**Document ID**: 05_10_99_1

**Date Received**: 1999-05-10 **Date Revised**: 1999-09-30 **Date
Accepted**: 1999-10-07

**Curriculum Topic Benchmarks**: S12.4.7, S15.4.1, S15.4.6, S16.4.2,
S16.4.5, S17.4.1,

S17.4.2, M5.4.4, M5.4.13, M5.4.16

**Grade Level**: \[9-12\] High School

**Subject Keywords**: computer model, numerical model, sports physics,

ski jump, jump, skiing

**Rating**: advanced

**Who Hits Harder: The Nordic Skier or Aerial Jumper?**

**By**: Donald P Wylie, Space Science and Engineering Center, University
of Wisconsin-Madison, 1225 W. Dayton Street, Madison WI 53717 e-mail:
don.wylie@ssec.wisc.edu

**From**: The PUMAS Collection http://pumas.jpl.nasa.gov

©1999, California Institute of Technology. ALL RIGHTS RESERVED. Based on
U.S. Gov\'t sponsored research.

# Abstract

Computer modeling is used to estimate physical quantities that are
difficult to measure, in this case, the landing shock experienced by ski
jumpers. The landing shock is difficult to measure because attaching
instruments to the skiers would affect their balance and possibly cause
them to fall or reduce their performance. Using basic principals of
physics, the flight trajectories of two forms of competitive ski
jumping, Freestyle Aerialist and Nordic Jumpers, were modeled and the
landing shocks compared. The model uses physical quantities such as the
takeoff inclination, takeoff height, the shape of the hill, and the
vertical drop to the landing. The landing shock is calculated from the
velocity of the skier and angle which he or she hits the hill. For
comparison to other sports, the landing shock is converted to a vertical
drop to a flat surface.

The model shows that Freestyle Aerialist have landing shocks equivalent
to a vertical drop of 20 feet to a flat surface. This is over twice the
landing shock of the Nordic Jumpers even though the Nordic Jumpers
travel at twice the velocity and fall twice as far.

**Introduction**

In the winter Olympics, two types of skiers, Nordic Jumpers and
Freestyle Aerialists, are known for their dazzling displays of aerial
maneuvers that most spectators find incredible and possibly dangerous.
The Aerial Jumpers in the Freestyle Skiing competition perform very high
leaps with twists and somersaults. The 1998 Olympic winner, Eric Bergous
(Figure 1), executed three complete rotations with four twists. This
required extensive time in the air, which he would later reconcile with
when gravity pulled him back to earth. At the Nordic Ski Jumps, Masahiko
Harada and Takanobu Okabe both flew 137 meters (449 ft) in distance.

Competition drives the skiers to fly higher and farther. The athletes
risk falling upon impact to win the competition. Fortunately, the
Federation of International Skiing[^1] (FIS) has imposed rules on the
designs of ski jumps to insure safety and minimize injuries. The FIS
specifications for Freestyle Aerial jumps is shown in Figure 2.

**Computer Model of Skier's Flight**

Nordic ski jumping has been modeled by Muller et al. (1995 and 1996).
They found that most jumpers have a very small impact on the landing
hill equivalent to a vertical drop of \< 1 meter, even though they fly
distances over 100 meters (328 ft.) with vertical drops over 30 m (98
ft.). The Nordic skiers experience light landings because they are
flying close to the hill which is shaped for their flight trajectory
(Figure 1).

The Freestyle Aerial Jumpers use a much higher trajectory to perform
rotations and twists. Judges award points for the heights of the jumps
along with the difficulty of the rotations and twists. Distance is not a
factor in Aerial Jumping as it is in Nordic Jumping.

![](media/image1.wmf){width="5.625694444444444in" height="2.65625in"}

Figure 1: Freestyle Aerial jumper Eric Bergous (left) and Nordic jumper
Masahiko Harada (right).

To answer the question, "How hard do the Freestyle Aerialists land?" a
smaller version of the Muller model was used. The impact of a skier on
the landing hill is not simply the vertical distance he/she drops
because they land on a steep slope (\~37°). The skier also does not stop
on the hill. The impact the skier feels is the force it takes to change
the skier's direction from their flight to the slope of the hill. This
is illustrated in Figure 3. The skier's direction of motion is shown by
the vector A-B. The hill has a slope from A-C. The skier's velocity
(V~A-B~) can be expressed as two orthogonal (right angle) components.
The component along the hill is V~A-C~ whereas the other component is
V~C-B~. Upon impact, the skier's velocity will change from V~A-B~ to
V~A-C~. The change in direction is angle (a) and the velocity component
V~C-B~ will be absorbed in the landing impact. The impact the skier
feels is the dissipation of V~C-B~.

![](media/image2.wmf){width="5.398611111111111in"
height="3.622916666666667in"}

Figure 2: Federation of International Skiing specifications for
Freestyle Aerial hills. Full details are available at
http://www.zip.com.au/\~birdman/FTP.html under "1997/98 Freestyle Course
Standards".

Using the triangle in Figure 3, the skier's velocity component
perpendicular to the face of the hill, V~C-B~, can be calculated from
the skier's velocity, V~A-B~, and the change in direction (a).

V~C-B~ = V~A-B~.SINE(a) (1)

![](media/image3.wmf){width="3.1256944444444446in"
height="3.1256944444444446in"}

Figure 3: The skier lands at A traveling toward B, but will abruptly
change direction toward C.

The flight trajectory model calculates the skier's velocity and position
for a series of small time steps (Figure 4). The model starts with a
speed and direction (velocity) at the point of takeoff and then
calculates how far the skier moves in each time increment. For this
model, time increments of 0.05 seconds were used which is about 0.5 m in
displacement. At the end of each increment, a new velocity has to be
calculated because forces have acted on the skier. The process proceeds
to a second step where a new position is calculated using the average
velocity of the time increment followed by a re-calculation of the
velocity. This process is successively repeated many times (60-100)
until the skier arrives on the landing hill.

![](media/image4.wmf){width="3.0930555555555554in"
height="2.1666666666666665in"}

Figure 4: The large numbers are time steps of 0.05 seconds. The
non-orthogonal blue arrows show the velocities at each point in time.
Orthogonal green arrows indicate velocity and distance component
vectors.

To simplify the calculations, the skier's displacement vector is
decomposed into horizontal and vertical components (Figure 4). The
horizontal component X increases to the right and the vertical component
Y increases upward. The skier's position at the end of each time step
is:

X~2~ = X~1~ + 0.05 × V~x~ (2)

Y~2~ = Y~1~ + 0.05 × (V~y1~+V~y2~) / 2 (3)

Subscripts 1 and 2 refer to the start and end of one time increment.

The velocities are changed by the accelerations of the forces on the
skier. Since the speed of the Aerial Jumper is relatively slow,
aerodynamic lift and drag are not considered. However, Muller had to
consider aerodynamic forces for the Nordic jumper because of the greater
velocities (23-27 m/s or 51-57 mph). For the Aerial jumper, we assume
the horizontal component (V~X~) is constant in time and gravity is the
only acceleration on the vertical component (V~Y~). Therefore, the
formulas used to calculate velocity components for the Aerial Jumper
are:

V~x2~ = V~x1~ (4)

V~y2~ = V~y1~ - G × 0.05 (5)

G is 9.8 m/s^2^ (32.3 ft/s^2^).

The model starts with velocities at the takeoff of V~x0~ and V~y0~, and
then calculates V~x1~ and V~y1~ using equations (4) and (5). The
position of point one, X~1~ and Y~1~, are calculated from equations (2)
and (3). Then V~x2~ and V~y2~ are calculated followed by X~2~ and Y~2~,
etc.

For the Aerial jumper we know the location of the takeoff point and the
inclination angle, but not the velocity. Aerial jumpers can choose their
velocity by selecting any starting position on the take off ramp. They
vary their velocity for the air time needed for their maneuvers. The
jumper's velocity can be estimated by assuming that the jumper intends
to land in the lower part of the landing slope. This will give the
maximum air time and also allow for the jumper to recover their balance
on the steep slope before skiing through the curve at the bottom of the
hill and stopping in the out run area.

A backward iterative method is used to find the takeoff velocity from
the landing point. It begins with a guess of the takeoff velocity and
running the model to find a landing point. If the skier lands too short
with this velocity, the velocity is increased and the model re-run.
Conversely, if the skier lands too long (or close to the bottom curve),
the starting velocity is decreased and the model is rerun. The takeoff
velocity can be found with a few iterations (\<10).

The largest takeoff kicker in Figure 2 has an inclination of 55° and is
positioned 8.1 meters back from the start of the landing slope, which
has a constant pitch of 37° and is 25 meters long. From successive
iterations we found that, at a takeoff velocity of 12.5 m/s (28 mph),
would send the skier 26 meters to within 10 meters of the bottom of the
slope. For this flight the skier had an air time of 3.0 seconds and rose
5.5 meters (18 ft) above the take-off. He/she dropped 18.7 meters (64
ft) from the apex height to the landing. The skier was traveling 20.5
m/s (45 mph) at an declination of 69° when hitting the hill. From Figure
3, the impact velocity, V~C-B~, is 11 m/s.

Muller converted the vector V~C-B~ to an equivalent drop from a
fictitious height to a flat surface so that comparisons could be made to
other sports. The conversion uses the two basic equations for
acceleration and distance in a gravitational field, V = G × T and H =
0.5 × G × T^2^ resulting in:

H = 0.5 × V~C-B~^2^ / G (6)

Using equation (6) the Aerial jumper's impact velocity, V~C-B~, is
equivalent to a vertical drop of 6.1 meters (20 ft). A comparison
between the landing impacts of Aerial and Nordic jumpers is given in
Table 1. Nordic jumper flights are calculated from a replica of the
Muller model, which includes lift and drag terms in equations 4 and 5
(see Muller et al., 1996 for details). Two landings are shown for each
type of skier, a short jump and a long competitive jump. The Aerial
jumps differ only in takeoff velocity whereas the Nordic Jumps differ in
the leg spring of the skiers and the aerodynamic forces in flight. The
first jump represents a junior skier weighting 80# with a vertical leg
spring of 9 inches, and aerodynamic lift of 40# and drag of 48#. The
second is an Olympic competitor weighting 130# with a vertical spring of
18 inches and aerodynamic lift of 78# and drag of 65#.

**Table 1. Jump Parameters**

Take off Jump Total vertical Landing Equivalent drop Flight

Angle Velocity Distance fall distance V~A-B~ V~C-B~ to flat surface Time

[Deg]{.underline} [m/s]{.underline} ([mi/hr)]{.underline}
[meters]{.underline} ([feet)]{.underline} [meters]{.underline}
[(feet)]{.underline} [m/s]{.underline} [m/s]{.underline}
[meters]{.underline} ([feet)]{.underline} [seconds]{.underline}

Aerial hill

55° 9.7 (21.3) 15 (49) 10 (33) 15.1 7.9 3.2 (10) 2.2

55° 12.5 (27.5) 26 (85) 19 (62) 20.5 11.0 6.1 (20) 3.0

Nordic hill

-10° 23.1 (50.8) 62 (203) 28 (92) 17.3 3.3 0.6 (2) 3.5

-10° 23.1 (50.8) 109 (358) 54 (177) 21.7 6.9 2.4 (8) 5.9

**Discussion**

The Aerial jumper model shows landing impacts of 3.2-6.1 m (10-20 ft) in
vertical drop whereas Nordic jumpers experience much smaller impacts of
0.6-2.4 m. A 20-foot fall could cause serious injury. It's about the
height of the roof on a two-story house. However, Freestyle Aerial
jumping has an exceptional record in that they have never had a
debilitating injury in 22 years[^2].

The reason for Aerial jumping's safety record is in the preparation of
the landing hills and the training and guidance by the coaches. Aerial
landing hills have to be soft with two or more feet of soft snow that is
never packed by skiers or grooming machines. Before competitions, hill
crews loosen the snow with shovels and during competitions, they add
extra snow between jumps. The competitors use different launching
kickers based on the skill of the skiers. Also, Freestyle Aerialists
start their training using water pools in the summer where air is
bubbled into the pools to soften the landings.

**Future Applications of the Model**

The model described here can be applied to other sports for the same
purpose. Two possible applications are water ski jumping and motorcycle
stunts. To calculate the landing impacts in these sports, you need to
find the inclination angle of the take-off ramp (\~5°-10°), the
velocity, the distance traveled, and the inclination of the landing
ramp. Aerodynamic drag also should be included in Equations 4 and 5.
With a model, you could try different ramp inclinations and velocities
to estimate the best jump.

The computer code for this example is written in Fortran and available
from the author. The author is a scientist by profession (meteorology)
with a son who competes in Nordic Jumping. He also is a volunteer member
of the hill crews at the Intervale Olympic Ski Jumps in Lake Placid, NY
and the Blackhawk Ski Club in Madison, WI, and a member of the National
Ski Patrol.

**References**

Muller, W., D. Platzer, and B. Schmolzer, 1995: Scientific approach to
ski safety, *Nature*, **375**, 455.

\_\_\_\_\_\_, 1996: Dynamics of human flight on skis: Improvements on
safety and fairness in ski jumping, *J. Biomechanics*, **29**,
1061-1068.

[^1]: Federation Internationale de Ski, Blochenstrasse 2, CH-3653
    Oberhofen/Thunersee, Switzerland and web site
    http://www.zip.com.au/\~birdman/fis.html.

[^2]: Peter Judge, Chairman, FIS Freestyle Subcommittee for Rules and
    Technical, e-mail PRJudge@compuserve.com