ORIGINAL ARTICLE

High-speed video image analysis of ski jumping flight posture

Masahide Murakami • Masato Iwase •

Kazuya Seo • Yuji Ohgi • Reno Koyanagi

� International Sports Engineering Association 2014

Abstract Ski jumping flight posture was analyzed for

achieving large flight distance on the basis of high-speed

video images of the initial 40 m part of 120-m ski jumping

flight. The time variations of the forward leaning angle and

the ski angle of attack were measured from the video

images, and the aerodynamic forces were calculated from

the kinematic data derived from the images. Some corre-

lations were investigated between the initial-speed cor-

rected flight distance and such parameters as the angles of

jumper, the initial transition time and the aerodynamic

force coefficients. The result indicated that small body

angle of attack was a key for large flight distance in the

initial phase of flight because of small drag force, and that

the most distinctive fault of beginners was too large body

angle of attack and ski angle of attack leading to aerody-

namic stall. Too small drag force does not give an optimal

condition for large flight distance because the lift force is

also too small. The ratio of the lift to the drag was larger

than 0.95 for advanced jumpers.

Keywords Ski jumping � Image analysis � Flight

distance � Jumper posture � Aerodynamic force coefficients

1 Introduction

Ski jumping, in particular the flight phase, can be quanti-

tatively treated from the aerodynamics point of view. Some

wind tunnel experiments [1–5], field experiments with the

aid of high-speed image analysis [4, 6] and a simulation

study based on a multisegment dynamic model [7] were

conducted to search for the condition of maximum flight

distance and safety. The wind tunnel test for a V-style

flight was conducted, and a database for the aerodynamic

force coefficients was constructed by Seo et al. [5]. This

was further used in a jumping flight optimization study [8].

The aerodynamic data measured in wind tunnel experi-

ments are the most objective and reliable. But there is not

necessarily accumulation of wind tunnel data enough to

respond to so wide range of variations of parameters and

the data are not available for all parameters. In particular,

they are lacking the cases where the ski-body angle is large

and is rapidly changing in a short time typically found in

the early flight phase from a takeoff to a quasi-steady flight.

Wind tunnel data are not always updated corresponding to

occasional changes in ski length and jump suit due to the

change of rules, and in jumping flight technique from a

parallel flight to a V-style flight. In the image analysis

studies, several angles to characterize the posture of jumper

and the angle of attack were derived, based on which the

optimum jumping posture was inductively discussed. In

relation to this kind of researches, the aerodynamic data

must be referred to in order to give a rational background to

the discussion. However, the optimization could be per-

formed only within the limit of available aerodynamic data

M. Murakami (&)

University of Tsukuba, Tsukuba 305-8573, Japan

e-mail: murakami@kz.tsukuba.ac.jp

M. Iwase

Graduate School of Systems and Information Engineering,

University of Tsukuba, Tsukuba 305-8573, Japan

K. Seo

Faculty of Education, Art and Science, Yamagata University,

Yamagata 990-8560, Japan

Y. Ohgi � R. Koyanagi

Graduate School of Media and Governance, Keio University,

Fujisawa 252-0882, Japan

Sports Eng

DOI 10.1007/s12283-014-0157-z

obtained in wind tunnel tests. For the case of simulation

studies, the situation is almost similar to this with respect to

the necessity of wind tunnel aerodynamic data.

In view of these researches, we attempted to derive the

aerodynamic forces exerted on a jumper on the basis of the

image analysis of high-speed video image of real jumping

flights [9, 10] to be complementary with the existing

researches. We expected that the aerodynamic force data

should have been obtained for the parameters under real

flight condition as the result of this study. We furthermore

considered that aerodynamic data even in the cases where

the parameters were rapidly changing on a large scale,

including aerodynamically unsteady situations, could be

acquired in the study. As a result, coaches on site could

more rigorously give the jumpers advice while the video

images of their flights and the corresponding aerodynamic

data are simultaneously shown to them. More scientific and

reasonable discussion could also be developed on the

correlations between the flight distance and such parame-

ters as angles characterizing the flight posture by showing

the aerodynamic data. It was one of the objectives of the

present study that the image analysis procedure was further

improved for higher accuracy by executing detailed on-site

survey of the jumping field, Hakuba Jumping Stadium, and

by applying a more accurate image correction based on the

survey data than our previous analyses [9, 10]. Further-

more, large amount of image analysis data of jumping

flight have been accumulated so that a statistical data

analysis can be made. As the second objective of this study

some correlations were investigated for the optimal flight

posture leading to large flight distance between the flight

distance and such key parameters as the angles of jumper,

the initial transition time to a quasi-steady flight and the

aerodynamic force coefficients. Furthermore, the jumping

performances are compared between advanced and begin-

ner jumpers to extract typical faults of beginner jumpers.

2 Methods

The flights of large-hill class jumping were recorded with a

high-speed video camera (the main camera) (Photron

FASTCAM 1024PCI) set on the top of the coach tower of

the normal-hill jumping field that is built almost in parallel

to the large-hill jumping field 50 m away. The location of

the camera is about 15 m downward from the takeoff point,

aside the deck of the normal-hill landing slope, and 65 m

away from the center line of the large-hill landing slope.

The field of view of the camera with a wide-angle lens with

a focal length of 28 mm covers the upper 40 m range of the

large-hill landing slope, which roughly corresponds to an

initial flight portion for 2 s. The video camera images were

taken at the rate of 250 frames/s. An auxiliary video

camera (Photron FASTCAM 1024PCI) was also used for

the purpose of calibration of the angle data of a jumper.

This camera was set on the top of the large-hill coach tower

20 m aside the deck of the large-hill landing slope. The

jumper image taken with this camera is more than six times

larger than that of the main camera so that the forward

leaning angle can be measured with an accuracy sufficient

enough to calibrate the angle data of the images taken with

the main camera. The subjects in the field tests consisted of

six jumpers of Japanese Championship level including a

number of Japanese Olympic Team jumpers, 10 jumpers of

university and high school students and a number of World

Cup higher rank jumpers. Total numbers of flights that

were recorded are about 100. However, for some flights,

the data of the takeoff speed and/or the flight distance were

lacking, and thus these data were not analyzed. As no wind

data were available during the field tests, the effect of wind

was ignored. As a matter of fact, wind was weak to mod-

erate during the tests in the mid summers of 2008 and

2009. The coordinate system, and the centre of gravity

(CG) and the angles characterizing the posture of a jumper

are shown in Fig. 1; a is the ski angle of attack, b the flight

angle, c the body angle of attack, h the jumper forward

leaning angle, U the flight velocity. The angle h is defined

as the angle between a ski and legs. The bend angle of

waist could not be measured owing to smallness of a

jumper in the high-speed video images. Among these

angles, h is readily measured from a single instantaneous

frame of the video record, while the other angles must be

calculated after an image analysis for several sequential

images of a jumper. It is because they are the angles that

are defined with respect to the flight velocity vector U of

the centre of mass of a jumper. The flight angle b is the

angle of U measured from the horizontal line, and a and care angles measured from the vector U. In order to derive

the flight vector, and consequently the flight speed and the

acceleration, the time sequence of the CG of a jumper

under flight is pursued in the video images. In this study,

Fig. 1 Coordinate system and the definition of angles characterizing

the posture of a jumper

M. Murakami et al.

the CG value was calculated as a kind of weighted average

of the positions of jumper waist, head and toe. The specific

weights of body segments of a jumper were calculated by

referring to the paper [11]. The image tracking for these

selected positions of a jumper was mostly accomplished

automatically with the aid of image analysis software

(TEMA lite), but partly with some manual assist for cases

where the target positions were hard to be distinguished

because they seemed as if they were blended into the

background. The mass of a jumper was assumed to be

70 kg including ski. The coordinate origin is the takeoff

point, and the x- and y-axes are taken horizontally and

vertically downward, respectively. The z-axis is taken to

the horizontal depth direction. We used a left-handed

coordinate system. The aerodynamic forces, the lift L that

is perpendicular to U and the drag D that is in the opposite

direction to U, are computed in terms of the accelerations

in the x and y directions, ax and ay, and the flight angle bafter the effect of gravity g is subtracted.

L ¼ m ax sin b� ay � g� �

cos b� �

ð1Þ

D ¼ �m ax cos bþ ay � g� �

sin b� �

; ð2Þ

where m is the jumper mass. The lift area SL and the drag

area SD are conventionally used instead of the lift coeffi-

cient and the drag coefficient because the projected area of

a jumper largely changes during a flight [1–3].

SL ¼ 2L=ðqU2Þ ð3Þ

SD ¼ 2D=ðqU2Þ; ð4Þ

where q is the air density, and U is the air speed or the

jumper speed.

A single fixed video camera with a wide-angle lens was

used instead of adopting a three-dimensional camera system.

Some image distortion that is unavoidable in our simple

camera system must be corrected before quantitative image

analysis. For this purpose, site survey had been executed for

24 points on the centre line of the large-hill landing slope by

using a laser-surveying instrument. For the purpose of cali-

bration of position information we took the images of staffs

with four markers stood right on the 24 points surveyed on the

centre line of the landing slope with the video camera situated

at the same location as the jump image recording. The four

markers were mounted in 1 m intervals in the vertical

direction on the staffs. Thus, the positions of total 96 spatial

points were identified in the air above the landing slope on the

video image, forming a pixel map. Based on these points in

the images, of which coordinate values are known, we con-

structed an piecewise interpolation formula system for

conversion from the pixel coordinate (X, Y) to the physical

x–y coordinate. They are polynomials of 2nd order of X and Y

for the y-coordinate, and of 4th order for the x-coordinate.

This calibration procedure was done every day prior to the

video shoot in order to deal with minute changes of the

camera position and camera angle. The maximum values of

overall errors in the positions in an image are estimated to be

37 cm in the x-direction and 5 cm in the y-direction. The

breakdown of these values is as follows: a spatial resolution is

40 mm (=40 m/1,024 pixels); the maximum deviation of a

jumper in flight from the center line of the landing slope is

estimated to be 1 m in the z-direction, which is an interpo-

lation from the deviation at the landing, within ;2 m. This

1 m deviation may cause an error in the x-position 30 cm; the

positioning accuracy in the interpolation procedure is about

2.4 cm in the x-direction and 0.8 cm in the y-direction,

respectively. For the data analysis, the first and the second

order time derivatives of the CG data were computed for the

velocity and acceleration, from which the aerodynamic forces

are derived. The first order numerical differentiation scheme

of the 3rd accuracy for the sequential data of the CG coor-

dinate fi is adopted for the calculation of the velocity.

df

dt¼ �fþ2 þ 8fþ1 � 8f�1 þ f�2

12Dtð5Þ

Here, f? 1 is the value f at the instant of i ? 1 and Dt is the

time interval of the sequential data. It is, however, obvious that

the scattering of the time-series data of the CG is very large.

Thus, for the calculation of the velocity the time-series of the

first order time derivative data were, first, low-pass filtered

with a cut-off frequency of 6 Hz, and then the weighted least

square regression with a higher order polynomial are piece-

wise applied for smoothing. For the calculation of the accel-

eration, the smoothed velocity data were numerically

differentiated, low-pass filtered and finally smoothed.

3 Results and discussion

3.1 Derivations of the aerodynamic force coefficients

and the angles characterizing a jumper posture

Typical examples of the time variations of SL and the ski

angle of attack a are shown in Fig. 2a for two ski jumpers,

a Japanese A-class jumper (flight distance of 131 m) as an

advanced jumper and a high school student jumper (102 m)

as a beginner jumper. The takeoff speed was almost same

for the two cases. It is seen that SL is larger for the

advanced-class jumper than that for the high school student

jumper throughout the record. It is interesting to note that

SL begins to decrease a little before 1 s for the advanced-

class jumper and around 0.7 s for the high school student

jumper. This must be a result of aerodynamic stall. The

stall started earlier for the high school student jumper

because of larger a as a result of excessive pitching motion

of his ski (a). As seen in the previous wind tunnel mea-

surement result [5], the stall angle is between 35� and 40�,

Image analysis of ski jumping flight posture

though it is hard to exactly define the stall angle owing to

the three-dimensional nature of a jumper body. The

symptom of stall, which is the decrease in the rate of

increase of the lift and the increase in the rate of increase of

the drag, begins to appear around 25 degree of the ski angle

of attack a for the high school student jumper. The results

of the body angle of attack c and the body forward leaning

angle h are shown in Fig. 2b and c respectively. For these

figures, the data were shown also for both the advanced-

class and the high school student jumpers. It is seen in

Fig. 2b that c is larger for the high school jumper to result

in larger drag due to his posture where it nearly stands up

against airflow. Figure 2c shows that h values are not so

different between the two jumps, though the flight dis-

tances are greatly different. However, the time tf at which a

flight entered a quasi-steady flight phase, which is the

phase of metastable flight without change of a large-scale

posture or angles, as can be simply defined in Fig. 2c, is

different for the two flights, 0.4 s for the advanced jumper

and 0.8 s for the beginner jumper. The jumping of

advanced jumper enters the quasi-steady flight phase ear-

lier than the high school jumper. In the early flight phase

before tf, the body nearly stands up against airflow resulting

in larger c causing larger drag. Therefore a quick transition

to a quasi-steady flight phase with small c is desirable for a

successful jumping flight. In a beginner’s jump the angle ccannot be made small even in the quasi-steady phase.

3.2 Correlations between the flight distance

and the jumper posture parameters

3.2.1 Calculation of the normalized flight distance

for initial speed compensation

It is understood that the aerodynamic force coefficients

vary as functions of jumper posture characterized by the

angles, a, c and h, which, in turn, govern the flight distance.

We investigate the correlations among them. One of the

most important measures for the evaluation of jumping

performance is the flight distance Dj, which is, however,

directly influenced by the takeoff speed U0. The magnitude

of U0 depends on the start point and the crouching posture

in the in-run and waxing, which were individually out of

control in the present field tests. Therefore, the flight dis-

tance must be compensated for the effect of U0 prior to the

detailed data analysis. For this purpose, the reference flight

distance, Dv, is calculated as a function of U0, which is the

flight distance without any effects of aerodynamic forces,

which is equivalent to the free flight distance of a point

mass in vacuum. The normalized flight distance, Dr, is

defined by Dr = Dj/Dv, which is considered to be almost

free from the effect of the takeoff speed U0. The specifi-

cation of the virtual jumping field for the calculation of Dv

is: 11 degree of the downward angle at the takeoff point,

the height of the takeoff point = 4 m, the landing slope

with an angle of 37.5� immediately follows the takeoff

point without a deck. It should be noted that the present Dv

value is slightly larger than the possible free flight distance

in Hakuba Jumping Stadium due to neglect of the deck part

of the landing slope in the model calculation. The calcu-

lation result is that Dv is roughly in proportion to U0 but the

inclination is only slightly larger at large U0: for example,

83 m for U0 = 23 m/s (=82.8 km/h), 103.5 m for

U0 = 26 m/s (=93.6 km/h).

3.2.2 Correlation results: a - Dr, c - Dr, h - Dr

and tf - Dr correlations

The results of a - Dr, c - Dr, h - Dr and tf - Dr corre-

lations are shown respectively in Fig. 3a–d, and the data of

the coefficient of determination for the linear regressions,

R2, are shown in Table 1. The values of a, c, h and tf were

Fig. 2 Time variations of the lift area SL and some angles of a jumper

posture for two typical cases (an advanced jumper and a beginner

jumper). Data are numerically smoothed out without applying

numerical low-pass filtering: a for the lift area SL and the ski angle

of attack a, b for the body angle of attack c, and c for the body

forward leaning angle h

M. Murakami et al.

measured at 1.4 s after takeoff, when every flight has been

about to enter a quasi-steady flight phase. The result sug-

gests that the angles a, c and h should be smaller for larger

flight distance. It is because smaller a, c and h lead to

smaller aerodynamic drag. It should be noted that the

relation c = h ? a holds as seen from the definition shown

in Fig. 1. This relation indicates that only two among c, hand a are independent. A typical jumping posture of

beginners is that with small h but large a, that is the posture

with small ski angle of attack but an almost upright body

position. In this case, c inevitably becomes large, and thus

Dr is not large even if a is small. In the case of large a and cin the early jumping flight phase, a jumper flies while

possibly undergoing the effect of aerodynamic stall, char-

acterized by large drag force. The angle h that is easily

quantified even from a still picture tends to be regarded as

being important. However, it is the parameters a and cdefined with respect to the vector U of the airflow around a

jumper, or equivalently of the jumper flight path, that are

decisive of the aerodynamic force exerted on a jumper. The

flight posture having small a and c, and thus small h, results

in longer flight distance. The posture having small h, which

is sometimes considered as a target of the skill that

beginners should acquire, is effective only if a is small. The

result of Fig. 3d indicates that quick completion of initial

movement subsequent to takeoff is important for larger Dj.

Immediately after takeoff, jumpers nearly stand by upright

posture against airflow resulting in larger h causing larger

drag. Therefore, quick transition to a quasi-steady flight is

desirable for a long flight distance.

It is interesting to compare the data between the

advanced and beginner jumpers. It is evident in Fig. 3 that

every parameter, a, c, h and tf for advanced jumpers, is, on

average, smaller than for beginners. The flight posture with

small a, c, h and tf assures the condition of a flight posture

with small drag. In particular for the parameters, c and h, of

advanced jumpers the data points are rather concentrated in

narrower and more relevant regime for larger Dr. Thus, the

correlations of these parameters with Dr are quite low.

Smaller data scattering of these parameters for advanced

Fig. 3 Correlation results. a for

a - Dr correlation, b for

c - Dr correlation, c for h - Dr

correlation and d for tf - Dr

correlation. The solid line is a

linear least square regression

line for all the data

Table 1 Coefficient of

determination for linear

regression results

Regression a - Dr c - Dr H - Dr tf - Dr f - c f - h f - Dr

Figure 3a 3b 3c 3d 5a 5b 5c

Coefficient of determination R2 0.5482 0.8376 0.7489 0.4691 0.7436 0.7092 0.7304

Image analysis of ski jumping flight posture

jumpers suggests that the flights are rather reproducible,

and thus, they can perform two good jumps in high prob-

ability in a jumping competition. Data scattering is possi-

bly attributed to the contribution from some uncontrolled

factors, such as wind, the direction and timing of takeoff,

unnecessary movement of arms, etc. It may be considered

that the jumping flight of beginners is far more affected by

such uncertain factors. The correlations of parameters, cand h, with Dr for beginners are higher than those of

advanced jumpers. The regression lines seem to indicate

the path along which the jump skill of beginners is

improved. Beginners may ultimately reach the data area of

advanced jumpers that exists on the natural extension of

each regression line. On the other hand, the data distribu-

tions of a and tf are rather multi-layered in data group

structures between advanced and beginner jumpers. The

data of advanced jumpers are always on the higher side of

Dr than those of beginners at the same a or tf value. That is,

if beginners are continuing training of the beginner level

aimlessly, they get used only to a skillful beginner at most.

To reach the level of advanced jumpers, making a highly

motivated training such as the leap of something as

described below appears to be necessary. It is suggested

that it is insufficient for large Dj for beginners to have

mastered to only make a or tf small. It is quite true that

acquisition of the skill for a jumping posture with small

body angle of attack c is a key for beginner jumpers to

progress to a higher step. However, two points should be

paid attention to for their training. The first is that fear is

followed in order to take a jumping posture with small cbecause the transition from takeoff to this flight posture

requires throwing the trunk forward. The second is that it is

necessary to understand that the nose-down moment

(pitching moment in the direction where the head is going

down) must be given to the trunk as an initial condition of

flight dynamics at the instant of takeoff to achieve a

jumping posture with small c, because the nose-down

moment produced aerodynamically in the air during a

jumping flight is quite small.

3.2.3 Effect of the angle between CG-boots line and ski

at takeoff

As pointed out in the last section, for small c, which is one

of the keys for large Dj, nose-down moment should be

given to jumper’s trunk in the takeoff motion. However,

the direct measurement of the pitching moment from the

high-speed video images could not be performed because

the nose-down rotation of a jumper trunk immediately

before the takeoff point is not visible because a jumper was

screened by the opaque safety board. Instead, we selected

the angle between the ski (or the slope line at the takeoff

point) and the CG-boots line of a jumper immediately after

takeoff, f, as a plausible measure of the nose-down

moment at the instant of takeoff. The definition of f is

illustrated in Fig. 4. The angle f was measured from the

images taken with the auxiliary camera. The correlations

are shown in Fig. 5a–c respectively for f - c, f - h and

f - Dr correlations, and the data of the coefficient of

determination, R2, for the linear regressions are shown in

Table 1. It is seen that there are strong correlations among

them. This result means smaller f is a condition for sub-

sequent smaller c. This may be the reason why f is strongly

correlated with Dr. This may be regarded as the evidence

that f plays a role as the measure of the nose-down moment

at the instant of takeoff. The data of advanced jumpers

exist in a narrower regime on the extension of corre-

sponding regression lines of beginners.

3.3 Consideration on the effects of aerodynamic forces

The correlations between the drag area SD and Dr, and the

lift area SL and Dr, shown in Fig. 6a and b respectively, are

examined to investigate the aerodynamic force effects. The

detailed statistical data of the linear regressions are pre-

sented in Table 2. It is seen that there are a number of

definite differences in these two functional relations

between the data of the advanced and the beginner jump-

ers. For the data of the advanced jumpers, there are almost

no correlations in both SD - Dr and SL - Dr relations in

the following two different aspects: As for SD, the data

points are distributed in a narrow region of the SD - Dr

plane. However for SL, the data show rather wide distri-

bution in SL, but Dr hardly depends on SL, and the data

points for the advanced jumpers are on the larger side of Dr

than those of the beginner jumpers. The data point distri-

bution in the SD - Dr diagram in Fig. 6a indicates that the

data for the advanced jumpers are around the extension line

Fig. 4 Definition of the angle f between the ski and the CG-boots

line

M. Murakami et al.

of the regression for the beginners. This seems to suggest

that beginners can arrive at the level of advanced jumpers,

if ordinary efforts are continued. It is, on the other hand,

seen in the SL - Dr diagram that the data of the advanced

jumpers are located above the data point row of the

beginner jumpers. This consequence may suggest that

beginners cannot arrive at the level of advanced jumpers

after just another continuation of training. Some special

device and effort for higher skill are required for beginners

to grow up to be advanced jumpers. Mastery of a jumping

posture with small body angle of attack c for beginner

jumpers as described in the Sect. 3.2.2 may be in line with

this. It is fair to conclude that there may be weak corre-

lations in SD - Dr and SL - Dr relations if the data of the

advanced and the beginner jumpers are taken into account

all together. This result seems to suggest that the posture

with the smaller SD and SL makes Dr the larger. If this

supposition were true, a crouching jumping posture would

be the best flight posture as it fulfills the condition of small

SD and small SL simultaneously. The requirement of small

SD is quite reasonable for larger Dr. But it does not seem a

guiding principle for larger Dr and rather seems even

unreasonable that the smaller SL is the better. This seem-

ingly unreasonable consequence originates from the vari-

ations of SD and SL as a function of ski angle of attack ashown in Fig. 7, where small SD inevitably makes SL small

except in the aerodynamic stall region. It is, on the other

hand, a plain fact of aerodynamics that SD should be small

and SL should be large for larger flight distance Dj. The best

performance would certainly be achieved at some inter-

mediate values of SD and SL.

It is, then, investigated how much values of SD and SL

should be for the maximum jumping flight distance. It is a

well-known fact in aerodynamics that the maximum L/D,

or equivalently the maximum SL/SD, results in the maxi-

mum flight distance as long as it is in a steady flight. In

Fig. 8 the correlation between SL/SD and Dr is shown, and

the statistical data of linear regression are shown in

Table 2. This result shows the correlation is significant

only for the beginners, where a weakly positive correlation

is recognized, while for the advanced jumpers there is

Fig. 5 Correlation results. a For f - c correlation, b for f - h correlation and c for f - Dr correlation. The solid line is a linear least square

regression line for all the data

Fig. 6 Correlation results. a for SD - Dr correlation and b for SL -

Dr correlation. The solid line is a linear least square regression line for

all the data

Image analysis of ski jumping flight posture

almost no correlation between them, and merely SL/SD

value is larger than 0.95. The result, in fact, indicates that

the maximum SL/SD does not always give the result of

maximum jumping flight distance. Of course, all the

jumping flights examined here being in the early flight

phase were never in a steady state. For further consider-

ation, the present data are plotted on a SD - SL polar

diagram as given in Fig. 9, where the data are classified

according to the magnitude of Dr. In this figure the data

presented in Fig. 7 are also plotted in the form of a polar

curve just as an example for the case of a jumping posture

of straight trunk without a bend of the waist with h = 10�

and a ski opening angle k of 25�, though this jumping

posture is not a typical one in the early flight phase. It is

seen in the result that the value of SD never exceeds 0.6 for

the advanced jumpers, as also evident from the result of

Fig. 6a. As a matter of fact, this result indicates one of the

decisive differences between the advanced and the begin-

ner jumpers as follows. For the advanced jumpers, SL/SD is

larger than 0.95, though it is not a definite guiding principle

for larger flight distance that the larger SL/SD is the better.

In other words, small SD is desirable, but too small SD is

not recommended as this condition makes SL also too

small. Some lift force is certainly necessary, which, toge-

ther with the condition that SD is small, is reflected in the

condition of SL/SD [ 0.95 for satisfactory jumping flight

distance. The straight trunk posture with too deep forward

leaning is not so good for longer jumping flight distance

because of small SL though SD is small. Furthermore, this

Table 2 Linear regression statistics on aerodynamic data

SD - Dr SL - Dr SL/SD - Dr

Figure 6a 6b 6c

All data

Coefficient of determination R2 0.5353 0.1962 0.3348

p level 2.16E-5 2.34E-2 1.96E-3

H0 (5 %) rejected Yesa Yes Yes

Advanced jumpers

Coefficient of determination R2 0.0121 5.88E-3 2.32E-3

p level 0.733 0.813 0.882

H0 (5 %) rejected Nob No No

Beginner jumpers

Coefficient of determination R2 0.5628 0.232 0.524

p level 3.14E-3 8.10E-2 3.40E-3

H0 (5 %) rejected Yes No Yes

Yesa: H0 (Dr is independent of SD) is rejected

Nob: H0 (Dr is independent of SD) is not rejected

Fig. 7 An example of the wind tunnel measurement data of SD and SL

plotted against a [5]. The forward leaning angle h is 10� and the ski

opening angle k is 25�

Fig. 8 SL/SD - Dr correlation result. The solid line is a linear least

square regression line for all the data

Fig. 9 Flight experimental data plotted on a SL - SD polar diagram.

The data are classified according to the magnitude of Dr. The straight

lines are those of constant SL/SD, and the solid curve is the polar curve

of the data presented in Fig. 7

M. Murakami et al.

posture is disadvantageous in a quasi-steady flight phase

due to small SL because the reattachment of separated flow

is hard to occur. It should be noted that all the flight test

data of Fig. 9 are aerodynamically inferior to the wind

tunnel data of Fig. 7 in the sense that the flight experi-

mental data points are all below the polar curve of the data

of Fig. 7. It is primarily because the forward leaning angles

h of the present flight data are larger than 10� and the data

were taken in the early flight phase before reaching a quasi-

steady flight phase.

4 Conclusions

The conclusions of this study are summarized as follows:

• The body angle of attack c, the ski angle of attack a, the

body forward leaning angle h, and the transient time to

a quasi-steady flight tf should all be small for larger

jumping flight distance, which is a necessary condition

for small drag force, in particular in the early flight

phase. Among these parameters, c is the most

influential.

• The distribution of these measurement data is clearly

distinguished between advanced jumpers and beginner

jumpers. The data of advanced jumpers are found in the

small value ranges of a, c, h and tf, and furthermore the

data scattering of advanced jumpers is smaller than

those of beginners. Typical fault of beginner jumpers is

too large c and a resulting in too large drag caused by

aerodynamic stall.

• In the early flight phase, a flight posture with small drag

is a key for long flight distance with the condition that

SL/SD is larger than 1.0.

Acknowledgments This research was financially supported by the

Grant-in-Aid for Scientific Research from the Japan Society for the

Promotion of Science (19650167).

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