Sports Biomechanics, January 2007; 6(1): 17–30

Bat speed, trajectory, and timing for collegiate baseball batters hitting a stationary ball NORIYUKI TABUCHI1, TOMOYUKI MATSUO2, & KEN HASHIZUME2 1

Graduate School of Human Sciences, Osaka University, Japan, and 2Graduate School of Medicine, Osaka University, Japan

Abstract The aims of this study were to examine whether batters hit stationary balls at the time of peak speed of the bat head and whether the impact occurs at the lowest point of the bat trajectory. Eight university baseball players hit three balls, each hung with a string; each ball was made of a different material and was different in weight. Bat movement was captured by four 240-Hz infrared cameras and analysed three-dimensionally. Time for peak speed of the bat head varied according to the conditions. When stationary balls of standard weight were used, the bat head was at maximum speed at impact with the ball; then, it decelerated drastically owing to the impact. In contrast, maximum speed was obtained after impact when lightweight stationary balls were used. The time – speed profile of the bat head before impact in the lightweight ball condition was identical with that in the standard weight ball condition. Regardless of conditions, the timing of the lowest point of the bat head was nearly identical for each batter and most participants hit the stationary balls at about the lowest point of the bat trajectory

Keywords: Ball mass, baseball, batting, dry swing, impact point

Introduction Human actions that require the impact of two objects are difficult to control. Because of the accuracy required, baseball batting is a good example of such an impact. To improve their skills, batters practise swinging the bat without a ball, a technique called the “dry swing”. Although it is reasonable to believe that hitting a pitched ball is a more effective form of practice, the dry swing has some advantages. For example, it can be performed in a small space by a single player and is unlikely to result in injuries. Because of these advantages, the dry swing is commonly used by amateur and professional baseball players. To make the dry swing more effective, players should imagine not only a pitcher, a pitched ball, and a batted ball, but also the results of batting performance such as base hit in front of the right fielder, double to left, and line-drive home run. However, it is difficult to imagine these conditions because there is no impact between the ball and bat. Thus, the most critical disadvantage of the dry swing is that it produces no, or only a little, knowledge of the result. A new approach that will provide not only the same advantages of the dry swing but also information about knowledge of results is desirable. For example, a batting simulator in virtual reality space has been developed. In such an environment, batters try to hit an Correspondence: N. Tabuchi, 1-17 Machikaneyama - cho, Toyonaka - city, Osaka, 560-0043 Japan. E-mail: [email protected]. osaka-u.ac.jp ISSN 1476-3141 print/ISSN 1752-6116 online q 2007 Taylor & Francis DOI: 10.1080/14763140601058409

18

N. Tabuchi et al.

imaginary ball and the system must estimate the results of the swing. To develop such a technique, it would be necessary to determine accurately the imaginary point of the bat –ball impact, together with the kinematic variables of the bat swing. Moreover, it is valuable for coaching of baseball to clarify the characteristics of the kinematic variables of the bat swing around the bat– ball impact. From the standpoint of physics, the faster the bat is swung, the greater the speed of the batted ball, when all other things are equal. A batted ball of higher speed results in better batting performance. Thus, it is reasonable to assume that impact should occur as the bat reaches its peak speed. However, as the bat speed decreases owing to the bat – ball impact in normal baseball batting, it is unclear whether batters hit the ball before the bat reaches its peak speed. Previous studies have shown that the bat was not at peak speed when it struck the ball. Welch and colleagues (Welch, Banks, Cook, and Draovitch, 1995) examined bat kinematics during tee batting. In their study, 29 male professional baseball players hit balls on a batting tee. The authors found that the maximum speed of the bat occurred 15 ms before impact with the ball. Messier and Owen (1984) instructed eight intercollegiate softball players to hit a pitched softball and demonstrated that the bat head reached a maximum speed 32 ms before impact. McIntyre and Pfautsch (1982) required 20 current or former college baseball players to hit a pitched baseball into two assigned areas of the field and demonstrated that peak speed of the bat head was reached 13 – 16 ms before impact. These results indicate that the bat struck the ball at sub-maximum speed and that the impact occurred in a deceleration phase after peak speed. Similar results have been obtained in tasks akin to baseball and softball batting. Elliott and colleagues (Elliott, Marsh, and Overheu, 1989) reported that the head of a tennis racket during a forehand stroke reaches maximum speed about 8 ms before the impact with a ball. William and Sih (2002) measured the speed of the head of golf clubs swung by amateurs with handicaps ranging from 0 to 36, and reported that the peak horizontal velocity of the head of the club was observed before impact and that the speed decreased slightly before impact. Plagenhoef (1971) reported that such patterns have been consistently found in all sports involving impact, such as karate, boxing, golf, and squash. Thus, the phenomenon has been regarded as evidence of the preparation made in anticipation of impact. However, some researchers have questioned these findings. They boted that filters used in earlier studies made the time of peak speed earlier (Levanon and Dapena, 1998; Nunome, Asai, Ikegami, and Sakurai, 2002). In a study of the relationship between mass properties and bat velocity, the speed of the sweet spot was analysed for the two frames before impact (Fleisig, Zheng, Stodden, and Andrews, 2002). They excluded the effect of inconsistent decreases of speed at impact for appropriate data processing. However, since they did not use data after the impact for analysis, the time shift of the peak speed timing owing to the filtering effect was unclear. Gray (2002) used the lowest point of the bat head as a criterion to identify the impact with a ball in a batting simulation experiment. The lowest point is the point at which the height of the bat head was at a minimum in its trajectory during the swing. However, Ted Williams, formerly one of the greatest professional baseball batters, said that the slight upswing is ideal since the trajectory of the bat matches that of the pitched ball (Williams and Underwood, 1986). This view was supported by Messier and Owen in an earlier study (1984). They reported that eight female fast-pitch softball batters swung a bat slightly downward, maximizing the downward velocity at an average time of 87 ms before impact, and then increasing the upward velocity at the instant of impact in all trials. In coaching, it is important to know where and when the bat impacts with the pitched ball during the swing trajectory. From the viewpoints of task validity and application to the real

Bat speed, trajectory, and timing

19

game, it may be desirable to investigate normal hitting, that is, with a pitched ball. It would be straightforward to examine the spatial relationship between the bat trajectory and the point of bat – ball impact. However, it would be difficult to investigate the temporal relationship between speed profile of the bat and the instant of bat –ball impact because of rapid deceleration of the bat owing to the impulse of the bat– ball impact. Therefore, it is necessary to know the characteristics of the swing without impulse at the bat – ball impact to elucidate batting mechanics. In addition, pitched balls have unavoidable variability in location and speed. Thus, stationary and lightweight balls were used to avoid the effect of variability, in an attempt to form the basis for applied studies of baseball batting. The aims of this study were to determine the timing of peak speed of the bat head and its impact location in a trajectory (with reference to the vertical axis) of the swing.

Methods Participants Eight male college baseball players, six right-handed batters and two left-handed batters, volunteered to participate in the study. They were aged 20.5 ^ 0.5 years (mean ^ s) and had played baseball for 9.8 ^ 2.5 years (range ¼ 6 –13). They signed an informed consent form after receiving a detailed explanation of the study.

Tasks The task was to hit three balls of different weight, ten times each, in three conditions: . Condition 1: normal baseball of standard weight (diameter ¼ 73 mm, mass ¼ 150 g). . Condition 2: polystyrene ball (diameter ¼ 70 mm, mass ¼ 8.0 g). . Condition 3: small polystyrene ball (diameter ¼ 15 mm, mass ¼ 0.2 g). The normal baseball (Condition 1) was compared with one of negligible mass (Condition 3) to determine the effect of impulse. However, it was possible that target size might affect the batter’s swing. Therefore, to confirm this influence, another ball, which was about the same size as a normal baseball but much lighter in weight (Condition 2), was used. In this study, we hung a ball with a string instead of putting it on a batting tee as Welch et al. (1995) did, to avoid deceleration of the bat owing to contact with the batting tee. Each ball was wrapped in reflective tape for motion tracking. An aluminium baseball bat, Buw League Super 405 (Mizuno Corporation, Osaka, Japan), was used (length ¼ 0.84 m, diameter ¼ 70 mm, mass ¼ 850 g). Three metal sticks with a reflective marker at the distal end were attached to the bat head; the mass of the bat including attached markers, sticks, and attachment tool was 900 g (see Figure 1). The mass of a stick was 6 g and that of a marker was 0.2 g. Each metal stick was 120 mm long and 5 mm in diameter. As three metal sticks were attached equiangularly, the centre of the three markers was the centre of the bat head. The mass of a metal attachment tool was 31 g. Participants reported that they did not feel uncomfortable during the bat swing. Although metal sticks and markers might cause a minor increase in drag, the effect of this was presumed to be insignificant. A ball was hung with a string, about 0.8 m long, from a horizontally extended bar at a height of about 1.5 m. In the horizontal plane, the ball was set at the centre of the home plate. In the vertical plane, the location of the ball was chosen according to the preference of each participant (0.70 ^ 0.06 m), which was determined during a practice

20

N. Tabuchi et al.

Figure 1. Bat with three markers. Three metal sticks with a marker were attached to the bat head (length of a stick ¼ 120 mm, diameter of a marker ¼ 20 mm).

session. Balls were hung at the same height for all conditions for each player and centred to ensure consistency in each condition. This hanging ball method was selected for two reasons. First, a horizontally extended bar was about an inch wide and it disturbed the visual field of the cameras. Secondly, when we used a short string from the bar in a pilot experiment, some participants reported that it was difficult to swing a bat forcefully because of anxiety about striking the bar with the bat. Thus, we decided to set the ball and the bar apart.

Procedures Participants were given sufficient warm-up time and they hit all types of ball during the practice session. During the practice session, they were also asked to determine their preferred foot placement for use throughout the study. The experiment consisted of six blocks of five trials that included two sets with each type of ball. The order of blocks was randomized and counterbalanced with the participants. Each participant was requested to hit a line drive towards a net (2.0 £ 2.0 m), which was located 3 m away from the participant as a target. In Condition 1, as the batted ball left the string after the impact, its direction was judged from the location where the batted ball hit the net. In contrast, as the batted ball was swung up in the air on the string in Conditions 2 and 3, its direction was judged from the location of the impact and its path immediately after impact. If the batters could not hit a ball towards the target or they felt that they were not hitting well, the trial was repeated.

Data collection and reduction Four 240-Hz infrared cameras (Qualisys, Inc., Gothenburg, Sweden) were used to capture the movements of the markers on the bat and the ball. Qualisys Track Manager (Qualisys, Inc., Gothenburg, Sweden) was used to track and analyse the individual trials. The measurement volume encompassing the swinging area was calibrated with the Dynamic Wand Calibration method using a Qualisys wand kit. Calibration of the camera system yielded a standard deviation in position of the system calibration of 1.0 mm within the entire recorded area. The frame immediately before the bat –ball impact was defined as time ¼ 0 and the time after the impact was defined as positive. In this study, we used a global reference frame: the X-axis was directed from the home plate to the pitcher’s plate; the Y-axis was directed from

Bat speed, trajectory, and timing

21

the ventral (front) to the dorsal side (back) of the right hitter; and the Z-axis indicated a vertical upward direction. The three-dimensional location of the centre of the bat head was calculated by taking the average coordinates of the three markers. Given the threedimensional coordinates of the bat head, the instantaneous velocity for each axis was calculated, using three-point finite differences. The speed (magnitude of the resultant velocity) and the height of the bat head were calculated from the unfiltered threedimensional coordinates. Additional filtering/smoothing was not used unless otherwise specified. Although the data were recorded for 2 s ( ¼ 480 frames) from the beginning of a swing, data from 20 frames before the impact to 20 frames after the impact ( ¼ 170 ms) were analysed. Incomplete data were eliminated from the analysis. No more than two trials in each condition for each participant in any single condition were eliminated. The following three variables were calculated to identify the kinematic characteristics of the bat movement: . Time for peak speed of the bat head: This represented the time at which the bat head reached peak speed. In detail, it was the time between the frame at which the speed of the bat head recorded its maximum and the frame immediately before the bat– ball impact. When this variable had a value of 0, these frames were coincident, and positive values showed that the bat speed was maximized after bat –ball impact. For example, when the bat head reached peak speed at frame 25 and the impact occurred between the 20th and 21st frame, time for peak speed of the bat head was calculated to be 20.8 ms (1/240 s times 5 frames). . Time for the lowest point of the bat head: This represented the time at which the bat head was at its lowest during the swing. In detail, it was the time between the frame at which the bat head was the lowest and the frame immediately before the impact of the bat and the ball. When this variable had a value of 0, these frames were coincident, and positive values showed that the height of the bat head was minimized after bat –ball impact. For example, when the bat head reached its lowest point at the 18th frame and the impact occurred between the 20th and 21st frames, the time for the lowest point of the bat head was calculated to be 2 8.3 ms (1/240 s times 2 2 frames) . Peak speed of the bat head: Peak speed of the centre of the bat head was calculated over 39 frames excluding both ends of 41 frames.

Statistical analysis A one-way repeated-measures analyses of variance (ANOVA) was performed among conditions to investigate the effect of ball conditions on three variables: time for peak speed of the bat head, time for the lowest position of the bat head, and peak speed of the bat head. Tukey’s post-hoc procedure was used to determine significant differences between all combinations of balls and variables. Statistical significance was set at P , 0.05 throughout. All statistical analyses were performed with SPSS 9.0J (SPSS Japan Inc., Tokyo, Japan) software.

Additional experiment To identify similarities or differences between hitting moving balls and stationary balls, we examined the kinematic characteristics of swing against both pitched balls and lightweight stationary balls. Four left-hand hitters playing in a college baseball league participated in this

22

N. Tabuchi et al.

additional experiment; none of these players took part in the main experiment. They were required to hit a ball under two conditions: . Condition A: to hit a small polystyrene ball, which was hung with a string (same as Condition 3 of the main experiment). . Condition B: to hit a pitched plastic ball. In Condition B, each participant was asked to hit a plastic ball (diameter ¼ 70 mm; mass ¼ 50 g) 100 times thrown at about 20 m/s from 12 m. Three-dimensional locations of the bat and the ball were recorded by the same methods as in the main experiment. Participants were requested to hit a line drive. In Condition B, three trials in which the location of the impact and the horizontal direction of the batted ball were similar to that in the stationary condition, Condition A, were selected. Three trials in Condition A were also selected. Time – speed profiles synchronized at the impact in both conditions were used for evaluation of similarity, using cross-correlation analyses with no time lag. The coefficient of cross-correlation was calculated using the data from 20 frames before impact to immediately before impact, because the impulse of bat –ball impact was affected solely in Condition B. For every combination of time and speed in Conditions A and B (3 trials £ 3 trials), the coefficient of cross-correlation was calculated. Then, these nine coefficients for each participant were averaged after Fisher’s z-transformation. The results of the additional experiments suggested that the kinematics of the bat head in Condition B were not significantly different from those in Condition A, when the impact points were close together. The trajectories of the bat head under Condition A and B in the horizontal plane are shown in Figure 2 and those in the sagittal plane are shown in Figure 3. Figures 2a and 3a show the highest coefficient of cross-correlation between both conditions for the speed and each velocity component (Table I). Figures 2b and 3b show the lowest coefficient of cross-correlation (Table I). All coefficients of cross-correlation were significant (P , 0.01). In addition, the speeds of the bat head immediately before impact were not significantly different (32.2 ^ 0.9 m/s under Condition A and 32.7 ^ 1.8 m/s under Condition B). Thus, the bat kinematics of hitting a stationary ball were not significantly different from those of hitting a moving ball.

Figure 2. Representative examples of the trajectory of the bat head in the X –Y plane. (a) The trajectory showing the highest coefficient of cross-correlation (Batter 4). (b) The trajectory showing the lowest coefficient of crosscorrelation (Batter 2) (see Table I). Solid lines show the bat trajectories when hitting a stationary ball. Dotted lines show those when hitting a pitched ball. As the location of the pitched ball varied slightly spatially, these were moved parallel so as to overlap impact points. A circle shows the overlapped impact point.

Bat speed, trajectory, and timing

23

Figure 3. Representative examples of the trajectory of the bat head in the X –Z plane. (a) The trajectory showing the highest coefficient of cross-correlation (Batter 2). (b) The trajectory showing the lowest coefficient of crosscorrelation (Batter 3) (see Table I). Solid lines show the bat trajectories when hitting a stationary ball. Dotted lines show those when hitting a pitched ball. As the location of the pitched ball varied slightly spatially, these were moved parallel so as to overlap impact points. A circle shows the overlapped impact point. Table I. Coefficients of cross-correlation on the speed and each velocity component for the bat head (mean ^s). Batter 1 2 3 4

Speed

X-component

Y-component

Z-component

0.999** ^ 0.000 0.997** ^ 0.002 0.997** ^ 0.003 0.998** ^ 0.001

0.999** ^ 0.000 0.995** ^ 0.005 0.998** ^ 0.002 0.999** ^ 0.000

0.962** ^ 0.036 0.956** ^ 0.032 0.942** ^ 0.101 0.978** ^ 0.025

0.895** ^ 0.071 0.953** ^ 0.024 0.606** ^ 0.319 0.774** ^ 0.076

** P , 0.01.

Results Time for peak speed of the bat head The mean times for the peak speed of the bat head under Conditions 1, 2, and 3 were 2 0.2 ^ 0.4 ms, 10.1 ^ 8.5 ms, and 12.0 ^ 7.4 ms, respectively (Figure 4). The mass of the stationary ball clearly influenced the time for peak speed of the bat head (F2,23 ¼ 16.2, P , 0.001). Tukey’s post-hoc test showed that the time for peak speed in Condition 1 was significantly earlier than that in Condition 2 (P , 0.01) and that in Condition 3 (P , 0.01). Most participants reached peak speed of the bat head after bat –ball impact when balls of lighter mass (Conditions 2 and 3) were used. However, peak speed was shown to be very close to impact when the standard weight ball (Condition 1) was used. For six of the eight participants, the mean and standard deviation under Condition 1 were zero. In other words, the peak speed of the bat head always occurred at impact.

Time for the lowest position of the bat head The mean times for the lowest position of the bat head under Conditions 1, 2, and 3 were 3.6 ^ 6.6 ms, 3.5 ^ 6.6 ms, and 3.5 ^ 6.0 ms, respectively (Figure 5). The mass of the stationary ball did not appear to affect the timing at which the bat head reached its lowest point (F2,23 ¼ 0.038, P ¼ 0.963). For most participants, the time for the lowest position was within ^ 2 frames (8.3 ms) of impact with the ball for each condition.

24

N. Tabuchi et al.

Figure 4. Mean time for peak speed of each condition (ms). Open bars are the means for Condition 1. Hatched bars are the means for Condition 2. Black solid bars are the means for Condition 3. Error bars represent standard deviation. Positive values indicate that the speed of the bat head was maximized after the bat–ball impact. ** P , 0.01.

Figure 5. Mean time for the lowest point of the bat trajectory of each condition (ms). Open bars are the means for Condition 1. Hatched bars are the means for Condition 2. Black solid bars are the means for Condition 3. Error bars represent standard deviation. Positive values indicate that the height of the bat head dropped at the bat–ball impact.

Peak speed of the bat head The mean peak speeds of the bat head under Conditions 1, 2, and 3 were 33.4 ^ 0.9 m/s, 33.6 ^ 0.8 m/s, and 33.4 ^ 1.0 m/s, respectively (Figure 6). The peak speed of the bat head for all participants was not significantly different among conditions (F2,23 ¼ 1.356, P ¼ 0.29).

Bat speed, trajectory, and timing

25

Figure 6. Mean peak speed of the bat head of each condition (m/s). Open bars are the means for Condition 1. Hatched bars are the means for Condition 2. Black solid bars are the means for Condition 3. Error bars represent standard deviation.

Discussion and implications Relationship between the bat– ball impact and speed of the bat head The time for peak speed of the bat head generally varied according to the conditions. Under Condition 1, the bat head nearly always reached peak speed at the time of bat– ball impact. When the lighter balls were used (Conditions 2 and 3; Figure 4), the peaks were observed 5 – 17 ms after bat –ball impact. The time –speed profiles of the bat head before bat– ball impact under all conditions were almost identical, and peak speeds of the bat head in Conditions 2 and 3 were observed after impact. That peak speed of the bat head was found at bat– ball impact in Condition 1 would appear to have resulted from the large impulse of the impact, and that the speed of the bat head would have continued to increase if the impact had not occurred. Batters did not hit the stationary lightweight balls when the bat was at peak speed or in the deceleration phase after the peak speed had been achieved. Instead, the impact took place during the acceleration phase before the maximum speed of the bat head occurred. As a typical example, the averaged raw data of the time – speed profile for one participant in Conditions 1 and 3 are shown in Figure 7. A sharp drop in the speed of the bat head was observed immediately after the bat– ball impact in Condition 1. The peak speed of the bat head, measured in Condition 1, was a result of the large impulse caused by the collision. If the impulse was small enough, the peak speeds occurred at about 17 ms after the impact. Thus, the bat – ball impact occurred as the speed of the bat head was increasing. Some previous studies of the impact of two objects indicated that bats or rackets obtained maximum speed at least 10 ms earlier than the instant of impact (Elliott et al., 1989; McIntyre and Pfautsch, 1982; Messier and Owen, 1984; Welch et al., 1995). This may result from improper filtering. Knudson and Bahamonde (2001) reported that it is likely that a velocity curve including impact may be over-smoothed as a result of filtering. Vint and Hinrichs (1996) also reported that applying a filter to the coordinate data of the entire swing, including the impact, would result in over-smoothing, which might cause the researcher to miss the acceleration of the bat immediately before impact. They suggested

26

N. Tabuchi et al.

Figure 7. A typical example of time–speed profiles of the bat head for Conditions 1 and 3 (averaged data of Participant 1). A solid line shows the speed (m/s) of the bat head, calculated from averaged non-filtered raw data, under Condition 1. A dotted line shows speed (m/s) of the bat head, calculated from averaged non-filtered raw data, under Condition 3. A broken line shows the speed (m/s) of the bat head calculated from filtered data under Condition 1, using a Butterworth filter with a cut-off frequency of 14.9 Hz.

that the slowdown occurred before the impact because the frequency of the coordinate data around the impact was substantially higher than that during the remainder of the swing. Figure 7 shows the difference between the raw data (the solid line) and the low-pass-filtered data using a Butterworth filter with a cut-off frequency of 14.9 Hz (the broken line) under Condition 1. Data were synchronized at impact (time ¼ 0) and averaged across trials in each condition. The time for peak speed of the bat head for the filtered data was shifted to 9.5 ^ 2.6 ms earlier than the raw data on average across the participants. The results support those of Welch et al. (1995), who adopted a method similar to that used in Condition 1 and reported that the peak speed of the bat head occurred 15 ms earlier. Welch et al. (1995) used a Butterworth filter with a cut-off frequency of 13.8 Hz for kinematic data with a sampling rate at 200 Hz. It would appear that the filtering effect caused the peak speed to occur earlier in their study. Since a larger impulse induces a greater time shift owing to the filtering effect, hitting a pitched ball is expected to have a much greater influence than hitting a stationary ball. McIntyre and Pfautsch (1982) also measured the speed of the bat head when hitting a pitched baseball and reported that the speed of the bat head reached its maximum at about 15 ms before impact. Though they used another method (cubic spline curve fitting) for smoothing the data, it was reported that the method produced a similar result to a Butterworth filter (Vint and Hinrichs, 1996). This time lag may be partly explained by the filtering effect as well. Messier and Owen (1984) measured the speed of the bat head at a sampling rate of 100 Hz when hitting a pitched softball and reported that the speed of the bat head reached a maximum at 32 ms before impact. Unfortunately, as they did not describe their filtering – smoothing method, we cannot clearly refer to the causes of this large time lag. Although there may have been other effects, it seemed that the lower sampling rate and the filtering effect, owing to the large impulse, affected the time – speed profile. In addition, the effect of target-directed hitting may have further influenced the results (McIntyre and Pfautsch, 1982). The participants in their

Bat speed, trajectory, and timing

27

study hit a baseball to a designated area, either opposite field (right field for a right-handed batter) or same field (left field for a right-handed batter). Similar time lags have also been observed in other target-directed hitting tasks, such as tennis (Elliott et al., 1989; Plagenhoef, 1971) and golf (William and Sih, 2002). This time lag was interpreted to mean that players might decrease the speed of the racket or club to prepare for an impact. In particular, the results of the golf study by William and Sih (2002) were unaffected by the filtering effect since they used raw data. The peak speed before the impact could be one of the characteristics of target-directed hitting. Further studies that try to quantify the influences of the filtering effect and the task demands on the time shift in the goal-directed hitting task are necessary.

Relationship between the bat– ball impact and the lowest point of the bat head Our results showed that the minimum height of the bat head was a variable that could be used to estimate the point of impact, which supports previous research (Gray, 2002; Welch et al., 1995). Welch et al. (1995) reported that a bat had negative vertical velocity (downward movement) before impact, proceeded to 0 m/s at impact, and had positive vertical velocity (upward movement) after impact in tee batting in baseball. This implies that the bat– ball impact occurred around the instant when the bat was at its lowest point (Figure 5). It is also noteworthy that each participant swung the bat head to the lowest point at a specific time relative to impact in all conditions and showed smaller standard deviations than those related to timing of peak speed of the bat head. Under Condition 3, the standard deviation of the time for peak speed of the bat head was 4.4 ^ 1.4 ms; in contrast, that for the lowest point of the bat was 2.5 ^ 0.5 ms. The standard deviation of timing of the lowest bat height was significantly smaller than that of peak speed of the bat head using the paired t-test (t7 ¼ 3.13, P , 0.05). Each participant appeared to have strong coupling between the bat – ball impact and the lowest point of the bat head. The vertical position of the bat head may be useful to determine the imaginary point of impact. In other words, the lowest point of the bat head is better than the peak speed for estimating the timing of the impact in a dry swing. However, it is necessary in some cases to modify the estimation depending on the batters. It appeared that poor batters were less able to recognize their optimal impact point. Batters who understand the relationship between the vertical position of the bat head and their own impact point may be able to improve their performance.

Effect of ball size on the peak speed of the bat head and the lowest point of the bat head The results of Conditions 2 and 3 were almost identical, which shows that ball size has little effect on the peak speed of the bat head and the minimum height of the bat head. A significant difference in the peak speed of the bat head was only observed for one participant. He obtained a higher swing speed under Conditions 1 and 2 than Condition 3. He may have paid too much attention to the spatial accuracy of the swing, which could have resulted in a reduction of speed. The finding could also be attributed to Fitts’ law (Fitts, 1954), which describes a speed –accuracy trade-off. None of the other participants, however, demonstrated the same phenomenon. It seems that the control strategy for the bat swing was unchanged relative to target size. Experienced batters have sufficient skill to control bat movement without any loss of speed of the bat head. It appears that they aimed at the centre of the ball even when hitting a larger ball (Conditions 1 and 2). This would be supported by the results of Watts and Bahill (2000). They calculated the error margin in baseball batting

28

N. Tabuchi et al.

and suggested that the batter must estimate the vertical position of the bat – ball impact within 12.7 mm to hit a line drive in fair territory. Reason why batters do not hit a ball at peak speed of the bat head Adair (1995) reported that the duration of the impact is about 1 ms and that most momentum transfer takes place in about 0.6 ms. During this short time, the momentum and kinetic energy exerted by muscle work are negligible. Therefore, batters have an advantage when they hit a ball at the instant of the peak speed of the bat head. Our participants did not do that. We anticipate other benefits for hitting balls while accelerating the bat head: one possible benefit is spatial accuracy. As described above, the spatial margin of the bat –ball impact requires a vertical accuracy of ^ 12.7 mm (Watts and Bahill, 2000). The vertical displacement of the bat head, during a certain short period, is minimized around the lowest position of the bat trajectory (Figure 8). This means that small temporal errors do not result in large spatial errors around the lowest position. In fact, our participants swung the bat head to its lowest point in the trajectory within an average of ^ 7.8 ms from impact with the ball. The vertical travel distance during this time was 6.6 ^ 3.1 mm; in contrast, that calculated around the instant of the peak speed of the bat head was 42.4 ^ 37.0 mm. As the task in this study was to hit a stationary ball, it seems reasonable to hit a ball at the instant when the bat head reaches its lowest point. On the other hand, when a batter hits a typical pitched fastball travelling at 42 m/s, the ball approaches the batter at an angle of 8.98 downward from the horizontal (Sawicki, Hubbard, and Stronge, 2003). This means that the pitched ball drops 51 mm around the home plate during the mean time (7.8 ms) between the lowest point of the bat head and the impact with the ball, as described above. Taking into the account the estimation of the vertical travel distance of the bat head, it seems reasonable to hit an approaching ball at the instant of peak speed of the bat because the strategy may generate less vertical spatial error. For the purpose of producing a maximum range in the trajectory of a batted ball, a slight upswing appears to

Figure 8. Change of the vertical position of the bat head (under Condition 3 of Participant 6). At about the instance of the bat–ball impact, the height of the bat head is less changeable than around the instant of peak speed. Although the two durations are identical, the vertical displacements of the bat head are quite different.

Bat speed, trajectory, and timing

29

be the best technique, as Williams and Underwood (1986) asserted. The physical model developed by Sawicki et al. (2003) estimated that an optimal strategy for achieving the maximum range of a batted ball is to swing a bat upward at an angle of 9.18 against a typical fastball travelling at 42 m/s. This estimation suggests that batters should match the angle of the swing with that of a pitched ball to hit a home run. Actually, our participants also swung the bat slightly upward at the instant of peak speed of the bat head (9.6 ^ 9.08). Therefore, if batters want to maximize the range of the batted ball, impact should occur around the time of the peak speed of the bat head, which will provide a higher speed of the bat head and a higher probability of hitting the pitch. However, this strategy may impose complicated processing on batters, as they should reconstruct a motor program with a difference in the movement of the bat at the point of impact on a pitch-by-pitch basis. In addition, from the viewpoint of temporal constraints, batters must react in less time because the speed of the bat head reaches its maximum after the bat head reaches its lowest point. That is, as movement time for achieving maximum speed of the bat head was longer than that for achieving minimum height of the bat head, the remaining decision time may be shorter for achieving maximum speed of the bat head. The batters in this study did not adopt a strategy to maximize the range of the batted ball. Their strategy was one with the least vertical displacement when hitting a stationary ball at a certain location. Combined with the results showing that the speed of the bat head was reduced only 1.4 m/s (4.0%) on average at the lowest point of the bat trajectory from the point of peak speed, it seems that batters had a preference for swing accuracy over swing speed. Implications for training and coaching Considering the spatial and temporal constraints on the swing as mentioned above, it is desirable that impact occurs when the bat head is at its lowest point and at peak speed. Only one participant performed accordingly. His career of playing baseball was outstanding and his batting skill was the best, according to a coach, among the participants in this study. On the other hand, another participant had the greatest time lag between the lowest point of the bat and the impact. In addition, he had the slowest speed of the bat head. These results suggest that these variables may be useful for improving batting performance. Limitations of this study In the additional experiment, while the profiles of speed and the horizontal velocity components (X- and Y-components) of the bat head showed very strong correlations between the hung ball and pitched ball conditions, that of the vertical (Z-) component showed a relatively lower coefficient (Table I). This result might be caused by variance of the pitched ball height. The accuracy of the vertical displacement is particularly important in baseball batting (Watts and Bahill, 2000). A little discrepancy of the height of the pitched ball may cause a different angle of downward movement of the bat head. Since this study investigated the relationship between the impact and the lowest point of the bat head or the peak speed of the bat head at one preferred stationary ball location, further study of the effect of ball location on this relationship is required. Though kinematic similarity between the bat swing to a pitched ball and that to a stationary ball was shown, it does not mean that hitting a pitched ball is the same as hitting a stationary ball. It is premature to apply our results directly to normal baseball batting. Furthermore, only

30

N. Tabuchi et al.

experienced players participated in this study. These results may not apply to all standards of players, from professional to youth players. Further studies with participants with a wide range of playing standards and those using pitched balls may also be needed for the generalization of the current results. Conclusion This study revealed that the bat head was at maximum speed at the time of impact with the ball when stationary balls of standard weight were used. However, the bat decelerated drastically after the impact. Its time – speed profile before the impact was identical with that in the lightweight ball condition in which the speed of the bat head increased after impact. This means that batters hit balls while accelerating the bat head. Data filtering might cause peak speed to be recorded before impact, as reported for the bat or racket in previous studies. The time for the lowest point of the bat trajectory was shown to have less variability in spite of the ball conditions. This phenomenon should be useful to estimate the impact point and timing during a dry swing.

References Adair, R. K. (1995). The physics of baseball. Physics Today, 48 (5), 26– 31. Elliott, B., Marsh, T., and Overheu, P. (1989). A biomechanical comparison of the multisegment and single unit topspin forehand drives in tennis. International Journal of Sport Biomechanics, 5, 350–364. Fitts, P. M. (1954). The information capacity of the human motor system in controlling the amplitude of movement. Journal of Experimental Psychology, 47, 381– 391. Fleisig, G. S., Zheng, N., Stodden, D. F., and Andrews, J. R. (2002). Relationship between bat mass properties and bat velocity. Sports Engineering, 5, 1–8. Gray, R. (2002). Behavior of college baseball players in a virtual batting task. Journal of Experimental Psychology: Human Perception and Performance, 28, 1134–1148. Knudson, D., and Bahamonde, R. (2001). Effect of endpoint conditions on position and velocity near impact in tennis. Journal of Sports Sciences, 19, 839–844. Levanon, J., and Dapena, J. (1998). Comparison of the kinematics of the full-instep and pass kicks in soccer. Medicine and Science in Sports and Exercise, 30, 917 –927. McIntyre, D. R., and Pfautsch, E. W. (1982). A kinematic analysis of the baseball batting swings involved in opposite-field and same-field hitting. Research Quarterly for Exercise and Sport, 53, 206–213. Messier, S. P., and Owen, M. G. (1984). Bat dynamics of female fast pitch softball batters. Research Quarterly for Exercise and Sport, 55, 141–145. Nunome, H., Asai, T., Ikegami, Y., and Sakurai, S. (2002). Three-dimensional kinetic analysis of side-foot and instep soccer kicks. Medicine and Science in Sports and Exercise, 34, 2028– 2036. Plagenhoef, S. (1971). Patterns of human motion. Englewood Cliffs: Prentice-Hall. Sawicki, G. S., Hubbard, M., and Stronge, W. J. (2003). How to hit home runs: Optimum baseball bat swing parameters for maximum range trajectories. American Journal of Physics, 71, 1152–1162. Vint, P. F., and Hinrichs, R. N. (1996). Endpoint error in smoothing and differentiating raw kinematic data: An evaluation of four popular methods. Journal of Biomechanics, 29, 1637– 1642. Watts, R. G., and Bahill, A. T. (2000). Keep your eye on the ball: Curve balls, knuckleballs, and fallacies of baseball. New York: Freeman. Welch, C. M., Banks, S. A., Cook, F. F., and Draovitch, P. (1995). Hitting a baseball: A biomechanical description. Journal of Orthopaedic and Sports Physical Therapy, 22, 193–201. William, K. R., and Sih, B. L. (2002). Changes in golf clubface orientation following impact with the ball. Sports Engineering, 5, 65–80. Williams, T., and Underwood, J. (1986). The science of hitting. New York: Simon and Schuster.