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ORIGINAL ARTICLE
Designing tomorrow’s snow park jump
James A. McNeil •Mont Hubbard •
Andrew D. Swedberg
Published online: 31 January 2012
ÓInternational Sports Engineering Association 2012
Abstract Recent epidemiological studies of injuries at
ski resorts have found that snow park jumps pose a sig-
nificantly greater risk for certain classes of injury to resort
patrons than other normal skiing activities. Today, most
recreational jumps are built by skilled groomers without an
engineering design process, but the Snow Skiing Com-
mittee (F-27) of the American Society for Testing and
Materials is considering the inclusion of recreational jumps
in their purview which may lead to a greater role for
engineering jump designs in the US in the future. Similar
efforts are underway in Europe as well. The purpose of this
work is to review the current state of the science of snow
park jumps, describe the jump design process, and outline
the role that modelling will play in designing tomorrow’s
snow park jumps.
1 Introduction
The last two decades have witnessed a dramatic evolution
in the skiing industry worldwide. The number of partici-
pants on snowboards has increased and now approaches
parity with skiers [1]. Based on data from the National
Sporting Goods Association, of 11.2 million snow-slope
participants in 2008, 5.3M skied only, 4.7M snowboarded
only, and 1.2M did both. Because the snowboarding
population is younger and demands more access to the
acrobatic aspects of sliding than previously, ski resorts
instituted and continue to experiment with snow terrain
parks which include jumps and other airborne features.
Epidemiological studies of injuries at ski resorts have
found that snow park jumps pose a significantly greater risk
for certain classes of injury to resort patrons than other
skiing activities [2–5]. In particular, due to increases in
flight maneuvers and associated landings, a corresponding
increase in the frequency and severity of head, neck and
upper-extremity injuries has been documented [2].
In a report widely publicized by the National Ski Areas
Association (NSAA), Shealy et al. [6] noted that over the
period from 1990 to the present when overall snow terrain
park use has risen, overall injury rates have actually fallen.
Yet, all studies that focus on actual terrain park injuries
show that terrain parks do indeed present a special hazard
to riders. The Shealy finding is most likely due to
improved ski equipment and an aging, and therefore more
cautious, demographic patron profile over the period
covered in his study. Early research recognized that
snowboard injury patterns differed from those of skiers
[7], and that snowboard injury rates could be as much as
six times higher than those of skiers [8]. The snowboard
injury rate was discovered to have doubled between 1990
and 2000 from 3.37 to 6.97 per 1,000 participant days [9].
In addition, jumping was found early to be the most
important cause of injury [8]. An increased risk of head
injury at terrain parks as compared to ski runs continued
through the end of the last decade [10]. A recent study of
snowboard injury rates specifically in terrain parks found
J. A. McNeil (&)
Department of Physics, Colorado School of Mines, Golden,
CO 80401, USA
e-mail: jamcneil@mines.edu
M. Hubbard
Department of Mechanical and Aerospace Engineering,
University of California, Davis, CA 95616, USA
e-mail: mhubbard@ucdavis.edu
A. D. Swedberg
Department of Mathematics, U.S. Military Academy,
West Point, NY 10996, USA
e-mail: Andrew.Swedberg@usma.edu
Sports Eng (2012) 15:1–20
DOI 10.1007/s12283-012-0083-x
that the risk of injury on jumps was highest of all terrain
park features [3]. Another study comparing ski and
snowboard injuries inside and outside terrain parks found
that the percentage of spine and head injuries inside the
terrain park was double that outside [4]. The principal
hazards may be succinctly summarized as landing ‘‘hard’’
and/or landing upside down.
Many snow-related head, neck and back injuries are
extremely serious. A succinct summary provided by Me-
yers and Misra [11] states: ‘‘spinal cord injuries (SCIs) are
among the worst ski outcomes.’’ A broad review of SCI
epidemiological trends by Jackson et al. [12] found an
increase in SCI caused by snow (as opposed to water)
skiing, and that snow skiing (presumably including snow-
boarding) had replaced football as the second leading cause
of SCI in the US. Ackery et al. [13] published a review of
24 articles between 1990 and 2004 from 10 countries and
found evidence of an increasing incidence of traumatic
brain injury and SCI in alpine skiing and snowboarding
worldwide. They noted that this increase coincided with
‘‘development and acceptance of acrobatic and high-speed
activities on the mountain’’.
By as early as the end of the 1990s, Tarazi et al. had
found that the incidence of SCI in snowboarders was four
times that in skiers and that jumping was the primary cause
of injury (77% of snowboarder SCIs occurred from jump-
ing) [1]. In a review by Seino et al. [14] of six cases of
traumatic paraplegia SCIs resulting from snowboard acci-
dents at a single institution over 3 years, researchers found
that they occurred to young men between the ages of 23
and 25, and that the primary fracture mechanism was a
backward fall from an intentional jump.
Although the above published studies are compelling,
today it is still difficult to get a precise snapshot of
overall terrain park injury statistics nation- or worldwide.
In the US, although the NSAA collects skiing and
snowboarding injury data, these data are not made pub-
licly available.
These most serious SCIs (resulting in paraplegia or
quadriplegia) exact a very large cost on society. That they
happen uniformly to young people and are permanently
debilitating means that there is an enormous economic and
social cost (see [15,16]). One would hope that these large
costs would lead to a more careful and scientific approach
to the design and fabrication of those terrain park features
primarily involved, but this has not been the case.
Although terrain parks have improved in many ways
over the years, the quality of the end product varies widely
from resort to resort. There do exist creditable training
programs, such as ‘‘Cutter’s Camp’’ [17], that are intended
to increase the knowledge and skill level of the groomers.
These have helped considerably, but space in these courses
is limited so such training is by no means universal. In
addition, rider education efforts such as Burton’s ‘‘Smart
Style’’ have been beneficial. Yet adoption of quantifiable
engineering design of terrain parks has been resisted. At
present, in the US at least, terrain park jumps are typically
fabricated at the individual resorts using little or no quan-
titative analysis or engineering design by staff with no
formal training in engineering analysis of the designs.
The reluctance of ski resorts to adopt an engineering
design approach may be traced to their risk management
strategy. As gleaned from the waiver forms attached to lift
pass agreements and recent presentations by ski industry
defense lawyers, one apparent component of the legal
strategy in the US is to assert that the responsibility for
safety resides with the patrons. Aside from minimum
diligence related to roping, signage, lifts, rental shop
operations, and marking man-made obstacles, ski resorts
are reluctant to acknowledge additional responsibility for
the safety of their patrons. This is not to say resorts are not
concerned about safety; they are very much so, but the
point is that as part of their risk management legal strategy,
resorts are reluctant to explicitly acknowledge additional
responsibilities. This strategy extends to terrain park jump
designs. Specifically, the NSAA asserts that, due to rider
and snow variability, terrain park jump ‘‘standards are
impossible’’ [18]. Thus, by this reasoning, engineering
design of winter terrain park jumps is likewise impossible,
which enables resorts to argue that they are not liable
should anything go wrong. In essence, the apparent legal
position of the industry is one whereby the resorts provide
(possibly unsafe) terrain park jumps for their patrons who,
in deciding to use them or not, bear the full responsibility
for the consequences.
While the participants must bear primary responsibility
for and control over their safety while using terrain park
jumps, employing engineering design principles to improve
the quality and safety of the jumps could prevent or miti-
gate the potentially tragic consequences of poor patron
decisions and otherwise minor accidents. Our central pre-
mise is that, although there is significant variability due to
snow conditions and rider decisions, these variations are
bounded in understandable ways that nevertheless allow
engineering designs that accommodate the variability or
render it irrelevant. Indeed, based partly on this view, the
Committee (F-27) on Snow Skiing of ASTM (previously
the American Society for Testing and Materials) is con-
sidering bringing recreational winter terrain park jumps
within its purview. It appears that engineering design
approaches may soon be applied to winter terrain park
jumps. Physical modelling of riders using terrain park
jumps will then become an important, if not essential,
component of this evolution.
2 J. A. McNeil et al.
Previous work on the modelling and simulation of snow
park jumping includes Bohm and Senner [19], Hubbard
[20], and McNeil and McNeil [21]. In this paper, the earlier
work is reviewed and expanded by treating the terrain park
jump as a design problem and exploring the central role
that modelling plays in the iterative design process. Several
of the most important concepts and design components
from a jumper safety point of view are identified and
suggestions made that can improve safety and could be
easily implemented.
The paper is organized around the iterative design pro-
cess intended to address the principal hazards of impact
and inversion. Section 2lays out the design problem of the
terrain park jump system, identifies its components, and
their interaction, and examines the physical constraints and
parameters involved. Section 3addresses the specific
hazard associated with impact in landing, introduces the
concept of the equivalent fall height (EFH) and uses it to
illustrate the potential impact hazards associated with
design decisions. This is illustrated using the standard
tabletop design revealing fundamental safety issues with
this nearly universal form. Section 4addresses the specific
hazard of inversion, and identifies the potential causes of
inversion including those that can be induced involuntarily
by the design of the approach and takeoff. Example cal-
culations of potential inverting rotations are presented.
Section 5examines how the previously developed material
can inform novel terrain park jump designs that limit EFH.
Specifically, the concept of the constant EFH landing
surface is modelled and example calculations are pre-
sented. This section concludes with an illustration of how
the calculation of jumper trajectories over each of the
terrain park features can be used iteratively to create
practical designs with reduced risk of impact and inversion
injury. The paper concludes with a summary.
2 Defining the engineering design problem
2.1 General design criteria and constraints
A terrain park jump is a system of interacting components
which can generically be labelled as the start, the
approach, the transition, the takeoff, the maneuver area,
the landing, and the run-out (see Fig. 1). Each component
has a kinematic input and output that can be modelled
using Newton’s laws. For example, the input speed for the
transition is the output speed of the approach and the output
speed of the transition is the input speed for the takeoff and
so forth. Terminology is still evolving, however, so the
same style jump can be called something different
depending on the region where it is built. (For the examples
used here, the jump style will be defined graphically.) The
most commonly built jump is the standard tabletop jump
shown schematically in Fig. 1. For the tabletop jump the
maneuver area is intended to be above the deck; however,
it is quite common for jumpers to land on the deck; so the
deck should be considered part of the landing as well. In
the so-called ‘‘gap’’ jump design, there is a deep open gap
between the takeoff and the landing. Gap jumps are com-
mon in professional competitive events, but are very unu-
sual at recreational resorts serving the general public and so
will not be treated here.
Typically the overall performance criterion for the jump
is the desired (horizontal) distance of the jump. The length
and topographical profile of the terrain park base or parent
surface provides a global constraint onto which the jump or
jumps must be placed. In general the pitch (the angle of
incline) of the terrain park base will not be constant and the
analysis of the performance will require numerical solu-
tion, but for the sake of illustration, the simple case of a
constant pitch for the approach, h
A
, will be treated. The
shape of the landing surface will be the subject of a more
in-depth analysis. For treating the common tabletop case
the pitch of the landing slope will be constant, h
L
. Given a
budget constraint of a certain volume of snow, the size of
the jump(s) will be constrained. Once the location and size
of the jump are determined, one can enumerate the other
performance criteria for the various jump components such
that the overall performance is met within the global
constraints.
Consider the first element of the jump, the start. The
start provides a generally flat staging area for the riders to
prepare themselves and their equipment and represents the
start point for their descent. Design considerations not
treated here include having the area be large enough to
accommodate the expected traffic flow and having limited
access to control the traffic through the jumps as it would
be undesirable to have several entry points that could
present a collision hazard.
The location of the start point relative to the takeoff
provides a limit on the rider’s speed that can be attained
before takeoff which is an important design parameter.
Assuming the rider starts from rest, the maximum speed at
takeoff is limited by the difference in elevation between the
start and the lip of the takeoff, i.e. vmax ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2gðystart ylipÞ
p
where gis the gravitational acceleration constant and y
start
and y
lip
are the elevations of the start and lip, respectively.
Of course, friction and air drag will lower this value by an
amount that depends on the friction coefficient between
the snow and ski/snowboard surfaces, the shape of the
approach, transition, and takeoff, as well as skier drag
coefficient and wind direction and speed. Example calcu-
lations for a range of example jumps and kinematic and
material parameters will be presented later.
Designing tomorrow’s snow park jump 3
Once leaving the start, the rider enters the approach or
run-in which provides an elevation drop that accelerates the
rider to some minimum speed needed for the jump. It is
common but not necessary for an approach to have a steep
(20–25°) initial ‘‘drop-in’’ region followed by a straight
slope of modest pitch (*10–15°) leading to the transition
before takeoff. The purpose of the transition is to provide a
smooth region from the downward approach angle to the
upward takeoff angle. Therefore, the transition must be
curved causing the rider to experience radial acceleration
and shifting balance points. Careful consideration must be
given to the amount of curvature to avoid excessive radial
accelerations, say greater than 2g, which could throw the
rider into an uncontrolled posture [22]. The takeoff pro-
vides the last surface of contact for the rider before
jumping and therefore the greatest of care must be given to
its surface to ensure that it is well groomed and straight. As
discussed in detail below, there should be no (concave)
curvature in the takeoff which can induce a potentially
dangerous inverting rotation. The maneuver area is the
region of the jump where the rider is assumed to be in the
air; so its properties are often assumed to be less important.
This would be true if all jumpers left the takeoff with some
minimum speed, but due to the many variables in snow
conditions and rider actions, it will often be the case that
riders will not maintain sufficient speed into the takeoff to
clear the maneuver area; therefore the maneuver area
should actually be treated as part of the landing area and its
properties examined.
The start of the run-out marks the end of the landing
area. The transition between these is called the bucket.
Through continuous use, the bucket will collect snow that
is dislodged from the landing area above by the jumpers.
This causes the bucket to creep up the landing area, slowly
decreasing the landing area’s total length. Designers should
accommodate this possibility by increasing the initial
design landing length by an amount depending on their
local snow conditions and expected use, as well as putting
in place monitoring and maintenance procedures for the
grooming crews.
2.2 Terminal speed and design considerations
for the landing area
The length of the landing area is a critical parameter since
overshooting the landing presents a particular impact haz-
ard due to the flatter angle of the run-out. Ideally, the
landing area should be designed to accommodate the
greatest takeoff speed possible as well as rider ‘‘pop’’
effects which can increase the jump distance. To model the
performance of the approach, transition, and takeoff, one
requires the dynamic equations describing the center-of-
mass motion of the rider while in contact with the snow.
Let y
A
(x) represent the vertical elevation of the snow sur-
face of the approach, transition, and takeoff where xis the
horizontal distance. Assume that the surface is smooth with
existing first and second derivatives up to the lip of the
takeoff. Assume also that the rider starts from rest and
makes no speed-checks, and ignore any motion transverse
to the downward direction. This will give a maximum
value for the takeoff speed for a given set of physical
parameters. The equations of motion for the center-of-mass
of the rider while in contact with the surface are given by
McNeil and McNeil [21]
d2r
~
dt2¼g^
yþð^
nl^
vÞNgv2^
v;ð1Þ
where gis the gravitational acceleration constant, r
~¼ðx;yÞ
is the position vector of the rider suppressing the transverse
(z) motion, ^
vis the unit velocity vector, ^
nis the unit vector
normal to the surface, Nis the normal force of the surface
on the rider, and gis the drag parameter defined below in
Eq. 2. The range of physical parameters used in this work
are given in Table 1. To treat wind, the velocity vector in
the (last) drag term is replaced by the air-rider relative
velocity, v
~w
~;where w
~is the wind velocity vector. The
drag force is given by Streeter et al. [23],
F
~drag ¼CdAq
2v2^
v¼mgv2^
v;ð2Þ
where Ais the cross-sectional area of the rider
perpendicular to the direction of travel, qis the density
HD
T
L
Approach
Transition
Takeoff
Deck
Landing
Bucket
Run-out
Start
(Maneuver area)
knuckle
Fig. 1 Geometry of the standard tabletop jump. Although used as an example extensively in this work, the authors do not endorse the tabletop
due to its safety problems discussed further in Sect. 3
4 J. A. McNeil et al.
of air, v
~is the velocity, and C
d
is the drag coefficient.
Hoerner [24] provides approximate values for the product
CdAfor various positions, i.e. for humans standing forward
(0.836 m
2
), standing sideways (0.557 m
2
), and tucked
facing forward (0.279 m
2
). The density of air, q;depends
on elevation, Y, and absolute temperature, T, according to
the approximate relation [25],
qðY;TÞ¼q0
T0
TeT0
TY
Y0;ð3Þ
where Yis the altitude above sea level in meters, T
0
is the
reference temperature (298.15 K), Tis the absolute tem-
perature (T=T
C
?273.15, where T
C
is the temperature in
Celsius), q
0
=1.1839 kg/m
3
, and Y
0
=8.772 910
3
m.
For example, this relation gives the air density at an
elevation of 3,000 m as qð3;000 mÞ¼0:928 kg/m
3
at
T
C
=-10C.
The normal force is given by McNeil and McNeil [21]
N¼mðgcos hAðxÞþjðxÞv2Þ;ð4Þ
where hAðxÞ¼tan1ðy0
AÞis the local value of the pitch
angle of the hill and j(x) is the local surface curvature,
jðxÞ¼ y00
AðxÞ
ð1þy0
AðxÞ2Þ3=2:ð5Þ
If there were no friction or drag, the maximum speed
would be determined simply by the difference in elevation
between the start and the lip of the takeoff, i.e. vmax ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2gðyAð0ÞyAðxlipÞÞ
pwhere x
lip
is the x-coordinate of the
lip of the takeoff. This provides an upper bound to the
takeoff speed. Of course, there will always be some friction
that reduces the maximum speed by an amount that
depends on the shape of the approach and takeoff and the
coefficient of kinetic friction which in practice varies
between roughly 0.04 and 0.12 [26,27]. (Note that this is a
practical range for the friction coefficient which can be
[0.12 for unusually wet circumstances.) For the special
case of a straight approach at angle h
A
(constant) with
respect to the horizontal, the energy lost per unit length is
just the constant frictional force, lmg cos hA:
Air drag also reduces the speed at takeoff and provides
the velocity dependent force that determines the terminal
speed. While the ideal is to have the length of the landing
accommodate the maximum takeoff speeds, snow budgets
may not allow for very long landings and an engineering
trade-off is needed. For the case of a single jump the start
point relative to the takeoff can be adjusted to limit the
takeoff speed. To calculate the appropriate start point for a
given set of conditions, one must solve the equations of
motion numerically. One could imagine having two (or
more) possible start points and opening the one best suited
for the snow conditions. For multiple jumps and for long
run-ins, one must consider the possibility of approaching
the terminal speed. For such cases the US Terrain Park
Council (USTPC) has adopted a landing length design cri-
terion based on a maximal speed of 80% of nominal ter-
minal speed [28]. For the special case of an approach with
constant pitch, h
A
fixed, the terminal speed is given by:
vT¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2mgðsin hAlcos hAÞ
CdAq
s:ð6Þ
For example, for a 15°slope using an average rider mass of
75 kg with a low coefficient of friction of l=0.04, an air
density of 0.928 kg/m
3
, and a drag-area of C
d
A=0.557 m
2
(appropriate to a rider facing sideways) Eq. 6gives a ter-
minal speed of v
T
=24.5 m/s. Front-facing skiers will have
larger drag-area and thus lower terminal speeds.
Note that the terminal speed also provides an upper
bound on the size of a jump depending on the shape of the
landing surface. Of course, it takes some distance before
terminal speed is attained. For the purposes of designing
the approach it is necessary to calculate the approach to
terminal speed by numerically solving Eq. 1. Figure 2
shows the approach to terminal speed for the range of
friction coefficients between 0.04 \l\0.12. From Fig. 2
for the low friction case, one can see that it takes
approximately 150 m for a rider to reach *20.3 m/s (80%
of the terminal speed); so any approaches less than this will
meet the USTPC design criterion for the entire range of
friction coefficients.
Finally, the width of the landing area must accommo-
date the possibility that the rider’s takeoff is not straight
down the jump but includes a component of takeoff
velocity transverse to the downhill direction. The approx-
imate transverse distance the rider will land from the
downhill center line of the jump will be given by the
product of the time-of-flight and this transverse velocity
component. Since the time-of-flight is given by the hori-
zontal jump distance divided by the horizontal component
of the takeoff velocity (ignoring drag), the transverse
deviation will be equal to the horizontal jump distance
times the ratio of the transverse to horizontal components
Table 1 Physical parameters
Parameter Symbol Units Value range
Acceleration of gravity g m/s
2
9.81
Mass of jumper mkg 75
Drag coefficient times
frontal area of jumper
C
d
Am
2
0.279–0.836
Density of air qkg/m
3
0.85–1.2
Coefficient of kinetic
friction
lDimensionless 0.04–0.12
Lift to drag ratio q
ld
Dimensionless 0.0–0.1
Designing tomorrow’s snow park jump 5
of the takeoff velocity. An approximate limit on this ratio is
given by the ratio of the width of the takeoff to the length
of the takeoff. In practice, this has not been shown to be a
significant safety concern as most riders land with a few
meters of the center line, but more on-hill data is needed to
better constrain the range of transverse takeoff velocities.
A useful guideline is to make the width of the landing area
at least three times the width of the takeoff at the design
landing point and taper it outward by at least 10from
there to the end of the landing area.
2.3 Jump approach design process
Like many engineering design processes, the jump design
process is an iterative one that intimately employs mod-
elling to inform design decisions. A hypothetical design
process might take the following steps. First, the perfor-
mance criteria are established. These may include the tar-
get jump distance, minimum radial acceleration in the
transition, minimum straight region for the takeoff, and
maximum permitted EFH (defined in the next section).
Next the constraints are listed, e.g. snow budget and base
area on which the jump is to be sited. On a topographical
cross sectional profile of the base hill, the snow base and
preliminary sketch of the jump with each component is
prepared with estimates of all relevant scales such as the
snow base depth, start area, length of approach, length and
radius of transition, length and width of takeoff, height of
lip above start of maneuver area, shape, length and width of
the landing area, bucket and run-out. Next, one models the
jump design with a mathematical representation of the
entire landing surface, defines a range for the physical
parameters such as coefficients of friction, drag, and lift, a
range for the rider’s mass and height, and a range for rider
actions such as ‘‘pop’’ speeds (jumping or dropping prior to
takeoff) and transverse takeoff angles (relative to straight
down the jump). This represents the complete parameter
space of interest.
From the mathematical representation of the jump sur-
face, the volume of snow required is calculated and com-
pared with the snow budget. If needed, adjustments to
component dimensions can be made, leading, perhaps, to
the conclusion that the desired jump size is not compatible
with the snow budget. For each set of physical and rider
parameters, Newton’s laws of motion are solved numeri-
cally as described below and the rider’s kinematics deter-
mined. By repeating the calculation for the various ranges
of parameters the entire space of possible rider outcomes is
mapped out and the ranges of key performance character-
istics calculated such as jump distances, radial accelera-
tions, and EFHs. These may then be compared to the
desired performance criteria. From this information design
changes can be made and the process repeated until the
performance criteria are met within the constraints or a
decision is made to revise either the performance criteria
(e.g. build a smaller jump), or revise the constraints (e.g.
increase the snow budget).
3 Impact hazard
3.1 Equivalent fall height
Clearly, one of the most important factors affecting the
relative safety of a jump is the total energy absorbed upon
landing. Several authors have used the concept of the
equivalent fall height (EFH) to characterize this important
parameter [20,29,30]. Suppose an object falls vertically
onto a horizontal surface from a height, h. Ignoring drag,
the speed at impact, v, is given by h=v
2
/2g. On a sloped
landing, the impact depends only on the component of the
velocity normal to the landing surface, and the relevant
energy relation then gives h=v
\
2
/2g. The EFH can be
made arbitrarily small by making the angle of the landing
surface closely match the angle of the jumper’s flight path
at landing. The component of the landing velocity normal
to the landing surface is given by v?¼vJsinðhJhLÞ;
where v
J
is the jumper’s landing speed, h
J
is the jumper’s
landing angle, and h
L
is the angle of the landing slope.
Thus, the EFH is,
h¼v2
Jsin2ðhJhLÞ
2g:ð7Þ
Due to friction and landing surface distortion, there will
be some energy loss associated with the component of
landing velocity parallel to the surface. As shown by
= 0.04
= 0.12
0 50 100 150 200 250
0
5
10
15
20
25
30
Fig. 2 The approach to terminal velocity as a function of distance
down an approach with a constant pitch angle of hA¼15for the
range of kinetic friction coefficients between 0.04 \l\0.12
obtained by numerically solving Eq. 1. The dashed lines represent
the terminal speeds for the limiting friction coefficient values
6 J. A. McNeil et al.
McNeil [31], if surface distortion effects are neglected, the
maximum energy change, DU;including the transverse
component is bounded by:
DU½2ð1lÞ2mgh:ð8Þ
This represents the maximum energy absorbed by the
jumper, assuming the landing surface absorbs none. Of
course, in all realistic cases the surface will experience
some distortion which will lower the amount of energy
absorbed by the jumper. This effect is carried to the
extreme with landing air bags that absorb almost all of the
energy. For snow-snowboard/ski surfaces the coefficient of
kinetic friction lies in the range 0.04 \l\0.12 [26]; so
the effect of including the tangential component of the
change in landing velocity on the absorbed energy is
generally small, and the total absorbed energy is well
characterized by just the EFH.
To calculate the EFH for an arbitrary jump shape, one
must solve the equations of motion for the jumper. The
general equations of motion governing the center-of-mass
motion including lift and drag are given in Ref. [31]:
d2r
~
JðtÞ
dt2¼g^
ygjv
~w
~jðv
~w
~qLD ^
sðv
~w
~ÞÞ;ð9Þ
where q
LD
is the lift to drag ratio. Assuming the rider
maintains a fixed orientation facing forward, ^
sis the unit
vector in the ‘‘sideways’’ direction, and the remaining
parameters are the same as in Eq. 1. In practice the lift
effect is very small (*1%) and depends on the orientation
and posture of the jumper in the air; so to simplify matters
and get a bound on the lift effect, the lift term in Eq. 9
assumes the jumper maintains his orientation with a
constant lift to drag ratio of qLD ’0:1 estimated from
the angle of descent of sky-diving snowboarders. Better
measurements of this parameter will be necessary for high
fidelity modelling. A jumper performing air maneuvers
could be modelled using a time-dependent drag/lift ratio,
but the net effect should be bounded by the constant q
LD
result. In general, these equations must be solved
numerically but, as shown by McNeil [31], for small to
medium-sized jumps ([*12 m) the drag can be ignored at
about the 10% level (and lift effects at the *1% level)
allowing for the closed form analytic solution that is
accurate to that level:
v
~
JðtÞ¼ðvJx;vJyÞ¼v0cos hT^
xþðv0sin hTgtÞ^
yð10Þ
r
~
JðtÞ¼ðx;yÞ¼v0cos hTt^
xþv0sin hTt1
2gt2
^
y;ð11Þ
where v
0
is initial speed at takeoff assuming the rider
adds no ‘‘pop’’. As shown by Hubbard [20] and McNeil
[31], one can treat the jumper’s ‘‘pop’’ by adding to
the ‘‘no-pop’’ initial velocity, v
~
0;an additional velocity
component normal to the takeoff surface, v
~
p¼fvpsin hT;
vpcos hT;0g:This will alter the initial velocity vector
(speed and direction) accordingly:
v0!v0þp¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
v2
0þv2
p
q;
hT!hTþp¼hTþdhpð12Þ
where dhp¼tan1ðvp=v0Þand where the (þp)-sub-
script denotes the initial conditions appropriate to the
‘‘pop’’ case. From field data taken by Shealy et al. [32],
‘‘pop’’ speeds have been shown to vary between
-2.6 \v
p
\?1.2 m/s [31].
From Eq. 11, one obtains the classic parabolic relation
for the jumper’s flight path, y(x),
yðxÞ¼xtan hTg
2v2
0cos2hT
x2:ð13Þ
As shown in Refs. [20,22,30], one can find the
jumper’s landing angle from the trajectory relation which
can be used in the definition of the EFH to derive the
following relation for the EFH, h(x), as a function of the
landing surface, y
L
(x):
hðxÞ¼ x2
4 cos2hTðxtan hTyLðxÞÞ yLðxÞ
sin2tan12yLðxÞ
xtan hT
tan1y0
LðxÞ
:
ð14Þ
where hLðxÞ¼tan1y0
LðxÞhas been used.
This expression for EFH characterizes the severity of
impact upon landing as a function of the horizontal distance
of every jump. In fact, once a rider leaves the takeoff with a
given initial velocity, his flight path, landing point and EFH
are determined. When drag and lift are neglected, simple
analytic expressions such as Eq. 14 for the EFH can be
obtained; however, it is straightforward to solve the Eq. 9
numerically to determine h(x) including drag and lift effects.
The most common recreational jump built is the standard
tabletop, shown schematically in Fig. 1. Most experienced
tabletop jumpers quickly learn that there is ‘sweet spot’ just
past the knuckle that is the ideal landing area resulting in a
light landing impact. Using Eq. 14, one can explore the
EFH for a variety of hypothetical tabletop jumps. The
tabletop jump is parametrized by the takeoff angle, h
T
, deck
length, D, takeoff lip height, H, and landing surface angle,
h
L
. The tabletop design is evaluated because of its current
widespread use and analytic simplicity; however, as shall be
shown below, this design is not optimal for limiting EFH.
For the special case of the tabletop jump the landing surface
is described by the function,
Designing tomorrow’s snow park jump 7
yLðxÞ¼ tan hLðxDÞH;xD
H;x\D:
ð15Þ
where His the height of the takeoff above the deck, Dis
the deck length and h
L
is the landing angle as shown in
Fig. 1. As shown by Swedberg and Hubbard [29], using
this form, one can obtain the EFH function specifically for
the tabletop jump:
One can examine the sensitivity of the EFH of the
tabletop jump by varying each of the relevant parameters,
{h
T
,H,D,h
L
}, keeping the remaining parameters fixed.
The resulting EFH function can also be compared to some
design criterion.
A natural question that arises is: how small a value for h
is small enough? To answer this question it is important to
realize that good designs do not limit themselves to best
case scenarios. As the EFH rises the probability of serious
injury rises with it. Clearly, subjecting a person to an EFH
of 10 m would be considered dangerous. Even if the
jumper is in complete control and lands in the best possible
body configuration, serious injury is likely to occur when
the EFH is this large. Similarly, an EFH of 3 m is probably
still too large, not because an athletic ski jumper under
ideal conditions could not possibly survive an impact with
this EFH, but rather because if anything deviated from the
best case scenario (slipping on takeoff, getting one’s skis or
board crossed in flight, etc.) then serious injury could still
occur. Another basis to inform an EFH criterion is the
ability of the jumper to control his descent after landing.
Recent experiments on jumpers have shown that above
about h=1.5 m, the ability of a young athletic human to
absorb the vertical impulse and maintain control without
the knees buckling is compromised. This is the basis for the
USTPC criterion of a maximum acceptable EFH of 1.5 m
[28]. Landing areas meeting this criterion shall be referred
to as the ‘‘soft-landing’’ region of the jump.
Figure 3a shows the sensitivity of the EFH to H, the
height of the lip of the takeoff, while fixing the remaining
parameters at: fhT¼20;D¼6:1mð20 ftÞ;hL¼30g:
For the case of H=1.0 m the EFH for landing on the deck
is below 1.5 m for the entire deck length. Beyond the deck
the EFH drops rapidly for landing on the sloped landing
area before increasing to 1.5 m at a distance of about 16 m.
Thus, for the low deck height of H=1.0 m, the EFH
exceeds the USTPC criterion value only beyond 16 m. This
constitutes a landing zone with a 10 m long soft-landing
region. For the larger deck height of H=2.0 m, the EFH
exceeds 1.5 m for the entire length of the deck before
dropping to acceptable levels after the knuckle but
increasing quickly again to 1.5 m at about 13 m. Thus the
soft-landing region for the large deck height case is more
narrow, about 7 m long, and in addition to the hazard of
landing short of the knuckle one would expect some
jumpers to land outside this soft-landing zone.
Figure 3b shows the sensitivity of the EFH to h
T
, the
takeoff angle, while fixing the remaining parameters at:
fH¼1:5m;D¼6:1m;hL¼30g:Since the lip height in
this case is 1.5 m already, in both takeoff angle cases the
EFH for landing on the deck exceeds 1.5 m the entire
length of the deck. For the hT¼10case, the EFH for
landing on the sloped landing area drops to a very small
value before increasing to 1.5 m again at a distance of
about 18 m. Thus, the soft-landing region for this case
is fairly long, *12 m. For the steeper takeoff angle of
hT¼30;the EFH rises quickly along the deck before
dramatically dropping to acceptable levels after the
knuckle but increasing to 1.5 m again at about 12 m. Thus
the soft landing region for this jump is about 6 m long and
one would expect some jumpers to land outside this zone.
Figure 4a shows the sensitivity of the EFH to h
L
, the
angle of the landing surface (here assumed constant), while
fixing the remaining parameters at: fH¼1:5m;D¼
6:1m;hT¼20g:The landing angle does not affect the
EFH for landing on the deck so for both cases the EFH for
landing on the deck rises with distance to about 1.8 m at
the knuckle. For the case of hL¼10the EFH for landing
on the sloped landing area starts at about 1.1 m just after
the knuckle and increases quickly to 1.5 m at a distance of
about 9 m. Thus, the soft-landing region for this case is
relatively short, only *3 m. For the steeper landing angle
of hL¼30;the EFH starts considerably lower and rises to
1.5 m at about 14 m. Thus, the soft-landing region for this
case is fairly long, *8 m; so it would appear that the
greater risk of having an EFH [1.5 m would arise from
landing before the knuckle.
Figure 4b shows the sensitivity of the EFH to D, the
deck length, while fixing the remaining parameters at: fh¼
1:5m;hT¼20;hL¼30g:The first case is for D=3m
(10 ft). The EFH rises from 1.5 to 1.6 m along the deck
httðxÞ¼ cos2hLðHDtan hLÞþ x2sin2ðhLþhTÞ
4 cos2hT½xðtan hTþtan hLÞþHDtan hL;xD
Hþx2tan2hT
4ðxtan hTþHÞx\D:
(ð16Þ
8 J. A. McNeil et al.
before dropping to about 0.4 m after the deck and then
rising to 1.5 m at a distance of 9 m. Thus, the soft landing
region for this case is about 6 m long. For the deck length
of D=6.1 m (20 ft), the EFH rises to about 1.8 m at the
knuckle before dropping to 0.2 m just after the knuckle and
then rising to 1.5 m at a distance of 14 m. Thus, the soft
landing region for this case is about 8 m long. For the deck
length of D=9.1 m (30 ft), the EFH rises to about 2.2 m
at the knuckle before dropping to *0.1 m just after the
knuckle and then rising to 1.5 m at a distance of *19 m.
Thus the soft-landing region for this case is quite long,
*10 m, but the EFH for landing before the knuckle sig-
nificantly exceeds the USTPC criterion value.
In most resorts the jumps are not engineered and new
jumps are empirically tested by professional staff riders.
Due to their considerable experience test jumpers are sel-
dom injured, yet such blind tests present a hazard to staff
that could be avoided with a relatively simple engineering
analysis such as presented here.
If a snow park jump can be designed with inherently
small values of EFH and large soft landing regions, then
many jump injuries can be avoided or their severity
reduced since energy absorbed upon landing is reduced. An
examination of the EFH for the tabletop jump shown in
Fig. 3and studied in detail by Swedberg and Hubbard [29]
shows a disturbing trend in that the EFH increases roughly
linearly as jump length increases despite adjustments to lip
height, H, deck length, D, and landing slope angle h
L
.As
these results show, landing just before the knuckle or well
after the soft landing region can result in large EFHs and
m
m
0 5 10 15 20 25
0
1
2
3
4
0 5 10 15 20 25
0
1
2
3
4
(a) (b)
Fig. 3 Tabletop EFH versus jump distance xfor avarying the takeoff
lip height, 1 BHB2 m, with h
T
=20°,D=6.1 m, and h
L
=30°
fixed, and bvarying the takeoff angle, 10°Bh
T
B30°, with
D=6.1 m, H=1.5 m, and h
L
=30°fixed. The dashed (green)
line marks the USTPC criterion value for the maximum allowable
EFH (1.5 m) (colour figure online)
0 5 10 15 20 25
0
1
2
3
4
(a)
D3m
D 6.1 m
D 9.1 m
0 5 10 15 20 25
0
1
2
3
4
(b)
Fig. 4 EFH versus jump distance xfor landing on a constant
downward slope landing for avarying the landing angle, 10°Bh
L
B30°, with D=6.1 m H=1.5 m, and h
T
=20°fixed, and
bvarying the deck length, D=3.0 m, D=6.1 m, and D=9.1 m
with h
T
=20°,H=1.5 m, and h
L
=30°fixed. The dashed (green)
line marks the USTPC criterion value for the maximum allowable
EFH (1.5 m) (colour figure online)
Designing tomorrow’s snow park jump 9
thus present a greater risk for an impact-related injury. The
straight landing used in tabletop designs will always have
an increasing EFH with jump distance suggesting that
alternative landing shapes should be seriously considered.
3.2 Interacting parameters: approach length, takeoff
speed and landing length
As discussed above, the primary impact hazards presented
by the standard tabletop jump are either landing too short
(before the knuckle) or too long, especially beyond the
landing area. Indeed, overshooting the landing area is one
of the least desirable outcomes of a jump. Of course, the
distance of a jump depends on the takeoff velocity (both
direction and magnitude). (As discussed above and in Refs.
[20,31], the effect of the rider adding ‘‘pop’’ merely
modifies the initial velocity according to Eq. 12.) As seen
from Eq. 1, the (no-pop) takeoff speed depends on the
shape of the surface up to the end of the takeoff ramp and
the physical parameters determining friction and drag. This
implies that there will be a coupling of the performance
characteristics of the components.
For example, the length of the approach will determine
the range of takeoff speeds which, in turn, will determine
the range of landing distances. In the examples below the
tabletop jump parameters were fixed as follows: fhA¼
15;LTO ¼4:0m;hT¼20;H¼1:5m;D¼9:1m;hL¼
30g;where L
TO
is the length of the takeoff ramp. The
physical parameters are listed in Table 1with the following
specific values: fCdA¼0:557 m2;q¼0:90 kg=m3;m¼
75 kgg:Figure 5a shows the takeoff speed versus the length
of the approach obtained by numerically solving Eq. 1for
this hypothetical tabletop jump. Figure 5b shows the
resulting total horizontal jump distance, x
J
, for the full
range of friction coefficients. These figures can be used to
inform the jump designer’s decision regarding the length of
the approach and length of the landing. For the low friction
case a modest approach length of *40 m will provide
enough takeoff speed to reach the start of the landing;
however, if the snow conditions change such as to increase
the friction coefficient, then there is a strong likelihood that
riders will land on the deck and be subjected to a large EFH
as discussed previously. To avoid this outcome for this set
of example parameters allowing an approach length of at
least *75 m will provide sufficient takeoff speed to reach
the landing area under the large friction conditions (without
having to add ‘pop’).
Consider next the length of the landing area. As an example
suppose an approach length of 100 m. Under large friction
conditions the landing distance is about 18 m, or 9 m (hori-
zontal) beyond the knuckle. Thus, having a landing length
(measured along the hill) of ð9:0=cos hLÞm¼10:4mwill
accommodate all jumpers under the high friction condition.
However, if the snow conditions change and the coefficient
of friction drops, there is a strong likelihood that jumpers
who fail to check their speed will land beyond the landing
area. Since the maximum jump distance for a 100 m
approach length for this takeoff angle under low friction
conditions is about 32 m, the landing length should be a least
ð32:0=cos hLÞm’37 m. If the snow budget is insuffi-
cient for that length for the landing area, the designer can try
a different takeoff angle or deck length or other para-
meter thereby continuing the iterative engineering design
process.
Next, the dependence of the peak radial acceleration
in the transition on the length of the approach, L
A
,is
0.04
0.12
0 50 100 150 200 250 300
0
5
10
15
20
25
30
(a)
0.04
0.12
050 100 150
0
10
20
30
40
50
(b)
Fig. 5 a Takeoff speed and bthe landing distance as a function of the
length of the approach, L
A
, for a standard tabletop jump for the full
range of friction coefficient, 0.04 BlB0.12. For both curves, the
tabletop jump parameters were fixed as follows: fhA¼15;LTO ¼
4:0m;hT¼20;H¼1:5m;D¼9:1m;hL¼30g;and the remain-
ing physical parameters were: fCdA¼0:557 m2;q¼0:90 kg=m3;
m¼75 kgg
10 J. A. McNeil et al.
examined. Figure 6a shows the peak radial acceleration in
the transition measured in units of the acceleration con-
stant, g, versus the length of the approach for the full range
of friction coefficients, 0.04 BlB0.12 with the remain-
ing parameters fixed as described previously. For the
transition radius of 15 m the radial acceleration exceeds the
USTPC recommended maximum of 2 g’s at an approach
length of about 90 m while the 20 m transition radius will
allow approach lengths greater than about 150 m.
Finally, the dependence of the EFH on the length of the
approach, L
A
, is examined. Figure 6b shows the EFH
versus the length of the approach for the full range of
friction coefficients, 0.04 BlB0.12 with the remaining
parameters (given in the figure caption) fixed. One can
clearly identify the approach length that provides sufficient
takeoff speed to clear the knuckle. For the low friction case
this occurs at an approach distance of only 45 m while for
the high friction case an approach length of about 95 m is
required. This figure clearly emphasizes the dilemma that
the jump designer faces when constrained to the simple
tabletop form. For both low and high friction cases there
exist regions where the EFH is below 1.5 m; however,
there is no single value of the approach length for which
the EFH is below 1.5 m for both friction extremes. One can
manage this situation by having multiple start points
depending on the snow conditions, but this requires that
resorts building such jumps be able to measure the friction
coefficient and devise plans to monitor it continuously.
These considerations strongly suggest that the jump
designer explore other options for landing shapes for jumps
of this size (9.1 m) or larger.
There is an insidious psychological effect that infects the
tabletop design as well. The large EFH arising from
landing short of knuckle on a tabletop creates a psycho-
logical avoidance response in jumpers. To avoid hard
knuckle landings jumpers tend to take the jump faster than
they might otherwise do which in turn increases the risk of
overshooting the landing with potentially even worse out-
comes than landing on the knuckle. The narrower the
region of ‘‘soft’’ landing, the greater the risk.
While the preceding analysis was performed for the
standard tabletop design, it should be emphasized that any
design shape can be similarly analysed.
4 Inversion hazard
One especially hazardous situation occurs when the jumper
lands in an inverted position which can lead to catastrophic
injury or death from spinal cord trauma. While jumpers
can execute inverted maneuvers intentionally, concave
curvature in the takeoff can lead to involuntary inversion.
Curvature in the takeoff can be intentionally built or can be
arise through heavy use. Understanding the dynamics of a
jumper riding over a takeoff with concave curvature is
necessary to developing designs and maintenance proce-
dures which mitigate, if not eliminate, this hazard.
McNeil [33] modelled the inverting effect of a curved
takeoff by treating the jumper as a rigid body which
approximates a stiff-legged jumper. This work is briefly
reviewed here. For a simple illustrative example, consider
the inverting rotation for trajectories calculated from the
R 15 m
R 20 m
0 50 100 150
0.0
0.5
1.0
1.5
2.0
2.5
3.0
(a)
0.04
0.12
0 50 100 150
0
1
2
3
4
5
6
(b)
Fig. 6 a Peak radial acceleration in the transition for R=15 m and
R=20 m as a function of the length of the approach, L
A
, for a
standard tabletop jump with the friction coefficient fixed at its lower
limit, l=0.04, and bequivalent fall height, h, as a function of the
length of the approach, L
A
with the friction coefficient varied between
0.04 BlB0.12. For both figures, the remaining tabletop jump
parameters were fixed as follows: fhA¼15;LTO ¼4:0m;hT¼
20;H¼1:5m;D¼9:1m;hL¼30g;and the remaining physical
parameters were: fCdA¼0:557 m2;q¼0:90 kg=m3;m¼75 kgg
Designing tomorrow’s snow park jump 11
standard tabletop jump which is the most widely used
design, although not built to any standard. The mathe-
matical simplicity of this jump shape allows for similarly
simple expressions for the quantities of interest. Results for
more general shapes are straightforward but may require
numerical solution. In the previous section it was shown
that the tabletop design is not optimal from the point of
view of its potential impact hazard as characterized by a
large EFH as discussed above (see also Refs. [20–22,29]).
However, the focus here is exclusively on the interaction of
the jumper with the takeoff. The other components of the
jump only affect the time of flight and impact as discussed
above. Ignoring drag, for a jump of horizontal distance of
x
L
the time of flight is given by:
tF¼xL
v0cos hT
:ð17Þ
The concavity of the takeoff is characterized by the
radius of curvature, R
TO
. Assuming the rider is a rigid body
during takeoff implies that the takeoff velocity is parallel to
the takeoff ramp surface at its end (the lip). The angular
speed for inversion induced by the curved takeoff is given
by the takeoff speed, v
0
, divided by the radius of curvature
x=v
0
/R
TO
where the inverting rotation is about an axis
normal to the plane of the jumper’s trajectory. This can be
understood by imagining a rider executing a full vertical
circle at constant speed who must rotate by 2pin each
revolution. Once the jumper leaves the surface, neglecting
drag and lift, no further torques can be exerted on the
jumper so his/her angular momentum is conserved. Under
the rigid body assumption, the moment of inertia is fixed so
the angular velocity is also fixed. Under these conditions,
the total inverting rotation is given by the angular velocity
times the time of flight,
/¼xL
RTO cos hT
:ð18Þ
One notes that the inverting rotation is proportional to the
horizontal jump distance and inversely proportional to the
radius of curvature; so smaller jumps can tolerate a smaller
radius of curvature. The magnitude of the total backward
inverting angle relative to the landing surface under the
rigid body assumption is thus,
U¼/þhTþhL:ð19Þ
Non-rigid body motion, arising for example by the rider
executing a maneuver in the air, will change the rider’s
moment of inertia and the rotational speed will not be
constant. Modelling such cases requires a detailed analysis
of the maneuver’s affect on the moment of inertia during
the jump. It should be noted that the most common
maneuver is the ‘‘grab’’ which will tend to reduce the
moment of inertia thereby increasing the inverting angle.
For jumps with relatively large takeoff and landing
angles hT¼hL30;very little additional inverting rota-
tion can result in a potentially disastrous landing on the
head or neck. In the example medium-sized tabletop
(*6 m) jump treated in Ref. [33] the takeoff was found to
have a radius of curvature of about 8.1 m. For an
approximate takeoff speed of 7.2 m/s, the inverting angle
was estimated to be about 63°which resulted in the (rigid-
body) rider landing in an inverted position at the potentially
dangerous angle of 126°with respect to the landing surface
normal.
What are the design implications? Clearly, curvature
should be avoided in the last few meters of the takeoff, but
how long should the straight section of the takeoff be? As can
be seen from Fig. 1the takeoff immediately follows the
transition which takes the rider from a predominantly
downhill direction to the upward direction of the takeoff. The
straight section of the takeoff ramp should be long enough to
allow the rider to recover from the transition. The USTPC
has proposed that the straight section of the takeoff be at least
the nominal design takeoff speed times 1.5 human reaction
times, or about 0.3 s. This is close to the standard used for
nordic jumps which is 0.25 s times the nominal takeoff
speed. For the medium jump treated in Ref. [33] the nominal
takeoff speed was about 7.2 m/s which implies a recom-
mended minimum straight section of takeoff of about 2.2 m.
5 Designing tomorrow’s terrain park jump
Previously it was shown that the impact risk associated
with landing is naturally quantified by the EFH, and that a
‘‘soft’’ landing arising from a correspondingly small EFH is
possible if the jumper path and landing surface have nearly
the same angle at the point of impact.
Although the most commonly used jump, the tabletop,
was used in the previous example calculations of impact
and inversion hazards, it was emphasized that this was not
intended to be an endorsement of this kind of jump. As
shown above, tabletop jumps have narrow ‘‘soft landing’’
regions and, even if built for such landings under one set of
snow conditions, these can change to a ‘‘hard landing’’
when the snow conditions change. Nevertheless, tabletops
continue to be built. This may be due their ease of con-
struction, the vast experience builders now have with that
form, and the fact that this is what everyone else is doing, a
form of safety in numbers. Tabletops are indeed relatively
simple to design and fabricate with the only design deci-
sions being the quantities: L
A
, the length of the approach
with some arbitrary curved transition; L
T
, the length of the
takeoff; h
T
, the angle of the takeoff; H, the height from the
lip of the takeoff to the (generally horizontal) deck surface;
D, the length of the deck surface; h
L
, the angle of the
12 J. A. McNeil et al.
intended landing surface; and L
L
, the length of the intended
landing region. Typically, the base or parent slope upon
which the jump is sited is fairly straight and inclined at
some average angle h
A
which is used for the approach
angle to conserve snow. The tabletop is relatively easy to
fabricate due to the many straight lines which can be
constructed readily with modern grooming equipment.
But, while it may be easy to build a landing region
consisting only of two straight line segments, as shown
above the resulting jump is not necessarily ideal from the
perspective of impact hazards. Simplicity of design (in the
sense of few choices to be made) thus carries an associated
penalty: it can subject the jumper to large impacts on
landing. In Sect. 3above, it was shown that the EFH
function associated with such a generic tabletop jump has
undesirable characteristics. Specifically, on medium to
large-sized jumps, the EFH is small only in a relatively
narrow range of takeoff speeds (and even this requires the
correct choice of the intended landing region surface angle).
Furthermore, the EFH can be dangerously large when
landing occurs at the end of the deck or beyond the intended
landing region. Thus tabletop jumps will have acceptable
EFH’s only in too narrow a region that can change dra-
matically with the snow conditions. This places large
demands on the jumper to manage the takeoff velocity
precisely within a narrow range and too large a penalty
(severe and possibly dangerous impacts) if the velocity is
outside this range for whatever reason. Indeed, as shown in
Sect. 3there may be conditions where the approach length
provides an acceptable takeoff speed under one set of snow
conditions, only to be unacceptable when the snow condi-
tions change. As shown previously, this sensitivity arises
from the constraint that the deck and landing surface be
straight. The possibility of relaxing this constraint is now
addressed by considering curved landing surfaces that, by
design, provide acceptable EFH independent of the takeoff
speed up to the limits of space and fabrication capabilities.
The benefits of such a surface are clear, but how might
this be done? The mathematical condition for a landing
surface shape that produces an acceptable EFH begins with
Eq. 7in which EFH vanishes when the jumper path and
landing surface have the same angle at the point of impact. A
more complete discussion of the theory behind calculation
of safer jump landing surface shapes is provided by Hubbard
[20]. In summarizing, a condition on the surface shape,
y
L
(x), is sought such that the EFH is limited to a specific
value hat all values of x. Indeed, Eq. 14 is such a condition,
although in that form it was used to calculate the EFH.
Solving for y0
LðxÞfrom Eq. 14, one can obtain an
expression for the derivative of the landing surface which
is now interpreted as a differential equation constraining
the landing surface,
y0
LðxÞ¼tan tan12yLðxÞ
xtan hT
þsin1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
h
x2
4ðxtan hTyLðxÞÞcos2hTyLðxÞ
s#:ð20Þ
In Eq. 20, which has been called the ‘‘safe slope
differential equation’’ [20], the first derivative of the
landing surface function is a function of two variables
(xand y
L
(x)) and three constant parameters (g,h
T
and h).
Since no jump can ever be definitively called ‘‘safe’’, in
this work such surfaces will be referred to as ‘‘constant
EFH’’ surfaces. Any surface y
L
(x) that satisfies this
differential equation imparts to an impacting jumper a
value of EFH equal to hno matter what the takeoff speed v
0
and consequent landing position. This surface has had its
impact safety designed into it through the specification of a
particular value hof EFH that parametrizes the surface.
The fact that the surface shape is insensitive to takeoff
velocity v
0
is made manifest by the absence of v
0
in Eq. 20.
Also note that Eq. 20 is general in that the EFH value,
h, could in fact be a (smooth) function of xthereby giving
the jump designer freedom to tune the EFH everywhere.
Shealy and others [18,32,34,35] have questioned the
practicality of building such a jump due to ‘‘uncontrollable
factors’’ such as jumper discretion (e.g. ‘‘pop’’) leading to
variations in takeoff angle and drag variations in flight,
snow variability (snowfall and melt), temperature varia-
tions, etc. However, recent studies by McNeil [31] and
Hubbard and Swedberg [30] of these so-called ‘‘uncon-
trollable factors’’ have shown, to the contrary, that these
factors are bounded in understandable ways that can be
addressed in the design. Thus the idea that such variability
precludes the use of engineering design is not supported.
Indeed, much of the present paper continues to illustrate
this fact: design of jumps limiting EFH can proceed while
explicitly accounting for such factors or rendering the
impact of such variability largely irrelevant.
To find specific instances of constant EFH surface
shapes one must solve Eq. 20 by numerical integration.
First, one must specify the values of the parameters h
0
and
h, and, since it is a first order differential equation, one
must also choose a specific boundary condition y
L
(x
F
)at
some value of x
F
. For technical reasons [22] related to the
behaviour of the equation at small values of x, it is nec-
essary to integrate Eq. 20 backward, rather than forward, in
x;sox
F
is taken to be the terminal point for the constant
EFH surface. The arbitrariness of the boundary condition
means that there is an infinite number of such solutions for
fixed hparametrized by y
L
(x
F
).
The choice of the implemented value of his up to the
designer, but the designer needs to make an informed and
scientifically supportable choice. This decision can be
Designing tomorrow’s snow park jump 13
defended and communicated to the users of the jump so
that they are better informed as to the risks they undertake
in using the jump. The USTPC has adopted a 1.5 m
maximum EFH based on the degree of knee flex in
snowboarders landing on flat surfaces. Regardless of the
choice, however, it is important to be clear; constant EFH
landing surfaces will not eliminate serious injuries from
jumping, but they can reduce the likelihood of such injuries
and decrease the severity of injury should something go
wrong. This can be especially important in the event the
jumper becomes inverted.
One of the consistent lessons learned from the epide-
miological studies is that beginners are at significantly
greater risk than the more skilled riders [13]. In part to
address this problem, resorts have developed ski and
snowboarding schools including ‘‘progressive parks’’ with
varying levels of difficulty and risk. Constant EFH landing
surfaces would be especially well-suited to progressive
terrain parks which are intended to develop riders from
beginner to more advanced skill levels. Currently, such
progressive parks start with simple rollers over which a
rider can ‘‘catch some air’’, and then progress to a series of
tabletop jumps with increasing deck lengths. The novice
rider is then confronted with the requirement of a quantum
leap in skill level in progressing from one tabletop to the
next larger size with considerably greater risk for injury.
The constant EFH surface, on the other hand, provides the
opportunity for continuous development of skill level from
the smallest to the largest jump length attainable on a slope
of fixed pitch while mitigating impact risks through the
properties of the constant EFH surface.
Figure 7shows two samplings (as leaves from a book)
of the infinite family of constant EFH landing surfaces for
the two values of h=1.0 and 1.5 m with h
T
=10°fixed.
Similarly, Fig. 8show two samplings of constant EFH
landing surfaces for the combinations of h=1.0 and 1.5 m
with h
T
=25°fixed. Each set of landing surfaces extends
over the range 0 \x\30 m, with the boundary conditions
for each constant EFH surface given by values of y
L
at
x
F
=30 m evenly distributed from -6m[y
L
[-12 m.
For each pair of design values of hand h
T
, the family of
surfaces asymptotically emanates from the same point,
called the ‘‘singular point’’ [22] where the argument of the
tangent function in Eq. 20 goes through p/2. Every one of
the landing surfaces in each of Figs. 7and 8produces the
same EFH (either h=1.0 m or h=1.5 m) independent of
jumper takeoff speed.
Note that the constant EFH landing landing surfaces in
Figs. 7and 8do not look like tabletops with cusps. Every
smooth constant EFH landing surface has a monotonically
decreasing slope (becoming more and more negative and
eventually steeper and steeper). All surfaces become stee-
per at larger values of x. Of course, there is a practical limit
to the steepness of the landing surface determined by the
capabilities of the grooming and fabricating equipment. As
discussed more completely below, most commonly used
grooming machines today can function only up to an
incline angle of about 30°.
Figures 7and 8display the sensitivity of the families of
constant EFH landing landing surfaces to the two design
parameters hand h
T
. Generally speaking, as the value of
hincreases (e.g. Fig. 7a, b), the family of constant EFH
curves shifts downward on the left, near x=0, and become
slightly ‘‘flatter’’. As the takeoff angle h
T
increases (e.g.
Figs. 7ato8a), the family becomes significantly ‘‘rounder’’,
without shifting downward near the singular point.
The constant EFH landing surfaces in the figures were
calculated by integrating Eq. 20 which assumes no drag. It
is important to emphasize that similar safe surfaces incor-
porating air drag can be calculated and that these differ
O
O
O
O
O
T10o
h 1.0 m
5 10 15 20 25 30
(a)
O
O
O
O
O
T10o
h 1.5 m
5 10 15 20 25 30
-10
-5
0
(b)
-10
-5
0
Fig. 7 Constant EFH landing surfaces for ah=1.0 m and
bh=1.5 m with h
T
=10°fixed. Also shown are sets of possible
jumper paths, the uppermost of which corresponds to a 26 m
horizontal jump distance requiring a takeoff speed of 18.9 m/s. The
circles mark the location of the boundary conditions used to integrate
Eq. 20 to obtain the constant EFH surface. The h
L
=30°point on the
constant EFH surfaces is identified by a dashed line beyond which
fabrication is impractical with present snow grooming equipment
without use of a winch
14 J. A. McNeil et al.
only marginally from the surfaces, shown in Figs. 7and 8,
that satisfy Eq. 20. However, it is not possible to show the
analytic differential equation for the surface analogous to
Eq. 20 in the presence of drag. The essential modification
is the following: during the integration of the constant EFH
surface backward, because of drag there is no analytic
expression for the takeoff velocity required to pass through
the present point on the surface, nor for the impact velocity
at that point. Instead, these velocities must be calculated
numerically with a shooting technique, making the calcu-
lations more laborious but no less accurate. Later the effect
of including drag on the EFH is shown to be very small for
surfaces calculated using Eq. 20.
Several practical design considerations must be kept in
mind. First, the present generation of snowcats without use
of a winch are not able to fabricate and easily maintain
snow surfaces steeper than roughly 30°. Thus only those
portions of the constant EFH landing surfaces satisfying
h
L
B30°, can be created and maintained in practice. It is
impractical to view the segments of the surfaces shown in
Fig. 8as desirable solutions in ranges where their angles
exceed 30°. In each surface shown in Figs. 7and 8, the
segments where the slope of the surface exceeds 30°are
indicated by dashed lines.
Secondly, although all the surfaces shown in each of the
figures are equally ‘‘safe’’ in the sense that they all have
h=1.0 or h=1.5 m, not all are appropriate for the same
parent slope. Unless the constant EFH landing surface
shape is pre-formed from the base slope, much of the shape
of the jump surface will need to be fabricated from snow.
Certain of the surfaces will take more snow budget than
others, another of the design tradeoffs mentioned above in
Sect. 2.3. So the ultimate choice from the family of landing
surfaces in Fig. 7might be the least expensive in terms of
snow budget.
Also shown in Fig. 7are sets of jumper paths, the
uppermost of which corresponds to a 26 m horizontal jump
distance requiring a design speed of 18.9 m/s. As discussed
more completely below, once a constant EFH landing
surface that fits the parent slope is chosen, it is essential
that the entire portion of it, out to the point of its inter-
section with the maximum design speed jumper path, be
used. Otherwise, some speeds below the maximum design
speed will not be accounted for in the design and it will be
possible that faster jumpers will be able to over-jump the
constant EFH landing region, landing on a portion of the
snow where the EFH has not been controlled though sur-
face shape. The design procedure now consists of choosing
one of the constant EFH landing surfaces (say from Figs. 7
and 8or another similar one) to build on the hill.
Thus far the focus has been mostly on the role of only
one of the constant EFH landing surface parameters, h. The
parameter h
T
can also play an important role in the design
in the following way. Figures 7and 8contain only four
families of constant EFH landing surfaces, but there is a
similar infinite set of constant EFH landing surfaces for
every other pair of values of takeoff angle h
T
and h. Thus
if, after the first try, none of the constant EFH landing
surfaces in Figs. 7and 8satisfies all the design constraints
(e.g. perhaps there is not enough room in the chosen resort
location to fit in the complete constant EFH landing surface
all the way to the maximum design speed point), then the
designer can select from another figure containing constant
EFH landing surfaces for the same value of hbut for a
different takeoff angle, say h=1 and h
T
=13°as illus-
trated below [20].
How does the design of the other components affect the
design of the constant EFH landing surface, especially in
the light of the fact that these surfaces are velocity-robust?
A substantial fraction of the serious SCIs incurred in terrain
O
O
O
O
O
T25o
h 1.0 m
5 10 15 20 25 30
-10
-5
-5
0
(a)
OOOOO
OOO
OOO
OOO
OOO
T25o
h1.5 m
510 15 20 25 30
-10
0
(b)
Fig. 8 Constant EFH landing surfaces for ah=1.0 m and
bh=1.5 m with h
T
=25°fixed. Also shown are sets of possible
jumper paths (dotted), the uppermost of which corresponds to a 26 m
horizontal jump distance at a takeoff speed of 15.35 m/s. The circles
mark the location of the boundary conditions used to integrate Eq. 20
to obtain the constant EFH surface. The h
L
=30°point on the
constant EFH surfaces is identified by a dashed line beyond which
fabrication is impractical with present snow grooming equipment
without use of a winch
Designing tomorrow’s snow park jump 15
park jumps occur as a result of over-jumping the intended
landing region. Thus, perhaps the most critical factor to
insure is that over-jumping cannot occur. The constant
EFH landing surfaces discussed here are able to insure that
EFH is limited to the design value of h, but only if the
entire surface out to the maximum design speed point is
employed.
The key connections are the ramp takeoff angle and the
design velocity. If the entire constant EFH landing surface
were able to be chosen as the design solution, then all
takeoff velocities would be accounted for. But often only a
finite segment of the constant EFH landing surfaces can fit
in the space restrictions in a given location. This implies
that only velocities up to a certain takeoff velocity are
protected against, in the sense that EFH is limited to honly
up to this speed. This ‘‘maximum design speed’’ must be
large enough to bound all reasonably possible speeds that
can be chosen by the jumper. If this is not the case, then
some manner of limiting the takeoff speed must be adopted
(such as limiting the start height) so that the constant EFH
region of the landing surface cannot be out-jumped.
Terrain park jumps are supposed to be fun. Young
athletic skiers and snowboarders jump because it is exhil-
arating. The exhilaration stems from two main factors:
flight time during which tricks and other maneuvers can be
performed, and air height, defined as the maximum vertical
distance between the jumper path and the landing surface at
any point along the path. An essential tradeoff in the design
is between safety and exhilaration. Finding the right
balance is the key: designing in an appropriate amount of
exhilaration while maintaining the maximum amount of
safety possible.
A previous study by Hubbard [20] has shown that
constant EFH landing jumps can still provide the flight
time and jump air height to be considered ‘‘fun’’. To a good
approximation, given a takeoff angle and EFH, both flight
time and air height increase roughly linearly as the distance
jumped increases, and this is nearly independent of the
particular member of the infinite surface family chosen for
that takeoff angle and EFH [20]. Considerable exhilaration
can be achieved while maintaining adequate safety since,
for example, when h=1.0 m, and h
T
=30°, a flight time
and air height of roughly 2 s and 3 m, respectively, can be
achieved on jumps about 30 m long.
Consider briefly the issue of rider variability. Perhaps
the most common rider variable is the so-called ‘‘pop’’, or
jump just before takeoff. This phenomenon can be treated
in design by altering the initial conditions as described in
Eq. 12 and was indirectly examined experimentally by
Shealy et al. [32] who provided the raw data used by
McNeil [31] to extract ‘‘pop’’ speeds for over 240 jumpers.
McNeil found that for the larger jump studied by Shealy,
et al. [32], the maximum positive ‘‘pop’’ speed added was
about 1.2 m/s. Figure 9shows one of the constant EFH
surfaces along with several trajectories of jumpers who
have ‘‘popped’’ the takeoff. Of course, the EFH for the
constant EFH surface is only constant if the rider leaves the
takeoff at the same angle as that of the takeoff ramp. Since
adding ‘‘pop’’ alters the takeoff angle, the EFH for the
resulting landings will be different. One sees that by using
a constant EFH surface designed for h=1.0 m and
h
T
=25°, the maximum EFH experienced by the rider
adding 1.2 m/s of positive ‘‘pop’’ is 1.44 m. This is still
within the USTPC maximum EFH criterion of 1.5 m. In
other words all trajectories on this surface with or without
‘‘pop’’ over the entire range of ‘‘pop’’ speeds measured in
the field will satisfy the USTPC criterion of EHF below
1.5 m. Also shown in Fig. 9is the jumper trajectory
including both positive ‘‘pop’’ and drag (using the same
parameters used in Sect. 2). Note that the two effects nearly
cancel resulting in a trajectory very close to the original (no
drag and no ‘‘pop’’) case.
5.1 Example of the jump design process using
modelling
As an illustration of the iterative design process, several
example designs are shown that employ some of the con-
cepts discussed above and illustrate some of the design
tradeoffs. Any such design begins with a vertical section
of the parent slope. For simplicity and to illustrate the
principles, first assume that the parent slope consists of a
constant slope equal to the approach angle, h
P
=h
A
=15°.
Generalization to any slope profile is straightforward. The
design question is then: Which candidate constant EFH
surface to choose and where and how to place it on the
parent slope?
The first important consideration is that to turn the
velocity vector, from basically ‘‘down’’ during the approach
to ‘‘up’’ at takeoff, requires space and, if the chosen surface
is not pre-formed from earth, also requires a considerable
investment in snow. The approach, transition, and takeoff
are the same for the standard tabletop and the constant EFH
jump. As can be seen from Fig. 1, the transition requirement
pushes the takeoff, maneuver and landing areas outward
away from the parent slope. Table 2lists the vertical
distance from the parent slope to the lip of the takeoff
(y
lip
-y
parent
), for sets of approach parameters, all assuming
an approach angle h
A
=15°. In general, the larger the
transition radius and takeoff angle, the more space is
required. Thus, the takeoff point needs to be positioned
significantly above the parent slope as shown in Fig. 10.
To prevent the landing surface from being over-jumped,
the jumper path corresponding to the highest speed
attainable (the maximum design speed) must intersect a
chosen constant EFH landing surface. Further, this surface
16 J. A. McNeil et al.
must be able to be built in practice. In other words, this
path must intersect the chosen landing surface above the
practical buildable limit h
L
B30°. Any landing surfaces
lying below, and not intersecting the maximum design
speed path, do not protect the jumper at all speeds up to the
maximum design speed.
If it is not possible, due either to lack of space or
inadequate snow budget, to select a landing surface that
protects at the maximum design speed, it will be essential
to limit the maximum takeoff speed by limiting the length
of the approach as described in Sect. 2. Thus this most
important design consideration that the landing surface not
be able to be over-jumped should be primary.
To be specific, consider the situation shown in Fig. 10a,
b showing a parent slope with angle h
P
=15°and ten
candidate constant EFH surfaces, all with h=1.0 m and
for h
T
=10°and h
T
=25°, respectively. The jumper path
corresponding to the maximum design speed is based on an
approach length of 100 m for the low friction case. From
Fig. 2this gives v’15:0 m/s =33.6 mph. In Fig. 10a the
maximum speed jumper path intersects five of the ten
candidate surfaces shown and therefore only these (highest)
five candidates protect the jumper at all speeds up to the
maximum design speed. The lowest of these candidates is
the cheapest from a snow budget point of view, since less
snow needs to be added to the parent slope to support the
shape of the surface. This gives a general design rule of
thumb: the most economical constant EFH surface to build
is the one that intersects the jumper maximum design speed
path at the buildability limit or at the parent slope.
Table 2 Height of takeoff above parent slope
Takeoff angle
h
T
(°)
Curvature radius of
transition R(m)
Height of takeoff
(y
lip
-y
parent
) (m)
10 15 3.205
25 15 6.295
10 20 3.690
25 20 7.506
1.2 m /s
0.0 m /s
-1.2 m /s
EFH 1.44 m
EFH 1.0 m
EFH 0.639 m
0 10 20 30 40
-10
-8
-6
-4
-2
0
2
4
Fig. 9 Large jump trajectories including rider ‘‘pop’’ landing on a
constant EFH landing surfaces for h=1.0, h
T
=25°, and the bound-
ary condition set at {x
F
=40 m, yLðxFÞ¼10 mg:The ‘‘pop’’ speeds
shown are v
p
={-1.2, 0.0, ?1.2} m/s resulting in EFHs of
{0.639, 1.00, 1.44} m, respectively. In other words all trajectories on
this surface with or without ‘‘pop’’ will satisfy the USTPC criterion of
EHF below 1.5 m. Also shown is the v
p
=?1.2 m/s jumper trajectory
including drag/lift (dashed). Note that the drag effect in this case nearly
cancels the effect on range from the positive ‘‘pop’’
h1.0 m
15 m /s
10
(TT) 27.6o
5 10 15 20 25 30
(a)
1.0 m
15 m/s
25o
(TT) 29.0
510 15 20 25
(b)
-15
-10
-5
0
-15
-10
-5
0
30
Fig. 10 The parent slope (brown) and ten constant EFH landing
surfaces for h=1.0 for ah
T
=10°and bh
T
=25°. Also shown are
the jumper paths (dashed) corresponding to the horizontal jump
distance at the maximum design speed for an approach length of
100 m (15 m/s). The constant EFH surface that intersects the parent
slope at this point is thicker (magenta). The buildable limit h
L
=30°
on the constant EFH surfaces is identified by transition to a dashed
line beyond which fabrication is impractical with present snow
grooming equipment without use of a winch (colour figure online)
Designing tomorrow’s snow park jump 17
Having chosen a landing surface that intersects the
maximum design speed jumper path, one defines the jump
length x
L
as the value of xwhere this intersection occurs.
The artificially constructed landing surface (for x\x
L
)
must thereafter rejoin the parent slope(for x[x
L
) but
since, by definition, there can be no landings in this region,
there is more flexibility in its shape. Although a mathe-
matical expression could be used to characterize the sur-
face this is not essential. One simply assumes that this
bucket region is a smooth transition that limits surface
curvature, and thus normal acceleration, to reasonable
values.
Occasionally there may be restrictions on the space
available for placement of the jump. For example, these
might be expressed as 0 \x\x
S
and y(0) \y\y(x
S
), the
interior of a rectangle inside which the jump landing sur-
face must lie. In such a case the intersection of the maxi-
mum design speed path with a constant EFH surface for the
specified value of hand chosen value of h
T
may not exist
within the rectangle, i.e. it may not be possible to build a
landing surface that limits EFH to hover a reasonable set
of takeoff speeds with the given takeoff angle.
As an example, having chosen h
T
=25°and h=1.0 m,
suppose it is desired to restrict space to x\x
S
=20 m. It
is clear from Fig. 10b that none of the constant EFH
landing surfaces intersect the maximum design speed
jumper path within this region, and thus none of these will
protect at the h=1.0 m level for the value of h
T
=25°.
Then, either the maximum design speed must be changed
by limiting takeoff velocity by restricting the approach
length, or another constant EFH surface must be chosen
above the ten shown, or the takeoff angle must be modified.
This last choice as a design option corresponds to using,
say, Fig. 10a as the design template instead, in which there
is one of the constant EFH surfaces that meets both the
space and the maximum design speed constraints.
A general design rule of thumb arises from this example:
it is always possible, by decreasing the takeoff angle, to
choose a constant EFH surface lying within any restricted
region that limits h to a given value. This shows that all
three of the design variables, h
T
,h, and the particular
surface chosen from the infinite family are important.
Figure 11a shows an example constant EFH surface
choice replacing an example tabletop surface (parameters
are given in the caption) along with sample trajectories for
a takeoff angle of h
T
=13°. In this example, at x*23 m
the constant EFH surface reaches the practical slope angle
limit of 30°. To be realistic, at this point the constant EFH
surface (solid) transitions to a constant slope (dashed)
surface with constant h
L
=30°until it intersects the parent
surface.
Figure 11b shows the resulting EFHs for the tabletop
and constant EFH landing surfaces. Since this constant
EFH landing surface was calculated assuming no drag, it
yields a value h=1 m for jumper paths without drag. By
design, the constant EFH landing surface provides a con-
stant 1.0 m EFH out to *23 m at which, as discussed
previously, the landing angle is a constant 30°, so the EFH
rises linearly as expected for a straight landing surface but
still only exceeds the USTPC criterion value of 1.5 m after
x’27 m, the maximum design jump distance and an
unusually long jump of over 27 m. The standard tabletop
surface exceeds the USTPC criterion for the entire length
of the deck before dropping briefly to acceptable values
between 9:1m\x\22 m. For x[22 m the EFH for the
tabletop rises approximately linearly to a value of about
2.4 m at x=30 m. Clearly the constant EFH surface
provides greater protection for the two hazardous situations
of landing short (on the deck) and landing ‘‘deep’’. If
holding the EFH to the USTPC criterion is deemed
important enough, perhaps it would be worth using a winch
to increase the angle of the constant EFH surface in the
relatively small 7 m long region 22.6 \x\30 m Also
shown in Fig. 11b is the small added effect of including
drag on the EFH for the constant EFH surface (dashed
line). Since this constant EFH landing surface was calcu-
lated assuming no drag, it yields a value h=1 m for
jumper paths without drag. The effect of drag on the
resulting EFH for this jump is modest.
Even though the straight tabletop landing surface enjoys
a lower EFH for part of the landing surface, the constant
EFH landing surface in Fig. 11a obviously protects the
jumper more effectively over a greater range of jump dis-
tances. Furthermore, the cost of the added protection in
terms of snow budget is quite small. The volume of snow
per unit width of the landing surface in these examples is
72.3 m
2
for the tabletop and 76.2 m
2
for the constant EFH
surface. Designers will need to include this economic fact
in their considerations. If snow budget becomes an
unmanageable constraint, an alternative approach is to
preform the hill to lower the snow required, but this option
removes some flexibility in moving or significantly altering
jump shapes in mid-season and probably incurs signifi-
cantly more earthmoving expense.
For the reasons explained above, building a jump
entirely above a constant slope can be expensive. Terrain
park personnel figured out long ago that taking advantage
of natural undulations in the parent surface can minimize
this expense. Essentially the jump is put at a location on the
parent slope that uses the natural topography rather than
snow to fill in the volume below the landing surface. This
strategy is effective with constant EFH landing surfaces as
well for tabletops.
Nevertheless, it may sometimes be desired to place a
jump on a constant slope. If the terrain is malleable (not
bedrock) the rough jump shape may be sculpted from the
18 J. A. McNeil et al.
earth by excavation rather than from snow each season.
The shape of the landing surface is formed from earth
rather than snow and thus need not require large amounts
of expensive artificial snow. An example of such topo-
graphical preforming would have the approach partially
submerged below the original parent surface, and the earth
from this cut would then be pushed downhill to form the
takeoff ramp and part of the landing surface. After the
snow is added, the entire jump would conform much more
closely to the parent slope, being sometimes above and
sometimes below the original parent slope, saving con-
siderably on the snow required to fine tune the shape and
provide enough base to ensure the parent slope is unlikely
to ever be exposed. This permanent fabrication of the jump
shape will likely be more expensive initially, but it would
require less additional snow and shaping each snow season
thereafter, and may be the most economical design option
especially in a region with limited natural snow and arti-
ficial snow budgets. Whether this is an economically fea-
sible option will depend on the details of the terrain park
jump location, base parent slope, annual snowfall and snow
making ability, grooming equipment and maintenance
resources, and crucially on the energy cost of artificial
snow.
6 Summary
Winter terrain park jumps have been shown to present a
special hazard to ski resort patrons for spine, neck, and head
injuries. Presently, such jumps are built without a quanti-
tative engineering design approach based on the assertion
by the NSAA and supported by some researchers that there
is too much variation in the conditions and rider decisions.
However, recent studies by the authors and others have
examined these factors and determined that while they do
indeed vary, they do so in an understandable and bounded
fashion that can be accommodated or rendered irrelevant by
the design. The role of modelling the behaviour of a jumper
executing a terrain park jump enables and informs an
intelligent design process that meets reasonable perfor-
mance criteria while satisfying the constraints of the parent
terrain, snow budget, and safety considerations. The paper
concluded with an illustrative example of the design of a
constant EFH jump on a hypothetical terrain park parent
slope. It was shown that, unlike the tabletop design, an
appropriately designed constant EFH landing surface can
satisfy the USTPC criterion for maximum EFH for all
values of the takeoff speed including rider ‘‘pop’’ effects.
These insights were made possible by the extensive use of
computer modelling in setting up, constraining, and solving
for the relevant design parameters.
There are several ways in which the present work may
be extended and improved. Many of the physical parame-
ters such as the range of drag and friction coefficients were
estimated from older published research of a generic nature
and better values appropriate to actual snowboarders and
skiers are needed. Once these physical parameters are
better determined, on-slope validation of the trajectory and
rigid body models would improve general acceptance of
this approach as well as lead to further insights to improve
the model. Finally, the modelling of trajectories is only part
of the story. Better modelling of the human factors related
to the range of rider actions prior to the takeoff, in the air,
and upon landing would greatly improve our understanding
of how riders interact with winter terrain park jumps.
T13o
Parent slope
Tabletop
Constant EFH
5 10 15 20 25 30
(a)
T
13
o
Tabletop
Constant EFH (no drag)
Constant EFH (w/drag)
0 5 10 15 20 25
0.0
0.5
1.0
1.5
2.0
2.5
3.0
(b)
-15
-10
-5
0
5
Fig. 11 a Comparison of an example standard tabletop (magenta)
and constant EFH (blue) landing surfaces along with example jumper
trajectories for h
T
=13°. The landing surfaces end at the run-out
when intersecting with the parent slope (brown)atx’27 m.
bComparison of the EFH values for the standard tabletop and
constant EFH landing surfaces. The tabletop has a lip height of 1.5 m,
a deck length of 30 ft, and a landing slope angle of 27.6°chosen to
intersect the parent surface at the maximum jumper path for the
design speed (15.0 m/s) based on an approach length of 100 m. The
constant EFH surface has h=1.0 m and continues until x’23 m at
which point the landing angle equals 30°. Beyond this point the
landing angle is constant at h
L
=30°(dashed line) until it intersects
the parent slope. The horizontal dashed (green)line in bmarks the
EFH = 1.5 m USTPC criterion value (colour figure online)
Designing tomorrow’s snow park jump 19
Acknowledgments The authors acknowledge useful discussions
with and helpful suggestions from J. Brodie McNeil, A.Wisniewski,
and a terrain park manager who requested anonymity.
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