A Ab bs st tr ra ac ct t
Using a simple physical model of ski jumping the
current study investigates the relationship between
a ski jumper’s position angle during the flight phase
and the jump length as well as other physics vari-
ables. The physical parameters of the jump (veloci-
ty, current drag, jump length) were calculated taking
into account height of the jumping hill, inclination of
the outrun, and position of the system jumper-skis
during the flight phase. The analysis of the jump
curve was made on the basis of the angle between
the flight trajectory of the common centre of gravity
and the x-axis (ϕ). For the analysis of the jumper’s
position during the flight phase the angle between
the upper part of the body and the flight trajectory of
the common centre of gravity of the body (ϑ) was
chosen. In the computer simulation of the jump
length two variants of the course of the angle ϑ
were used. The flight velocity was very similar in
both variants. Immediately after take-off, the result-
ant velocity v(t) and its horizontal component v
x
(t)
slightly decreased. After that the resultant velocity
increased constantly due to the constant increase
in the vertical component v
y
(t). Due to the current
drag the horizontal flight component v
x
(t)
decreased constantly during the entire flight phase.
The current drag F
D
(t) and lift components F
L
(t)
rose to a greater extent in the changeable ski
jumper’s position angle ϑ. 
Key words: ski jumping, physical model, computer
simulation, position angle
1 1  F Fa ac cu ul lt ty y  o of f  N Na at tu ur ra al l  S Sc ci ie en nc ce es s, ,  M Ma at th he em ma at ti ic cs s  a an nd d    
E Ed du uc ca at ti io on n, ,  U Un ni iv ve er rs si it ty y  o of f  S Sp pl li it t, ,  C Cr ro oa at ti ia a
2 2  F Fa ac cu ul lt ty y  o of f  S Sp po or rt t, ,  U Un ni iv ve er rs si it ty y  o of f  L Lj ju ub bl lj ja an na a, ,  S Sl lo ov ve en ni ia a
* Corresponding author: Faculty of Natural Sciences,
Mathematics and Education, University of Split, Nikole
T esle 12, 21000 Split, Croatia
T el.: +385 (0) 21 385 009, fax: +385 (0) 21 385 431  
E-mail: mile@pmfst.hr
I Iz zv vl le eč če ek k  
Cilj raziskave je bil z enostavnim fizikalnim mode-
lom smučarskega skoka ugotoviti povezanost med
skakalčevim kotom med fazo leta in dolžino skoka
ter nekaterimi drugimi fizikalnimi parametri. Fizikalni
parametri skoka (hitrost, sila upora, dolžina skoka)
so bili izračunani z upoštevanjem višine skakalnice,
nagiba pristajalnega dela skakalnice in pozicije sis-
temska skakalec - smuči med fazo leta. Analiza
krivulje skoka je bila narejena na osnovi kota med
linijo leta glede na skupno težišče in x-osjo (ϕ). Za
analizo pozicije skakalca med letom je bil izbran kot
med zgornjim delom telesa in linijo leta glede na
skupno težišče telesa (ϑ). V računalniški simulaciji
dolžine skoka smo uporabili dve varianti kota ϑ.
Hitrost leta je bila je bila v obeh različicah podobna.
Takoj po vzletu sta se hitrost v(t) in njena horizontal-
na komponenta v
x
(t) nekoliko zmanjšali. T emu sledi
stalno naraščanje hitrosti, ki gre na račun stalnega
povečevanja njene vertikalne komponente v
y
(t).
Horizontalna komponenta leta v
x
(t) je zaradi sile
upora skozi celoten let stalno upadala. Komponenti
sil upora F
D
(t) in dviga F
L
(t) s spreminjanjem kota
pozicije skakalca ϑ naraščata. 
Ključne besede: smučarski skoki, fizikalen model,
računalniška simulacija, kot skakalca
Faculty of Sport, University of Ljubljana, ISSN 1318-2269
7 70 0
R RE EL LA AT TI IO ON NS SH HI IP P    B BE ET TW WE EE EN N    J JU UM MP P    L LE EN NG GT TH H    A AN ND D    T TH HE E
P PO OS SI IT TI IO ON N    A AN NG GL LE E    I IN N    S SK KI I    J JU UM MP PI IN NG G
R RA AZ ZM ME ER RJ JE E    M ME ED D    D DO OL LŽ ŽI IN NO O    S SK KO OK KA A    I IN N    K KO OT TO OM M
S SK KA AK KA AL LC CA A    P PR RI I    S SM MU UČ ČA AR RS SK KI IH H    S SK KO OK KI IH H
M Mi il le e    D Dž Ž e el la al li ij ja a  
1 1
* *
N Ni ik ko ol la a    R Ra au us sa av vl lj je ev vi ić ć  
1 1
B Bo oj ja an n    J Jo oš št t  
2 2
Kinesiologia Slovenica, 9, 1, 70-79  (2003)

I IN NT TR RO OD DU UC CT TI IO ON N
Ski jumping is a very complex skill involving several phases such as: inrun, take-off,
flight and landing. The main purpose of the inrun and take-off phases is to give the
jumper optimal values of the velocity and to produce a favourable body position angle.
The main purpose of the flight phase is to achieve the optimal flight velocity, maximise
the horizontal component and minimise the vertical component of the flight velocity
(Vaverka, 1987). In the last several decades, biomechanics of skiing has demonstrated
considerable growth, evolving from an exercise in the filming of human movement to an
applied science with a vast array of measurements and modelling techniques. The ear-
liest analytical model of ski jumping was made by Straumann (1927), who used a
model for the wind tunnel measurements of drag forces. Since then, many approaches
to studying the physics of ski jumping have been adopted, but only one model of ski
jumping that takes into account the changes in the athlete’s position during the flight
phase has been presented in literature (Jin, Shimizu, Watanuki, & Kobayashi, 1995;
Muller, Platzer, & Schmolzer, 1996; Schmolzer, & Muller, 2002). There are only a few reli-
able studies of the effects of the athlete’s position angle during the flight phase (Muller,
Platzer, & Schmolzer, 1996; Schmolzer & Muller, 2002).
When a ski jumper moves through the air, as shown in Figure 1, he is exposed to the grav-
itational force and the dynamic fluid force in the air. The magnitude of the dynamic fluid
force is proportional to the air density ρ, the surface area A of the jumper in the air, and
the square of the ski jumper’s velocity v v  relative to air (Halliday, Resnick, & Walker, 2001).
(1)
For practical reasons, the dynamic fluid force that results from motion in the air is com-
monly divided into two components: the drag force and the lift force. The drag force F
D
is the component of the resultant dynamic fluid force that acts in opposition to the rela-
tive motion of the jumper with respect to air. This force tends to decrease the relative
velocity. The drag force is produced by two different means: surface drag and form
drag. Surface drag is the sum of the friction forces acting between air molecules and
the surface of the jumper. Form drag is the sum of the impact forces resulting from the
collisions between the air and the jumper. The lift force F
L
is the dynamic fluid force
component that acts perpendicularly to the relative motion of the jumper, and causes a
change in the direction of the relative motion of the jumper.
If during the flight phase an athlete’s body is oriented at the angle ϑ towards the direc-
tion of motion, the magnitude of the drag force can be expressed as follows
(2)
where A A
⊥
and A A  
||
are surfaces of the frontal and the longitudinal areas of the jumper. C C
D D
is the coefficient of drag. In addition, the magnitude of the lift force F
L
is
(3)
Relationship between jump length and the position angle in ski jumping Kinesiologia Slovenica, 9, 1, 70-79 (2003)
7 71 1

where C C
L L
is the coefficient of lift. The horizontal component F
x
of the resultant force that
acts on the ski jumper during the flight phase is
(4)
where ϕ is the current angle of the velocity with respect to the x-axis (see Figure 1). The
vertical component F
y
of the resultant force is
(5)
where m is the mass of the ski jumper, and g = 9.8 ms
-2
is the acceleration caused by the
Earth’s gravitational force.
On the basis of vertical and horizontal components F
x
and F
y
of the resultant force and
the inrun and take-off velocities, it is possible to calculate the components of the
instantaneous jumper’s velocities (v
x
,v
y
) and positions (x, y) by using the following dif-
ferential equations:
(6)
The resultant force influences the athlete, his velocity, position and the position angle
change during the flight phase.
M ME ET TH HO OD D
Using a simple physical model of ski jumping the current study investigates the rela-
tionship between a ski jumper’s position angle during the flight phase and the jump
length as well as other physics variables (Figure 1).
Relationship between jump length and the position angle in ski jumping Kinesiologia Slovenica, 9, 1, 70-79 (2003)
7 72 2

F Fi ig gu ur re e  1 1: :  T Th he e  g gr ra av vi it ta at ti io on na al l  f fo or rc ce e  F F
g g
a an nd d  t th he e  c co om mp po on ne en nt ts s  o of f  t th he e  d dy yn na am mi ic c  f fl lu ui id d  f fo or rc ce e  ( (d dr ra ag g  F F
D D
a an nd d  l li if ft t  F F
L L
f fo or rc ce es s) )  a ac ct ti in ng g
o on n  a a  s sk ki i  j ju um mp pe er r. .  v v  i is s  t th he e  s sk ki i  j ju um mp pe er r’ ’s s  v ve el lo oc ci it ty y  r re el la at ti iv ve e  t to o  a ai ir r, ,  ϕ ϕ i is s  t th he e  a an ng gl le e  o of f  t th he e  v ve el lo oc ci it ty y  w wi it th h  r re es sp pe ec ct t  t to o  t th he e  x x- -a ax xi is s  a an nd d
ϑ ϑ  i is s  t th he e  a an ng gl le e  b be et tw we ee en n  t th he e  u up pp pe er r  p pa ar rt t  o of f  t th he e  b bo od dy y  a an nd d  t th he e  f fl li ig gh ht t  t tr ra aj je ec ct to or ry y. .
In the computer simulation series A
⊥
= 0.15 m
2
, A
||
= 1.2 m
2
have been taken as the sur-
faces of the frontal and longitudinal areas of the ski jumper, and the equipment m = 65
kg has been taken as his mass. According to the International Civil Aviation
Organisation (Dubs, 1987) the air density ρ in normal atmosphere conditions equals
1.225 kgm
-3
at mean sea level, 1. 15 kgm
-3
at an elevation of 650 m, and 1.0 kgm
-3
at 2000
m. Since jumping hills are located at the elevation of about 2000 m, the air density of ρ =
1.0 kgm
-3
has been applied. For the approach and take-off velocities (due to the ath-
lete’s jumping force) ν
x0
= 27 ms
-1
and ν
y0
= 2.5 ms
-1
have been used (Virmavirta,
Kiveskas, & Komi, 2001). The profile of a typical K = 120 m jumping hill with a maximum
gradient of α = 37 .5
o
has been used (H(K) = 60 m, N(K) = 103.9 m). For the drag and lift
coefficients C
D
= 1 and C
L
= 1 have been applied as in [3]. A fixed time step integration
of ∆ t = 0.001 s has been used.
Figure 2 shows the profile hill used in the simulation and the flight trajectories. The solid
line corresponds to the simulation with a changeable jumper’s position angle (the
longest simulated jump), and the dashed line correspond to the fixed one, ϑ = 35
0
. For
the longest jump (133.4 m) the flight time obtained by simulation is 4.72 s, and for the
flight with a fixed position angle of ϑ = 35
0 0
the simulated flight time is 4.51 s.
Relationship between jump length and the position angle in ski jumping Kinesiologia Slovenica, 9, 1, 70-79 (2003)
7 73 3

F Fi ig gu ur re e  2 2: :  T Th he e  j ju um mp pe er r’ ’s s  p po os si it ti io on ns s  d du ur ri in ng g  t th he e  f fl li ig gh ht t  p ph ha as se e. .  T Th he e  s so ol li id d  l li in ne e  c co or rr re es sp po on nd ds s  t to o  t th he e  s si im mu ul la at ti io on n  w wi it th h  a a  c ch ha an ng ge e- -
a ab bl le e  j ju um mp pe er r’ ’s s  p po os si it ti io on n  a an ng gl le e  ( (t th he e  l lo on ng ge es st t  s si im mu ul la at te ed d  j ju um mp p) ), ,  a an nd d  t th he e  d da as sh he ed d  l li in ne e  c co or rr re es sp po on nd ds s  t to o  t th he e  f fi ix xe ed d  o on ne e, ,  ϑ ϑ = =  3 35 5
o o
. .
Figure 3 presents the jump length d calculated for the variable jumper’s position angle ϑ. 
F Fi ig gu ur re e  3 3: :  T Th he e  j ju um mp p  l le en ng gt th h  d d  a as s  a a  f fu un nc ct ti io on n  o of f  t th he e  j ju um mp pe er r’ ’s s  p po os si it ti io on n  a an ng gl le e  ϑ ϑ w wi it th h  r re es sp pe ec ct t  t to o  t th he e  f fl li ig gh ht t  p pa at th h. .
If the position angle ϑ is fixed relative to the flight path during the whole flight phase, the
simulated jump length varies from d = 59.6 m to d = 126.2 m, and reaches the peak at a
fixed jumper’s position angle of ϑ = 35
o
. However, even a longer jump length d can be
reached, if the position angle ϑ changes properly during the flight phase. The longest
jump length d = 133.4 m has been simulated when the jumper’s position angle varied
Relationship between jump length and the position angle in ski jumping Kinesiologia Slovenica, 9, 1, 70-79 (2003)
7 74 4

with time nearly linearly, from 18
0 0
to 54
0 0
, as indicated by the solid line in Figure 4. Optimal
position angle as a function of time has been obtained by using the iteration proce-
dure, starting with the constant value of the position angle, and then in the next iteration
step changing it as a function of time, choosing the optimal values. These values have
been then used as starting values for the next iteration step. The last iteration step has
been made when the optimal position angle as a function of time resulted in the jump
length close enough to the value recorded in the previous iteration step. The dashed
line in the same figure represents the current angle č of the flight path with respect to
the x-axis under equal simulation conditions.
F Fi ig gu ur re e  4 4: :  T Th he e  a at th hl le et te e’ ’s s  p po os si it ti io on n  a an ng gl le e  ϑ ϑ a as s  a a  f fu un nc ct ti io on n  o of f  t ti im me e  t t  a at t  w wh hi ic ch h  t th he e  l lo on ng ge es st t  j ju um mp p  l le en ng gt th h  o of f  d d  = =  1 13 33 3. .4 4  m m  h ha as s
b be ee en n  o ob bt ta ai in ne ed d. .  T Th he e  d da as sh he ed d  l li in ne e  r re ep pr re es se en nt ts s  t th he e  c cu ur rr re en nt t  a an ng gl le e  č č  o of f  t th he e  f fl li ig gh ht t  p pa at th h  w wi it th h  r re es sp pe ec ct t  t to o  t th he e  x x- -a ax xi is s. .
The difference between the variant of the computer simulation for establishing the
parameters of the ski jumper’s jump carried out at a fixed angle ϑ = 35
0 0  
and the variant
carried out at the changeable angle arises in the first phase of the flight. About 0.3 sec-
ond after the take-off the angle is ϑ = 35
0 0
(Jošt, Vaverka, Kugovnik, & Čoh, 1998). The
angle increases slightly and amounts to ϑ = 45
0
approximately in the second third of
the flight. On landing the angle ϑ can equal 55
0
.
R RE ES SU UL LT TS S  A AN ND D  D DI IS SC CU US SS SI IO ON N
We have used differential equations (6) together with the expressions (4) and (5) for the
horizontal and vertical components of the resultant force that acts on a ski jumper. 
The jumper’s current velocity v(t) and its horizontal v
x
(t) and vertical components v
y
(t)
are shown in Figure 5. The solid lines represent the values that have been simulated for
the variable jumper’s position angle ϑ that corresponds to the longest simulated jump
and the dashed lines correspond to the fixed position angle of ϑ = 35
0
.
Relationship between jump length and the position angle in ski jumping Kinesiologia Slovenica, 9, 1, 70-79 (2003)
7 75 5

F Fi ig gu ur re e  5 5: :  T Th he e  c cu ur rr re en nt t  a at th hl le et te e’ ’s s  v ve el lo oc ci it ty y  v v( (t t) )  a an nd d  i it ts s  c co om mp po on ne en nt ts s  v v
x x
( (t t) ), ,  v v
y y
( (t t) )  w wh he en n  t th he e  p po os si it ti io on n  a an ng gl le e  ϑ ϑ v va ar ri ie es s  w wi it th h  t ti im me e  a as s
i in n  F Fi ig gu ur re e  4 4. .  T Th he e  s so ol li id d  l li in ne es s  r re ep pr re es se en nt t  t th he e  v va al lu ue es s  t th ha at t  h ha av ve e  b be ee en n  s si im mu ul la at te ed d  f fo or r  t th he e  v va ar ri ia ab bl le e  j ju um mp pe er r’ ’s s  p po os si it ti io on n  a an ng gl le e  ϑ ϑ t th ha at t
c co or rr re es sp po on nd ds s  t to o  t th he e  l lo on ng ge es st t  s si im mu ul la at te ed d  j ju um mp p  a an nd d  t th he e  d da as sh he ed d  l li in ne es s  c co or rr re es sp po on nd d  t to o  t th he e  f fi ix xe ed d  p po os si it ti io on n  a an ng gl le e  o of f  ϑ ϑ = =  3 35 5
0 0
. .
As presented in Figure 5, the basic velocity of motion of the common centre of gravity of
the ski jumper v(t) and its horizontal component v
x
(t) decreases immediately upon
take-off. In the ultimate stage of the take-off the vertical component of the resultant
velocity v
y
(t) reaches the trough. After that an upward trend is present in v
y
(t) during the
entire phase of the flight up until landing. In the final stage of the flight the vertical com-
ponent of the velocity v
y
(t) is greater than the horizontal component v
x
(t). When the
jumper’s position angle ϑ changes with time as indicated in Figure 3, the velocity of
motion v(t) shows a minimum at the flight time t = 0.68 s, whereas its horizontal compo-
nent v
x
(t) decreases evenly. A slight downward movement is recorded in the horizontal
component of velocity v
x
(t) throughout the flight, which is in agreement with the empiri-
cal findings of the research carried out by Jošt and colleagues (1998) in 1996 on the
Innsbruck hill. The difference between the 8 m and 7 4 m marks beyond the edge of the
take-off was approximately 2 ms
-1
, which meant slightly less than 10% of the absolute
value measured at the first mark. In this section of the Innsbruck hill the vertical compo-
nent v
y
(t) rose by about 10 ms
-1
. The resultant velocity v(t) grew by about 2 ms
-1
and in
both cases exceeded the horizontal and vertical components. The landing velocity v(t)
is 33. 1 ms
-1
and its perpendicular component to the hill is 7 .0 ms
-1
. It corresponds to the
landing velocity of a jump onto a horizontal plane from the height of 2.5 m.
The current drag and lift components of the dynamic fluid force are presented in Figure 6.
Relationship between jump length and the position angle in ski jumping Kinesiologia Slovenica, 9, 1, 70-79 (2003)
7 76 6

F Fi ig gu ur re e  6 6: :  T Th he e  c cu ur rr re en nt t  d dr ra ag g  F F
D D
( (t t) )  ( (s so ol li id d  l li in ne es s) )  a an nd d  l li if ft t  c co om mp po on ne en nt ts s  F F
L L
( (t t) )  ( (d da as sh he ed d  l li in ne es s) )  o of f  t th he e  d dy yn na am mi ic c  f fl lu ui id d  f fo or rc ce e  i in nf fl lu u- -
e en nc ci in ng g  t th he e  a at th hl le et te e. .  Z Ze er ro o  t ti im me e  ( (0 0) )  r re ep pr re es se en nt ts s  t th he e  s st ta ar rt t  o of f  t th he e  f fl li ig gh ht t  p ph ha as se e. .
The current drag reaches the minimum in the final stage of the take-off or in the first
phase of the flight. Later both forces increase evenly in the first second of the flight but
no so rapidly. During the first and the second seconds of the flight, when the flight angle
changes significantly, both forces grew more. The current drag F
D
(t) increases almost
entirely linearly up until landing. The rise in lift components F
L
(t) is almost linear all the
way to the point when the ski jumper prepares for landing. During landing the lift com-
ponent F
L
(t) increases minimally.
The horizontal and vertical components of the resultant force that acts on the ski
jumper are presented in Figure 7. The solid lines correspond to the simulation with a
changeable athlete’s position angle, as shown in Figure 3 (the longest jump), and the
dashed lines correspond to the fixed one, i.e. ϑ = 35
0
. 
Relationship between jump length and the position angle in ski jumping Kinesiologia Slovenica, 9, 1, 70-79 (2003)
7 77 7

F Fi ig gu ur re e  7 7: :  T Th he e  c cu ur rr re en nt t  h ho or ri iz zo on nt ta al l  F F
x x
( (t t) )  a an nd d  v ve er rt ti ic ca al l  c co om mp po on ne en nt ts s  F F
y y
( (t t) )  o of f  t th he e  r re es su ul lt ta an nt t  f fo or rc ce e  t th ha at t  a ac ct ts s  o on n  t th he e  a at th hl le et te e. .  T Th he e
s so ol li id d  l li in ne es s  c co or rr re es sp po on nd d  t to o  t th he e  s si im mu ul la at ti io on n  w wi it th h  a a  c ch ha an ng ge ea ab bl le e  a at th hl le et te e’ ’s s  p po os si it ti io on n  a an ng gl le e, ,  a as s  s sh ho ow wn n  i in n  F Fi ig gu ur re e  3 3  ( (t th he e
l lo on ng ge es st t  j ju um mp p) ), ,  a an nd d  t th he e  d da as sh he ed d  l li in ne es s  c co or rr re es sp po on nd d  t to o  t th he e  f fi ix xe ed d  o on ne e, ,  i i. .e e. .  ϑ ϑ = =  3 35 5
0 0
. .
Figure 7 shows that the resultant force F
x
(t) stays nearly constant. The vertical compo-
nent of the resultant force F
y
(t) decreases quite evenly during the entire flight with a lin-
ear trend, which is represented inversely in Figure 7. The vertical component of the
resultant force is much greater than the horizontal component F
x
(t). This difference is
particularly marked in the first and middle stages of the flight of a ski jumper.
This model is somewhat limited. Detailed studies of other parameters dependencies
are required - such as the jumper’s mass and its surfaces of the frontal and the longitu-
dinal areas - which we plan to carry out in the near future.
R RE EF FE ER RE EN NC CE ES S
Dubs, F. (1987). Aerodynamik der reinen Unterschallströmung [Aerodynamics of Pure Subsonic Currents]. Basel:
Birkhduser Verlag.
Halliday, D., Resnick, R., & Walker, J. (2001). Fundamentals of Physics. New Y ork: John Wiley & Sons.
Jin, H., Shimizu, S., Watanuki, H. K., & Kobayashi, K. (1995). Desirable gliding styles and techniques in ski jumping.
Journal of Applied Biomechanics 11, 460-47 4.
Jošt, B., Vaverka, F ., Kugovnik, O., & Čoh, M. (1998). Differences in selected Kinematic flight parameters of the most
and the least successful ski jumpers of the 1996 World Cup Competition in Innsbruck. Biology of Sport, Vol. 15, 4,
245 - 251.
Muller, W., Platzer, D., & Schmolzer, B. (1996). Dynamics of human flight on skis: improvements on safety and fair-
ness in ski jumping. Journal of Biomechanics 29, 1061-1068.
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7 78 8

Schmolzer, B., & Muller, W. (2002). The importance of being light: aerodynamic forces and weight in ski jumping.
Journal of Biomechanics 35, 1059-1069.
Straumann, R. (1927). Vom Skiweitsprung und seiner Mechanik: Jahrbuch des Schweizerischen Ski Verbandes
[From Ski-jumping and its Mechanics: Yearbook of the Swiss Ski Union]. Berri: Selbstverlag des SSV.
Vaverka, F . (1987). Biomechanika skoku na lyžich [Biomechanics of ski-jumping]. Olomouc: UP .
Virmavirta, M., Kivekas, J., & Komi, P.V. (2001). Take-off aerodynamics in ski jumping. Journal of Biomechanics 34,
465-470.
A Ac ck kn no ow wl le ed dg gm me en nt ts s
The authors gratefully acknowledge the financial support of the Croatian Science Foundation. 
Received: 25 September 2002 - Accepted: 20 May 2003
Relationship between jump length and the position angle in ski jumping Kinesiologia Slovenica, 9, 1, 70-79 (2003)
7 79 9