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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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