A Ab bs st tr ra ac ct t
A physical model of downhill skiing has been con-
structed with the aim of determining the influence of
a skier's body mass on their performance. The
present article investigates the relationship
between a skier's body mass and the shortest time
needed for a downhill skier to complete the dis-
tance.
The accuracy of the physical model used in the
study is essentially determined by the accuracy of
the input parameters, but their errors do not notice-
ably affect the reliability of this comparative simula-
tion study. The computer simulations show that the
total time needed to complete the whole distance
decreases as the skier's body mass increases.
Slightly better total times have been simulated for
higher skier's body masses. When a skier's body
mass is 80 kg instead of 75 kg, the total time is about
0.4% shorter, i.e. approximately 0.6 s for the profile
used in these simulations. The only factor that is
causing this is the drag force influencing the skier,
which has more effect on skiers with lower body
mass.
Key words: downhill skiing, physical model, body
mass, performance, computer simulation
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 ns s, ,  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
Educations, 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
Fizikalen model smuka je bil izdelan s ciljem določi-
ti vpliv telesne mase smučarja na njegov rezultat.
Članek preučuje razmerje med telesno maso
smučarja in najboljšim časom, ki ga je smučar
potreboval, da je presmučal določeno razdaljo.
Natančnost fizikalnega modela, ki je bil uporabljen
v raziskavi, v osnovi določajo vhodni podatki, ven-
dar napake ne vplivajo bistveno na zanesljivost te
primerjalne simulacijske raziskave. Računalniške
simulacije kažejo, da je skupen čas, potreben za
premagovanje razdalje, tem krajši, čim vecja je
telesna masa smučarja. Pri smučarjih z večjo
telesno maso je bil v simulaciji dosežen nekoliko
boljši skupen čas. Če je telesna masa smučarja   80
kg namesto 75 kg, se skupen čas skrajša za prib-
ližno 0.4 %, kar pomeni okoli 0.6 s za profil, ki je bil
uporabljen v omenjenih simulacijah. Edini dejavnik,
ki je za to odgovoren, je sila vleke, ki deluje na
smučarja in ki ima večji vpliv na smučarje z manjšo
telesno maso.
Ključne besede: smuk, fizikalen model, telesna masa,
rezultat, računalniška simulacija
Faculty of Sport, University of Ljubljana, ISSN 1318-2269 Kinesiologia Slovenica, 9, 1, 18-24 (2003)
1 18 8
I IN NF FL LU UE EN NC CE E    O OF F    B BO OD DY Y    M MA AS SS S  O ON N  P PE ER RF FO OR RM MA AN NC CE E    I IN N
D DO OW WN NH HI IL LL L    S SK KI II IN NG G
V VP PL LI IV V    T TE EL LE ES SN NE E    M MA AS SE E    S SM MU UČ ČA AR RJ JA A  N NA A  N NJ JE EG GO OV V
R RE EZ ZU UL LT TA AT T    P PR RI I    S SM MU UK KU U
M Mi il le e    D Dž že 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
M Mi il la an n    Ž ž v va an n
2 2

I IN NT TR RO OD DU UC CT TI IO ON N
Extremely low body mass is becoming widespread in many sports, such as bicycle racing,
horse racing, gymnastics, long-distance running and ski-jumping, but there are some sports in
which higher body mass gives athletes some advantages. One of them is downhill skiing,
which is a very complex skill.
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 earliest analytical model of skiing 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 skiing have been adopted, but
there are only a few reliable studies of the effects of an athlete's body mass on their perform-
ance.
Using a physical model, the present article investigates the relationship between a skier's
body mass and the total time needed to complete the whole distance.
M ME ET TH HO OD DS S
T Th he e  m mo od de el l
F Fi ig gu ur re e  1 1: :  T Th he e  f fo or rc ce es s  t th ha at t  e ex xe er rt t  o on n  t th he e  d do ow wn nh hi il ll l  s sk ki ie er r
The gravitational force F
g
, the normal contact force F
N
, the drag component F
D
of the dynamic
fluid force, and the components of the friction force (longitudinal F
F
and transversal F
F    
), exert-
ed on a downhill skier. The x- and y-axes of the coordinate system are in the horizontal plane.
When a skier descends, as shown in Figure 1, the gravitational force F
g
, the normal contact
force F
N
, the longitudinal F
F  
and transversal F
F
components of the friction force, and the drag
component F
D
of the dynamic fluid force are exerted on him. The magnitude of the drag force is
proportional to air density ρ, the surface area A of the skier in the air, and the square of the
skier's velocity ν relative to air (Halliday, Resnick, & Walker, 2001; McGinnis, 1999).
Body mass in downhill skiing: physical model Kinesiologia Slovenica, 9, 1, 18-24 (2003)
1 19 9
||
F
⊥
||
F
⊥

(1)
where C
D
is the coefficient of drag. The drag force F
D
acts in opposition to the relative motion of
the skier with respect to air and that force tends to decrease the magnitude of the relative veloc-
ity. It is produced by two different factors: surface drag and form drag. Surface drag is the sum
of the friction forces acting between air molecules and the surface of the skier. Form drag is the
sum of the impact forces resulting from the collisions between the air and the skier.
The friction force F
F
is a contact force that acts between and is parallel to the surfaces in con-
tact, i.e. skis and snow. The magnitude of the friction force is proportional to the normal contact
force pushing the two surfaces together
(2)
where µ is the coefficient of friction and F
N
is the magnitude of the normal contact force. The
friction force has been divided into two components: longitudinal and transversal 
, where and , respectively, are the longitudinal and transversal coefficients of
friction. The longitudinal component of the friction force acts in opposition to the skier's motion
with respect to snow and that force tends to decrease the magnitude of the velocity. The trans-
versal component of the friction force acts as centripetal force parallel to the contact surfaces
and perpendicular to the skier's velocity, and its effect results in a change of the direction of the
skier's motion.
The horizontal components F
x
and F
y
of the resultant force that acts on the skier are
(3)
where ϕ is the angle of the horizontal component of the velocity at that instant with respect to
the x-axis, and α is the angle of the velocity with respect to the horizontal plane (see Figure 1).
The vertical component F
z
of the resultant force is
(4)
where m is the skier's body mass and g = 9.8 ms
-2
is the acceleration caused by the Earth's grav-
itational force.
Using the horizontal F
x
, F
y
and vertical F
z
components of the resultant force, we have calculat-
ed the components of the skier's velocities (v
x
, v
y
, v
z
) and positions (x, y, z) at that instant using
the following differential equations:
Body mass in downhill skiing: physical model Kinesiologia Slovenica, 9, 1, 18-24 (2003)
2 20 0
,
2
1
2
Av C F
D D
ρ =
,
N F
F F
), cos(
) sin(
2
1
), sin( ) cos( ) cos(
) cos( ) cos(
2
1
||
2
||
2
ϕ µ ϕ µ ρ
α ϕ α µ ϕ α µ ρ
N
N D y
N N
N D x
F
F A v C F
F F
F A v C F
⊥
⊥
−






+ − =
+ +






+ − =
), cos( ) sin( ) sin(
) cos( ) sin(
2
1
||
2
α ϕ α µ ϕ α µ ρ
N N
N D z
F F
F A v C mg F
+ +






+ + − =
⊥
N F
F F
|| ||
N F
F F
⊥ ⊥ ||
F
⊥

(5)
In addition to the resultant force exerted on the skier, skier's velocity and position change during
downhill skiing as well.
S Si im mu ul la at ti io on n
We have used differential equations (5) together with the expressions (3) and (4) for the hori-
zontal and vertical components of the resultant force that acts on a downhill skier. In the com-
puter simulation series, for a skier's body mass of 75 kg, for the minimal and maximal values of
the frontal skier's surfaces we have applied A
min
= 0. 12 m
2
, A
max
= 1.0 m
2
. We have used minimal
surface when the skier tends to increase the magnitude of the velocity crouching in an "egg
position" (at the beginning and after each curve), and maximal surface when he tends to
decrease the velocity (before each curve). Surfaces of other body masses used in computer
simulations have been adjusted in line with the skier's body mass (A 
~
m
2/3
). The air density ρ,
according to the International Civil Aviation Organisation (Dubs, 1987), 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 the elevation of skiing hills is about 1000 to 2500 m, the following expo-
nential function was used to describe the air density as a function of elevation 
(6)
where ρ
o
= 1.225 kgm
-3
is the air density at mean sea level and is the constant.
For the longitudinal and transversal coefficients between the surface of the skis and the snow
we have chosen and . The transversal friction force acts perpendicularly to
skis, allowing skiing in curves. The profile of a typical skiing hill (see Table 1) with an average
gradient of α = 13.5
O
(α
min
= 6
O
, α
max
= 42
O
) has been applied (                                                                         ). For the
drag coefficient we have used C
D
= 1 as in (Schmölzer, & Müler, 2002). Fixed time step integra-
tion of                                       has been used.
T Ta ab bl le e  1 1: :  T Th he e  p pr ro of fi il le e  o of f  t th he e  s sk ki ii in ng g  h hi il ll l  u us se ed d  i in n  t th he e  s si im mu ul la at ti io on ns s  ( (t th he e  t to ot ta al l  l le en ng gt th h  o of f  t th he e  c co ou ur rs se e  i is s  4 44 40 00 0  m m) )
Body mass in downhill skiing: physical model Kinesiologia Slovenica, 9, 1, 18-24 (2003)
2 21 1
,
0
kz
e
−
=ρ ρ
.              ,             ,
          ,          ,
z y x
z z
y y
x x
v
dt
dz
v
dt
dy
v
dt
dx
m
F
dt
dv
m
F
dt
dv
m
F
dt
dv
= = =
= = =
-1 4
m 10 015 . 1
−
⋅ = k
m 028 , 1 z , m 270 , 4 x
s 0001 . 0 t
07 . 0
||
1 . 1
⊥

R RE ES SU UL LT TS S
F Fi ig gu ur re e  2 2: :  L Le ef ft t: :  t th he e  t to ot ta al l  t ti im me e  n ne ee ed de ed d  t to o  c co om mp pl le et te e  t th he e  w wh ho ol le e  d di is st ta an nc ce e  a as s  a a  f fu un nc ct ti io on n  o of f  t th he e  s sk ki ie er r' 's s  b bo od dy y  m ma as ss s. .  R Ri ig gh ht t: :
r re el la at ti iv ve e  d di if ff fe er re en nc ce e  o of f  t th he e  t to ot ta al l  t ti im me e. .
For various body masses the total time needed to complete the whole distance has been calcu-
lated and displayed in Figure 2 left. Figure 2 right shows the
relative difference of the total time as a function of the skier's body mass. Shorter times have
been obtained in the case of higher body masses. For example, when a skier's body mass is 80
kg instead of 75 kg, the computer simulations show that the total time will be about 0.4% shorter,
i.e. approximately 0.6 s for the hill profile used in the simulations. The drag force is the only factor
that causes this difference in the total time, which tends to decrease the magnitude of the veloc-
ity and is more important for skiers with lower body masses, as indicated in Figures 3 and 4.
F Fi ig gu ur re e  3 3: :  T Th he e  d dr ra ag g  f fo or rc ce e  f fo or r  t th hr re ee e  d di if ff fe er re en nt t  b bo od dy y  m ma as ss se es s  ( (7 70 0  k kg g  - -  d do ot tt te ed d  l li in ne e, ,  7 75 5  k kg g  - -  f fu ul ll l  l li in ne e, ,  a an nd d  8 80 0  k kg g  - -  d da as sh he ed d  l li in ne e) ), ,
w wh hi ic ch h  a ac ct ts s  o on n  a a  s sk ki ie er r  d du ur ri in ng g  t th he e  f fi ir rs st t  7 70 00 0  m m  o of f  s sk ki ii in ng g
Body mass in downhill skiing: physical model Kinesiologia Slovenica, 9, 1, 18-24 (2003)
2 22 2

F Fi ig gu ur re e  4 4: :  T Th he e  d dr ra ag g  f fo or rc ce e  d di iv vi id de ed d  b by y  t th he e  s sk ki ie er r' 's s  b bo od dy y  m ma as ss s  ( (7 70 0  k kg g  - -  d do ot tt te ed d  l li in ne e, ,  7 75 5  k kg g  - -  f fu ul ll l  l li in ne e, ,  a an nd d  8 80 0  k kg g  - -  d da as sh he ed d  l li in ne e) )
Figure 3 shows the drag force for three different body masses (70 kg - dotted line, 75 kg - full line,
and 80 kg - dashed line), which acts on the skier during the first 700 m of downhill skiing. Figure
4 shows the drag force divided by the skier's body mass. Obviously, the magnitude of the drag
force is higher in the case of skiers with higher body mass, but, as shown in the Figure, the drag
force divided by body mass has slightly lower magnitude in the case of higher body masses,
thus causing higher speeds of skiers with higher body masses.
F Fi ig gu ur re e  5 5: :  T Th he e  m ma ax xi im ma al l  m ma ag gn ni it tu ud de e  o of f  v ve el lo oc ci it ty y  r re ea ac ch he ed d  d du ur ri in ng g  d do ow wn nh hi il ll l  s sk ki ii in ng g  a as s  a a  f fu un nc ct ti io on n  o of f  a a  s sk ki ie er r' 's s  b bo od dy y  m ma as ss s
Figure 5 shows the maximal magnitude of the velocity reached during downhill skiing as a func-
tion of a skier's body mass. Higher values were obtained in the case of higher body masses.
The possible hill-profile dependence of the results obtained in the simulations requires
detailed studies, which we plan to carry out in the near future.
Body mass in downhill skiing: physical model Kinesiologia Slovenica, 9, 1, 18-24 (2003)
2 23 3

R RE EF FE ER RE EN NC CE ES S
Dubs, F. (1987). Aerodynamik der reinen Unterschallströmung [Aerodynamics of Pure Subsonic Currents]. Basel:
Birkhäuser Verlag.
Halliday, D., Resnick, R., & Walker, J. (2001). Fundamentals of Physics. New Y ork: John Wiley & Sons.
McGinnis, P .M. (1999). Biomechanics of Sport and Exercise. Champaign, IL: Human Kinetics.
Schmölzer, B., & Müler, 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 Skijumping and its Mechanics (Yearbook of the Swiss Skiing Union)]. Bern: Selbstverlag des SSV.
A Ac ck kn no ow wl le ed dg gm me en nt t
The authors gratefully acknowledge the financial support of the Croatian Science Foundation.
Received: 16 November 2002 - Accepted: 4 June 2003Faculty of Sport, University of Ljubljana, Slovenia
Body mass in downhill skiing: physical model Kinesiologia Slovenica, 9, 1, 18-24 (2003)
2 24 4