IOURNAL OF APPLIED BIOMECHANICS, 1995, 11, 81-102 O 1995 by Human Kinetics Publishers, Inc.

Angular Momentum and performance in the Triple Jump:

A Cross-Sectional Analysis

Bing Yu and James G. Hay

The purposes of this study were (a) to determine the magnitude of the angular momentum elite triple jumpers possess during each of the three phases of a triple jump, and (b) to identify those components of the angular momentum that are closely related to the actual distance of the triple jump. Angular momentum about each of three orthogonal axes at the takeoff of each of the last stride, hop, step, and jump was computed from the smoothed 3-D coordi- nate data of 21 body landmarks and joint centers and normalized to body mass (m,) and standing height (h,). Linear and nonlinear regression analyses were conducted to examine the relationships between angular momenta and actual distance. The results suggested that the estimated optimum magnitude of this side-somersaulting angular momentum is 0.0069 mb h,2 kg . m2 . s-' toward the side of the free leg, that the side-somersaulting angular momentum needed at the takeoff of the step should be obtained during the support phase

, of the hop; and that the change in the side-somersaulting angular momentum during the support phase of the step should be minimized.

The triple jump is one of three jumping events in track and field. It consists of an approach run followed by (a) a hop, that is, a takeoff from one foot and a landing on the same foot; (b) a step, a takeoff from one foot and a landing on the other foot; and finally (c) a jump, a takeoff from one foot and a landing on both feet in the sand. Because it involves three consecutive touchdowns and takeoffs at high speed and a change in the support leg, the triple jump is technically more demanding than long and high jumps, which involve only one touchdown and takeoff each.

Although the techniques employed by elite triple jumpers have received increased attention from researchers in biomechanics over the last decade, a recent review of the literature revealed that most studies on triple jump techniques have been limited to two-dimensional analyses of those linear motions that determine the actual distance of a triple jump (Hay, 1992). Only one of the papers located for this review contained data on the angular momentum of triple

Bing Yu and James G. Hay are with the Biomechanics Laboratory, Department of Exercise Science, University of Iowa, Iowa City, IA 52242.

82 Yu and Hay

jumpers, and that data concerned only one of the three principal axes of the body (Hillman, 198 1).

It is believed that a triple jumper's angular momentum has a profound influence on the jumper's ability to maintain balance during the execution of a triple jump, that the need to maintain balance affects the triple jumper's ability to maintain anteroposterior velocity and produce vertical velocity, and that this in turn affects the official distance (Hay, 1992). The purposes of this study were (a) to determine the magnitudes of three orthogonal components of the angular momentum possessed by elite triple jumpers during each of the three phases of a triple jump, and (b) to identify those components of the angular momentum that are closely related to the actual distance of the triple jump.

Methods

Data Collection

Subjects. The subjects in this study were the 13 finalists in the men's triple jump event at the 1992 United States Olympic Trials in New Orleans, Louisiana (Table 1). The videotaped trial in which the longest official distance was recorded for each subject was selected for analysis.

Videotaping Protocol. A direct linear transformation procedure with panning cameras (Yu, Koh, & Hay, 1993) was used to collect three-dimensional

Table 1 Subjects and Performances

Distance of analyzed ,

Height Mass jump Subject (m) (kg) (m) Place Comments

Charlie Simpkins Mike Conley John Tillman Ray Kimble Reggie Jones Kenny Harrison Robert Cannon Willie Banks Joe Greene Erick Walder Greg Harper Tyrone Scott Reggie Jackson

Mean SD

1 2nd, Olympic Games, 1992 2 1992 Olympic Games champion 3 4 5 6 Ist, World Championships, 1991 7 8 World record holder, 17.97 m 9

10 11 12 13

83 Triple Jump

(3-D) coordinates of 21 body landmarks. This 3-D data collection procedure requires that at least two cameras be used to record the positions and orientations of a control object and the performances of the subjects during the event, and that each camera be panned by rotating about a single fixed axis. The same lens setup has to be used for each camera throughout the videotaping.

Two S-VHS video cameras were used to record the positions and orienta- tions of the control object and the performances of the subjects at a frequency of 60 Hz. An overhead view of the camera placements is shown in Figure 1. A marked tape was placed along each side of the approach runway and the pit. These marked tapes were used to determine the instantaneous orientation of each camera (Yu et al., 1993). A control object with 68 control points distributed in a sphere of 2.44 m diameter was placed and recorded in 10 consecutive positions along the runway and in the pit. The space covered by the control object at each position is referred to as an individual control volume. The 10 consecutive individual control volumes defined a total control volume (the sum of the space covered by all individual control volumes) of approximately 24.4 m x 2.44 m x 2.44 m, with the 24.4 m dimension parallel to the approach runway. This total control volume covered the space in which the last two strides (one stride consists of a support phase followed by a flight phase) of the approach run, the hop, the step, and the jump were performed. A global reference frame was defined with the x-axis parallel to the runway and pointing in the jumping direction, the y-axis perpendicular to the x-axis and pointing to the left side of the runway, and the z-axis perpendicular to the surface of ihe runway and pointing upward.

. . Inddual control volumes

Figure 1 - Overhead view of camera and control object placements.

84 Yu and Hay

Da fa Reduction

The videotape records of the control object and each of the selected trials were digitized with the aid of an S-VHS videocassette recorder, a 14-in. color monitor, a microcomputer, and Peak2D computer software (Peak Performance Technolo- gies, Denver, CO). The control object and two marks on the marked runway tape were digitized in nine fields from each camera, for each individual control volume. The number of control points digitized in each field varied from 49 to 61. The videotape record of each selected trial from each camera was digitized at a sampling frequency of 60 Hz, from two fields before the touchdown of the second last stride to two fields after the landing in the pit. In each digitized field, 21 body landmarks and joint centers defining a 14-segment model of the human body (Clauser, McConville, & Young, 1969) and two marks on the marked runway tape were digitized. Control volume calibrations, mathematical time- synchronization of the digitized two-dimensional (2-D) data from the two cam- eras, and the transformation from digitized 2-D data to real-life 3-D coordinate data were all conducted on a microcomputer using a set of computer programs written for these purposes. The mean resultant calibration error (the deviation between calculated position and known position) in each individual control vol- ume ranged from 9 to 15 mm with a mean value of 12 mm.

The real-life 3-D coordinate data of 21 body landmarks and joint centers were smoothed using a second-order Butterworth digital filter (Winter, Sidwall, & Hobson, 1974). The optimum cutoff frequency, F,, was estimated from sampling frequency, F,, using an equation proposed by Yu (1988):

The sampling frequency in this study was 60 Hz. The corresponding estimated optimum cutoff frequency was 7.4 Hz.

The actual distance (D,) of each trial was defined as the sum of the official distance and the distance lost at the board, that is, the difference between the x-coordinate of the pit edge of the board and the mean of the x-coordinates of the toe of the takeoff foot during the support phase of the hop.

The 3-D coordinates of the center of gravity of the whole body (G) were calculated using the basic segmental procedure described by Hay (1985) and the segment inertial data of Hinrichs (1990). Anteroposterior (x) and lateral (y) velocities of the center of gravity of the whole body at a takeoff (vy(,) and vzto)) were calculated using the equations

where x(:o) and ygo) are the x and y coordinates of G at a takeoff, xgd, and ygd) are the x and y coordinates of G at the subsequent touchdown, and At is the duration of the flight phase. The vertical velocities of G at a takeoff and at the subsequent touchdown (vz, and v$,) were calculated using the equations

Triple Jump 85

where zg,,, and zgd, are the z coordinates of G at the takeoff and subsequent touchdown, and g is the gravitational acceleration. The change in velocity of G during a support phase was equal to the velocity of G at the takeoff minus that at the preceding touchdown. The velocities of G at the touchdown and takeoff of the hop, the step, and the jump were calculated using Equations 1 through 4.

The angular momentum of the whole body about three orthogonal axes, x', y', and z', which were parallel to the corresponding axes of the global reference frame and passing through G, was calculated using the method presented by Dapena (1978) and the segment inertial data of Hinrichs (1990) and Whitsett (1963). The x', y', and z' axes will be referred to here as the side-somersaulting, somersaulting, and twisting axes, respectively, even though they only approxi- mated these three principal axes of the body-the side-somersaulting (antero- posterior), somersaulting (lateral), and twisting (longitudinal) axes. The three angular momentum components of the whole body about the x', y', and z' axes are similarly referred to as side-somersaulting angular momentum, somersaulting angular momentum, and twisting angular momentum, respectively.

The average value of each component of the angular momentum during a flight phase was considered to be the best estimate of that component of the angular momentum at the preceding takeoff or the subsequent touchdown (Hay, Wilson, Dapena, & Woodworth, 1977). Each of the three components of the angular momentum was calculated for the takeoffs of the last stride, the hop, the step, and the jump. The change in each component of the angular momentum of the whole body during a support phase was defined as the difference between the angular momentum during the subsequent and preceding flight phases.

To facilitate the interpretation of the velocity and angular momentum data, the directions of the lateral velocity and the angular momentum about the x' and z' axes were adjusted with respect to the takeoff leg of the hop. After the adjustment, the anteroposterior velocity was considered positive if it was directed to the positive direction of x-axis; the lateral velocity was considered positive if it was directed to the same side of the takeoff leg for the hop; the vertical velocity was considered positive if it was directed to the positive direction of the z-axis; the side-somersaulting angular momentum was considered positive if it acted to rotate the top of the body to the side of the takeoff leg for the hop; the somer- saulting angular momentum was considered positive if it acted to rotate the top of the body forward; and the twisting angular momentum was considered positive if it acted to rotate the front of the body to the side of the takeoff leg of the hop.

To eliminate the effects of variations in body mass and height and thus to emphasize the effects of technique, the angular momentum data were normalized for body mass and standing height using the equation

where H' is the normalized angular momentum, H is the original calculated angular momentum, and mB and h, are the mass and standing height of the subject, respectively.

The change in the velocity of G in a given direction during a given support phase was calculated as the difference between the corresponding velocities at the takeoffs immediately after and at the touchdown immediately before the support phase. The change in the normalized angular momentum of the body

86 Yu and Hay

about a given axis during a given support phase was calculated as the difference between the corresponding angular momentum at the takeoffs immediately after and immediately before the support phase.

Reliability Test

To test the reliability of the data collected, each of the selected trials was digitized twice and two sets of 3-D coordinate data of 21 body landmarks were obtained from two different sets of digitized 2-D data. All variables of interest were calculated twice from these two different sets of 3-D coordinate data. Because the best available estimate of the true value of any of the variables measured was the mean of the test and retest values of that variable, one half of the absolute difference between these test and retest values was considered as the error in that variable in that trial. The results of this reliability test showed that the maximum relative errors were generally less than 10% and that the correlation between the values of each variable of interest obtained in test and retest was generally above .99, indicating a high degree of reliability.

Data Analysis

The data analysis procedure consisted of the computation of means and standard deviations and the performance of linear and nonlinear regression analyses. The nonlinear correlation coefficient was used if the magnitude of the correlation coefficient obtained in the nonlinear analysis was significantly higher than that obtained in the linear analysis. Such use was consistent with the expectation that an optimum value existed for many of the independent variables considered, or, in other words, that relationships between dependent and independent variables were likely to be nonlinear.

The 0.1 level of confidence was chose to indicate statistical significance after considering the consequences of Type I and Type I1 errors. Since the purpose of the study was to identify those components of angular momentum that truly have an effect on performances in the triple jump, it was concluded that retaining a false null hypothesis (Type I1 error)-and thus dismissing an angular momentum that was trul-iimportant to the performance-was the greater ofthe two possible evils. Accordingly, a level of confidence was chosen that gave relatively little chance of a Type I1 error.

The Bonferroni procedure (Wilkinson, 1988) was used to guarantee that the chosen level of confidence was not exceeded for a given group of tests. The critical p value to achieve statistical significance for an individual test (p) within a group of tests was determined using the equation

p = [ l - (1 - p')""],

where p' is the selected overall level of confidence for the given group of tests, and n is the number of the tests in the given group of tests. In this study, 21 individual t tests were conducted in the descriptive statistical analysis and 66 individual tests of significance of correlations were conducted. The critical p values for each individual t test and test of significance of correlation were .005 and .0015, respectively. The corresponding magnitudes o f t and r were 4.04 and .78, respectively.

Triple Jump

Results and Discussion

Angular Momentum in the Triple jump

The angular momentum and normalized angular momentum for each subject at each takeoff are presented in Tables 2 through 4. The mean and standard deviation of the angular momentum at each takeoff are presented in Figures 2 through 4.

Side-Somersaulting. The mean value of the side-somersaulting angular momentum was found to be significantly different from zero at the takeoffs of the last stride and step, but not significantly different from zero at the takeoffs of the hop and jump (Figure 2).

The side-somersaulting angular momentum at each takeoff reflected the specific technical demands of the following flight phase. Consider a triple jumper, viewed from the rear, at the instant of takeoff to the last stride of the approach (Figure 5a). During the flight phase of the last stride, this jumper must lower his left leg, raise his right leg, and keep his trunk erect, so that he touches down on the board in an appropriate position (Figure 5b). This lowering of one leg and raising of the other requires a certain amount of counterclockwise side-

Table 2 Side-Somersaulting Angular Momentum (kg . mZ - s-') and Normalized Side-Somersaulting Angular Momentum (10-2s-')

Side-somersaulting Normalized side-somersaulting angular momentum angular momentum

Last Last Subject stride Hop Step Jump stride Hop Step Jump

Charlie Simpkins Mike Conley John Tillman Ray Kimble Reggie Jones Kenny Harrison Robert Cannon Willie Banks Joe Green Erick Walder Greg Harper Tyrone Scott Reggie Jackson

Mean 3.71* -0.40 -3.58* 0.70 1.45* -0.02 -1.40* 0.26 SD 1.17 0.83 1.56 2.04 0.45 0.33 0.63 0.82 Maximum 5.66 1.00 -0.13 2.58 2.08 0.41 -0.05 1.81 Minimum 2.02 -1.92 -5.91 -4.20 0.77 -0.79 -2.43 -1.77

*Significantly different from zero ( p < .005).

88 Yu and Hay

Table 3 Somersaulting Angular Momentum (kg . m2 - s-') and Normalized Somersaulting Angular Momentum (10-2s-')

Somersaulting Normalized somersaulting angular momentum angular momentum

Last Last Subject stride Hop Step Jump stride Hop Step Jump

Charlie Simpkins Mike Conley John Tillman Ray Kimble Reggie Jones Kenny Harrison Robert Cannon Willie Banks Joe Green Erick Walder Greg Harper Tyrone Scott Reggie Jackson

Mean SD Maximum Minimum

--

*Significantly different from zero @ < .005).

somersaulting angular momentum, indicated by the curved arrow in Figure 5b. If the jumper has exactly this amount of side-somersaulting angular momentum at the takeoff of the last stride, he can amve at the board with his trunk erect. However, if he has zero side-somersaulting angular momentum at the takeoff of the last stride, the subsequent lowering of one leg and lifting of the other evokes a contrary angular reaction in other parts of his body. This reaction may be manifested in a clockwise rotation of the upper body away from the erect position (Figure 5c) or in a lesser clockwise rotation of the entire body away from the erect position (Figure 5d), indicated by the upper curved arrows in the figure. In either case, the jumper arrives at the board in an unfavorable position from which to initiate the support phase of the hop.

It is perhaps of interest to note here that this account of the jumper's need for side-somersaulting angular momentum at takeoff of the last stride is consistent with the findings of Hinrichs (1987) concerning the side-somersaulting angular .. momentum and the distribution of the side-somersaulting angular momentum among segments during the flight phase of running.

The takeoff leg for the step is the same as for the hop and, thus, there is no need to change the takeoff leg during the flight phase of the hop. In light of

Triple Jump 89

Table 4 Twisting Angular Momentum (kg . m2 . s-') and Normalized Twisting Angular Momentum (lO-'s-')

Twisting Normalized twisting angular momentum angular momentum

Last Last Subject stride Hop Step Jump stride Hop Step Jump

Charlie Simpkins Mike Conley John Tillman Ray Kirnble Reggie Jones Kenny Harrison Robert Cannon Willie Banks Joe Green Erick Walder Greg Harper Tyrone Scott Reggie Jackson

Mean SD Maximum Minimum

*Significantly different from zero ( p < .005).

this, it is not surprising that, on average, the subjects possessed little side- somersaulting angular momentum at the takeoff of the hop. During the flight phase of the step, a triple jumper has to do essentially the same thing that was done during the flight phase of the last stride. A certain amount of side- somersaulting angular momentum toward the side of the free leg is therefore needed at the takeoff of the step.

Somersaulting. The mean value of the somersaulting angular momentum was significantly greater than zero at the takeoff of the last stride, hop, step, and jump (Figure 3). The mean value of the somersaulting angular momentum at the takeoff of the step and jump was significantly different from the corresponding values at the preceding takeoff.

The magnitude of the somersaulting angular momentum possessed at any given takeoff may have some relationship with the linear motions of segments during the succeeding flight phase. Koh and Hay (1990) reported that elite triple jumpers used active landings at the end of each of the hop and step phases (or, more specifically, at the touchdowns of the step and jump). The term active landing refers to a landing in which the athlete employs a pawing or backward sweeping motion of the landing leg to reduce the forward anteroposterior velocity

Yu and Hay

Jump

Figure 2 - Means and standard deviations of the side-somersaulting angular mo- mentum at the takeoff of the hop, step, and jump (kg . mZ . s-I). (Solid arrows indicate means significantly different from zero. The subjects are viewed from rear.)

Triple Jump

Jump

Figure 3 - Means and standard deviations of the somersaulting angular momentum at the takeoff of the hop, step, and jump (kg . mZ . s-'). (Solid arrows indicate means significantly different from zero.)

Yu and Hay

Last stride

Jump

Figure 4 - Means and standard deviations of the twisting angular momentum at the takeoff of the hop, step, and jump (kg . m3 . s-I).

Triple Jump

Figure 5 - (a) Body position at the takeoff of the last stride. Body position at the touchdown of the hop (b) with enoughside-somersaulting angular momentum toward the side of the free leg at the takeoff of the last stride, (c) without enough side- somersaulting angular momentum toward the side of the free leg at the takeoff of the last stride (trunk is rotated during the flight phase of the last stride), and (d) without enough side-somersaulting angular momentum toward the side of the free leg at the takeoff of the last stride (whole body is rotated during the flight phase of the last stride).

of the foot prior to impact (Koh & Hay, 1990). The results of Koh and Hay's study indicate that using an active landing at the end of a flight phase may benefit actual distance by minimizing the loss in the forward anteroposterior velocity of the body during the succeeding support phase.

Moving the landing leg backward relative to G during a flight phase requires that the landing leg possess some forward somersaulting angular momentum about G. This means that an athlete needs to possess extra forward somersaulting angular momentum at the takeoff preceding the flight phase in order to use an active landing without changing the motions of the other segments relative to G; it also means that the activeness of the landing may depend on the magnitude of the somersaulting angular momentum the athlete possessed at the takeoff of the preceding flight phase. In the present study, the somersaulting angular momentum at the takeoff of the hop was significantly greater than that at the takeoff of the step. This result is consistent with the activeness measures of the

94 Yu and Hay

hop and step landings reported by Koh and Hay (1990). Koh and Hay found that the step landing was less active than the hop landing.

There are at least two possible explanations for this consistency between the somersaulting angular momentum at the takeoff of the step and the degree of activeness of the subsequent step landing: (a) There was not enough somersaulting angular momentum during the flight phase of the step to permit a very active landing, or (b) elite triple jumpers decreased the forward somersaulting angular momentum they possessed during the support phase of the step because an active step landing was not important in determining actual distance. The first of these explanations seems more likely to be true than the second one.

The mean values of the somersaulting angular momentum at the takeoff of each of the three support phases of the triple jump were qualitatively consistent with those found by Hillman (1981) in a study of 15 male subjects. The mean values of the somersaulting angular momentum obtained for the hop, step, and jump in the present study were 9.0, 3.2, and 6.3 kg . m2 . s-', respectively; the corresponding mean values reported by Hillman were 11.5, 1.3, and 6.9 kg . m2 s-I. There are at least three possible explanations for the quantitative differences

between these two sets of mean values: (a) differences in the heights and masses of the subjects in the two studies, (b) differences in the technical levels of the subjects in the two studies, and (c) systematic errors involved in using moment of inertia data about principal axes in 2-D calculations of angular momentum.

The first of these possible explanations cannot be evaluated because no body mass and standing height data were presented in Hillman's study. The second may explain the quantitative difference because the mean value of the actual distances of the subjects in the present study (16.86 m) was significantly greater ( p < .001) than that in Hillman's study (15.86 m). The third may also be a reasonable explanation. Hillman used published data for the moments of inertia about the transverse axes of the body segments. Because these axes are often not perpendicular to the central vertical plane of the athlete's motion-and thus to the "film plane" of a camera set to record motion in sideviewduring the execution of a triple jump, the true values of the moments of inertia about axes perpendicular to the film plane may often have been different from those published moments of inertia about the transverse axes of the segments (Hay et al., 1977). No corrections were made to the segment moments of inertia for out-of-plane motion in Hillman's 2-D angular momentum calculations. It is possible, therefore, that the somersaulting angular momentum obtained in Hillman's study suffered from greater errors than did the same quantity obtained in the present study.

Twisting. None of the mean values of the twisting angular momentum at the takeoff of the last stride, the hop, the step, and the jump was significantly different from zero (Figure 4). This suggested that, on average, elite male triple jumpers tend to minimize the twisting rotation at each takeoff.

Angular Momentum and Actual Distance

No significant relationship was found between any angular momentum component and the actual distance, except the side-somersaulting angular momentum at the takeoff of the step. A significant, nonlinear correlation (r = .86) was obtained

Triple Jump

Normalized side-eomarsaulting ar momentum at the takeoff of the sts%ad,a)

Figure 6 - Relationship between normalized side-somersaulting angular momen- tum at the takeoff of the step and actual distance of triple jump.

between the normalized side-somersaulting angular momentum at the takeoff of the step and actual distance (Figure 6). The regression equation was

where H',,,,, is the normalized side-somersaulting angular momentum at the takeoff of the step. This equation was differentiated with respect to H',(,,,,, equated to zero, and solved for H',,,, to determine the optimum value of H'x,st,p,. It was found that the optimum value of the normalized side-somersaulting angular momentum at the takeoff of the step was -0.0069 s-', and the corresponding absolute side-somersaulting angular momentum was -0.0069 mb hb2 kg . m2 . s-'. In other words, the optimum value of the absolute side-somersaulting angular momentum at the takeoff of the step is 0.0069 m, hb2 kg . m2 . s-I toward the side of the free leg.

Most of the subjects possessed more than the estimated optimum magnitude of side-somersaulting angular momentum toward the side of the free leg at the takeoff of the step (Figure 6). No subject possessed a side-somersaulting angular momentum toward the side of the takeoff leg at the takeoff of the step (Figure 6). These results suggested that, in general, obtaining side-somersaulting angular momentum toward the side of the free leg at the takeoff of the step is not a problem for most elite triple jumpers, but limiting its magnitude may be.

96 Yu and Hay

Changes in Angular Momentum

The means and standard deviations of the changes in the angular momentum during the support phases are presented in Figures 7 through 9. Correlations of each component of angular momentum at each takeoff with the corresponding angular momentum during the preceding flight phase, and with the change in the corresponding angular momentum during the preceding support phase, are presented in Table 5.

Side-Somersaulting. The change in the side-somersaulting angular mo- mentum during the support phase of the step had a significant, positive correlation with the corresponding angular momentum at the takeoff of the step, and the change in this component of angular momentum during the support phase of the jump had a significant, positive correlation with the corresponding component of angular momentum at the takeoff of the jump (Table 5). These results suggested that, at least for the elite triple jumpers in this study, the difference in the side- somersaulting angular momentum at each of the step and jump takeoffs was due primarily to the change in the corresponding angular momentum during the preceding support phase, and was due little if at all to the corresponding angular momentum during the preceding flight phase. This means that what a triple jumper does during each of the support phases of the step and jump is important in determining an appropriate magnitude of side-somersaulting angular momen- tum at the succeeding takeoff.

AH, = -3.75 ( * 1.10 ) AH, = 4.28 ( * 3.26 )

HOP Jump

Figure 7 - Means and standard deviations of the change in the side- somersaulting angular momentum dur- ing the support phases of the hop, step, and jump (kg . mZ . s-'). (Solid arrows indicate means significantly different from zero.)

Triple Jump

Figure 8 - Means and standard devia- tions of the change in the somersaulting angular momentum during the support phases of the hop, step, and jump (kg . mZ . s-I). (Solid arrows indicate means significantly different from zero.)

Somersaulting. A significant positive correlation was found between the change in somersaulting angular momentum during the support phase of the jump and the corresponding angular momentum at the takeoff of the jump (Table 5). This result suggested that, at least for the elite triple jumpers in this study, the difference in the somersaulting angular momentum at the takeoff of the jump was due primarily to the change in the corresponding angular momentum during the preceding support phase.

Twisting. No significant correlation was found between the change in the twisting angular momentum during any support phase and the corresponding angular momentum at the takeoff succeeding the support phase (Table 5). This result suggested that the change in the twisting angular momentum about G during a support phase was not the major determinant of the magnitude of the corresponding angular momentum at the takeoff of the support phase.

Changes in Angular Momentum and Actual Distance

A significant nonlinear correlation (r = .86) with the actual distance was obtained for the change in the normalized side-somersaulting angular momentum during the support phase of the step (AH',,,,,,) (Figure 10). The best regression equation for actual distance as a function of AH',,,,,, was

D, = -2359.73 AH',,,tep,Z + 17.46.

Yo and Hay

Jump

- _ - - _ _ Figure 9 - Means and standard devia- Y' \ tions of the change in the twisting angular

momentum during the support phases of the hop, step, and jump (kg . m2 . s-I).

Table 5 Correlations Among Angular Momenta and Changes in Angular Momenta in the Triple Jump

Triple Jump

Change in normallzed aide-somersaulting angular momentum during the support phase of the step (rad/s)

Figure 10 - Relationship between the change in the normalized side-somersaulting angular momentum during the support phase of the step and the actual distance.

This equation was differentiated with respect to AH',(,,,, equated to zero, and solved for AH',(,,,, to determine the optimum value for AH',(,,,. It was found that the optimum value of AH',(,,,, was zero. This result suggests that triple jumpers should minimize the change in the side-somersaulting angular momentum during the support phase of the step, and that the optimum side-somersaulting angular momentum at the takeoff of the hop should be the same as that at the takeoff of the step, that is, 0.0069 mb hb2 kg . m2 - S-' toward the side of the free leg of the hop. These findings indicate that in terms of side-somersaulting angular momentum, triple jumpers should prepare themselves for the landing at the start of the jump phase as early as the takeoff of the hop.

The change in the side-somersaulting angular momentum during a support phase may depend on (a) the precise placement of the support foot relative to G in the lateral direction at the touchdown of the support phase, (b) the rotation of the trunk about an anteroposterior axis through the joint between vertebrae L4 and L5 during the support phase, (c) the rotation of the whole body about the subtalar joint during the support phase, and (d) the swing of the free limbs. The results of a recent study on the function of the free limbs in the triple jump (Yu, 1993) showed that the swing of the free limbs had little, if any, effect on the change in the side-somersaulting angular momentum during support phases of the triple jump.

A recent study on human gait (MacKinnon & Winter, 1993) demonstrated

100 Yu and Hay

that, in normal walking, control of the side-somersaulting rotation of the body about its anteroposterior axis during the stance phase is achieved primarily by the placement of the support foot relative to G, and that the synergistic roles of the side-somersaulting rotations of the upper body and the whole body during the support phase serve to correct errors in foot placement. The ground reaction force is much greater during the support phase of the step of a triple jump than during the stance phase of normal walking. A small error in the placement of the support foot at touchdown of a support phase in a triple jump may cause a large error in the moment of the ground reaction force about the anteroposterior axis through G and thus a large error in the change in the side-somersaulting angular momentum. This suggests that the placement of the support foot probably plays a critical role in the change in the side-somersaulting angular momentum of the whole body about G during each support phase of the triple jump.

Relationships Between Changes in Angular Momentum Components

No significant relationship was found between the change in a given normalized angular momentum component and the change in either of the other two normal- ized components during the same support phase. There are two possible interpreta- tions for this result. First, each angular momentum component can be adjusted independently during each support phase. In other words, rotation about one axis can be adjusted without affecting the rotations about the other two axes. Second, the effects that adjusting one angular momentum component have on the other two components differ among triple jumpers so that no consistent pattern is evident in a simple correlational analysis.

The first interpretation seems unlikely to be true. The angular momentum of the whole body is the sum of two terms: transfer and local terms for individual segments of the body. The sum of the transfer terms constitutes the major part of the angular momentum of the whole body in triple jumping. The transfer term of the angular momentum of each segment relative to G, H',, can be expressed as

Hg = r, x m, v,,

where r, is the location vector of the center of gravity of segment S relative to G, m, is the mass of segment S, and v, is the velocity of the center of gravity of segment S relative to G (Greenwood, 1988). The side-somersaulting, somer- saulting, and twisting components of the transfer term of H', can be expressed as

where H',, Hty, and Hi are, respectively, side-somersaulting, somersaulting, and twisting components of Hts; x,, y,, and z, are the x, y, and z components of the transfer term of r,; and v,,, v and v,, are x, y, and z components of v,. It can be seen from the preceding equation that a change in one component of the transfer term will inevitably affect at least one of the other two components. Therefore, the second interpretation is more likely to be true; that is, the effects

Triple Jump 101

of an adjustment in the angular momentum about one axis on the angular momen- tum about the other two axes differ among triple jumpers.

Relationships Between Changes in Angular Momentum and Velocity

No significant relationship was found between the change in a normalized angular momentum value and the change in any of the three components of the velocity of the body's center of gravity during the same support phase. There are also at least two possible interpretations of this result. First, the angular momentum of the body can be adjusted without affecting the linear motion of the body, and vice versa. Second, the effects that adjusting angular momentum of the body has on the linear velocity of G differ among triple jumpers so that no consistent pattern is evident in a simple correlational analysis.

It can be seen from the equations for the three components of H', that a change in any component of linear velocity of the center of gravity of S affects two components of Ht,. In other words, the changes in angular momentum and the change in linear velocity are related. Thus, once again, the second interpretation is more likely to be true.

Summary

The results obtained in this study appear to warrant the following conclusions:

The magnitude of the side-somersaulting angular momentum about G pos- sessed by a triple jumper at the takeoff of the step is closely related to the actual distance of the jump, and logically the relationship is causal. The optimum magnitude of this angular momentum appears to be about 0.0069 mb hb2 kg . m2 . s-' toward the side of the free leg. The side-somersaulting angular momentum of the whole body about G needed at the takeoff of the step should be obtained during the support phase of the hop. The change in this angular momentum during the sup- port phase of the step should be minimized. The change in the side-somersaulting angular momentum of the whole body about G during each of the support phases of the step and jump is closely related to the side-somersaulting angular momentum of the body at the corresponding takeoffs. The change in the somersaulting angular momentum of the whole body about G during the support phase of the jump is closely related to the somersaulting angular momentum at the takeoff of the jump. The change in the twisting angular momentum of the whole body about G during a support phase is not closely related to the twisting angular momentum at the corresponding takeoff.

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