equal to v_{T}, and v_{T} ≈ _{}

_{}

_{}

_{}

_{}

_{}

_{}

_{}

_{}

v_{w}. In this way the team can keep the continuity of the baton transition period speed, and the team can keep the speed v_{w} throughout the race. Besides, the transmitter and receiver should keep the same speed in order to maintain stable transmitting in the baton transition period. In other words, when the transmitter and receiver have no relative speed the baton transition is the most ideal.

figure 1 shows the ideal state of baton transition: as baton transmit, the initial speed should be quick (v_{T} ≈ v_{w }) and stable, and the speed should be keep v_{w} throughout the race. The curve in the transition period is slight decrease in punctuate, and the transmitter and receiver’s related speed is almost zero. The whole curve should be coherent. When the value of v is high and flat, the time t_{c} is the most short.

figure 2 shows general level team’s baton transition state: in order to maintain the stability of baton transition the team should cost some baton transition period speed. At first, the transmitter should slow down the speed, then the receiver run slowly even though they already start to run. The receiver should look back while waiting the transmitter, and they would receive the baton when the speed v_{T} ≈ v_{w} 50%, than they should speed up to v_{w}. Therefore, it will produce a large displacement in the long baton transition period, and which will influence the total score t_{c}. In this curve, the value v has larger fluctuations which are not like figure 1 in which the value of v is high and flat. Thus, the total time t_{c} is larger than that is in figure 1. However, it has some benefits, such as high stability, low risk in baton transition failure. Thus, this is the common method that average level of team may use

As shown in figure 3, in a state of rest, the baton receiver holds still on the line until he receives the baton. This kind of baton transition would greatly slow down the speed even though baton transition is very stable. Because the receiver’s speed should start from zero and then he should accelerate. In this way, total time in the relay race means to add up 4 the hundred meter results. In addition, for transmitter’s slowing speed, the formula would become 1 + 1 + 1 + 1 < 4. Then, it will lose the meaning of relay race, so this method is not acceptable.

The present case study based on the Athletics World Championships 4 × 100 m relay race final. The result of this final is very close to the result of 29^{th} 4 × 100 relay race world record in 2008. The following data of the table is from “China Sport Science and Technology” (Arjmandi et al., 2010).^{ }

The world top relay race runners’ v-t curves (Shen, 1997) are between figure 1 and figure 2. They use very short time in baton transition so that the initial speed v_{T} is almost near to v_{w} the explanation is shown in tables 1-3. The data in tables 1-3 is from Tokyo World Championships men’s 4 × 100 m relay race final in 1991. In table 1, there are cumulative time of 100 m, 200 m, 300 m, 400 m, the rank and the average speed (v_{A}) of 4 team members; In table 2, there are times of each 100 m and the average speed v_{w} of hundred meter; In table 3, there are the former three runners’ time that spend in take-over zones and the average speed v_{T} in the baton transition period; In table 4, there are data comparisons of various speeds.

As _{A} of each 100 m in 4 × 100 m relay race, the 400 m’s total time t4 is divided by 4. For instance, American team’s average time of each 100 m is t_{A} = 37.50/4 = 9.38 s, and the slowest team’s average time is 9.88 s that is very near to Jamaican athlete Usain Bolt’s 100 m world record 9.69 s at the Beijing Olympic Games in 2008. Thus, each relay race team member’s average rate is better than their own hundred meter rate, and they accomplish 1 + 1 + 1 + 1 > 4 (Kennedy et al., 1990). Then, we can divide 400 m by t_{4} and get the 4 runners’ average speed v_{A}, for instance, American team’s average speed is v_{A} = 400/37.50 = 10.67 (m/s). Other teams’ average speeds are 10.56, 10.50, 10.41, 10.38, 10.34, 10.12 m/s, respectively.

In _{A }, the last three runners’ hundred meter time is added together and then divided by three, for instance, the American relay race team’s the last three runners’ each hundred meter’s average time is t_{A} = (8.91 + 9.22 + 9.07)/3 = 9.07 s. Then, we can divide 100 m by the average time and get the last three runners’ average speed. Since the last three runners do not have starting in the relay race, the average speed can be regarded as their hundred meter rate V_{w}. The American team’s last three runners’ average speed is V_{w} = 100 m/9.07s = 11.03 m/s, and the other teams’ last three runners’ average speed are 10.94, 10.87, 10.71, 10.71, 10.66, 10, respectively.

In table 3, there are the former three runners’ time that spend in take-over zones (each take-over zone shall be 20 m long, and the time that spend in take-over zones are from the transmitter enters the take-over zone

. Cumulative time of 100 m, 200 m, 300 m, 400 m, the rank, and v_{A}

Country | 100 m t1 | 200 m t2 | 300 m t3 | 400 m t4 | Average time of 100 m t_{A} | Average speed of 100 mv_{A} = 400 m/t_{A} (m/s) |
---|---|---|---|---|---|---|

America | 10.30 (1) | 19.21(1) | 29.21(1) | 37.50(1) | 9.38(1) | 10.67 |

French | 10.46 (3) | 19.50(2) | 28.62(2) | 37.87(2) | 9.47(2) | 10.56 |

British | 10.50 (4) | 19.70(4) | 28.95(3) | 38.09(3) | 9.52(3) | 10.50 |

Niger | 10.41 (2) | 19.57(3) | 29.03(4) | 38.43(4) | 9.61(4) | 10.41 |

Italy | 10.51 (5) | 19.72(5) | 29.03(4) | 38.52(5) | 9.63(5) | 10.38 |

Soviet Union | 10.53( 6) | 19.77(6) | 29.30(6) | 38.68(6) | 9.67(6) | 10.34 |

Canada | 10.53( 6) | 20.12(7) | 29.70(7) | 39.51(7) | 9.88(7) | 10.12 |

. times of men’s 4 × 100 m’s each 100 m and the average speed v_{w}

Country | Each 100 m’s average speed v_{A} (m/s) | The last 3 runners’ each 100 m’s average speed v_{w} (m/s) | Average speed in the baton transition period v_{T} (m/s) | v_{T}/v_{w} percentage % |
---|---|---|---|---|

America | 10.67 | 11.03 | 10.62 | 96.28 |

French | 10.56 | 10.94 | 10.93 | 99.91 |

British | 10.50 | 10.87 | 10.73 | 98.71 |

. The former three runners’ duration in take-over zones and the average speed v_{t }in the baton transition period

Country | 1^{st} take-over zone | 2^{nd} take-over zone | 3^{rd} take-over zone | Total time in the former 3 take-over zone | Average time △t in each take-over zone | Average speed V_{T} in baton transition period |
---|---|---|---|---|---|---|

America | 1.81 | 1.82 | 2.02 | 5.65 | 1.88 | 10.62 |

French | 1.88 | 1.79 | 1.82 | 5.49 | 1.83 | 10.93 |

British | 1.91 | 1.83 | 1.85 | 5.59 | 1.86 | 10.73 |

. Data comparison of various speeds and the ratio of baton transition period speed to speed that is en route (v_{T}/v_{w})

Country | 0 - 100 m | 100 - 200 m | 200 - 300 m | 300 - 400 m | The last 3 runners’ each 100 m’s average time t_{A} | The last 3 runners’ each 100 m’s average speed v_{w} m/s | |||
---|---|---|---|---|---|---|---|---|---|

America | 10.30(1) | 8.91(1) | 9.22(2) | 9.07(1) | 9.07(1) | 11.03 | |||

French | 10.46(3) | 9.04(2) | 9.12(1) | 9.25(3) | 9.14(2) | 10.94 | |||

British | 10.50(4) | 9.20(4) | 9.25(3) | 9.14(2) | 9.20(3) | 10.87 | |||

Niger | 10.41(2) | 9.16(3) | 9.46(5) | 9.40(5) | 9.34(4) | 10.71 | |||

Italy | 10.51(5) | 9.21(5) | 9.31(4) | 9.49(6) | 9.34(4) | 10.71 | |||

Soviet Union | 10.53(6) | 9.24(6) | 9.53(6) | 9.38(4) | 9.38(6) | 10.66 | |||

Canada | 10.53(6) | 9.59(7) | 9.67(7) | 9.72(7) | 9.66(7) | 10.35 | |||

to the receiver cross out the take-over zone) and the average speed v_{T} in the baton transition period. As seen in table 3 the runners’ duration in the take-over zone are less than 2 s. The average speed in the baton transition period V_{T} = △x/△t. In this way, we can calculate the v_{T} of the teams of America, French, and British, they are 10.62, 10.93, 10.73 m/s, respectively.

Various speeds are listed in table 4 which clearly shows the relationship between speeds in each period. It shows us quantitative data about speeds so that we can easily compare and analyze the different speeds (Ahmad et al., 2013).

Firstly, as shown in table 4, the teams’ each 100 m’s average speed are 10.67, 10.56, and 10.50 m/s, respectively, which are higher than Bolt’s 100 m world record 9.03 m/s (v_{A} = 100 m/9.69s) It proves that we cannot get highest speed in single member’s effort.

Secondly, through calculating the last three runners’ 100 m averages speed in the 4 × 100 m relay race, we can get the exact data of speed v_{w} subtract from the highest speed. And table 4 explains the relationship between v_{T}, v_{A} and v_{w}, that is v_{T} < v_{A} < v_{w}, thus, we can clearly see that v_{w} is the fastest (Arjmandi et al., 2010).

Thirdly, the ratios of baton transition period speed to fastest speed are shown in table 4. The average speed of baton transition period are 10.62, 10.93, and 10.73 (m/s), respectively. The ratio of baton transition period speed to speed that is en route are 96.28%, 99.91%, and 98.71%, respectively (He, 2011).

1) The baton transition period is the most important factor in the success of 4 × 100 m relay race.

2) In the moment of baton transition period, the speeds of transmitter and receiver must be the fastest, and their relative speed is very tiny. In other words, when the transmitter and receiver have no relative speed the state of baton transition is the most ideal, as we can see in figure 1.

3) No deceleration during the baton transition period may effectively improve the results of the game, but it demands higher baton transition skills.

This research was supported by the Humanities and Social Science Research Youth Foundation of the Ministry of Education in China (Grant No.12YJC880005) and “the Fundamental Research Funds for the Central Universities (Grant No.SWU1309230)” and “the Fundamental Research Funds for the Central Universities (Grant No.SWU1309231)”. Chongqing Higher Education Reform Project (No.120388) and the Humanities and Social Science Research Youth Foundation of the Ministry of Education in China (Grant No.13YJC890014). Corresponding author Chang Jin-dong is a lecturer in the College of Physical Education at Southwest University; e-mail: changlai@swu.edu.cn. Here the authors would like to express heartfelt thanks.