Design of Terrain Park Jump Landing Surfaces for Constant Equivalent Fall Height is Robust to Uncontrollable Factors

Transcription

1 Design of Terrain Park Jump Landing Surfaces for Constant Equivalent Fall Height is Robust to Uncontrollable Factors Mont Hubbard Department of Mechanical and Aerospace Engineering, University of California, Davis, California, USA Andrew D. Swedberg Department of Mathematics, U.S. Military Academy, West Point, New York, USA (Dated: May 13, 2012) Epidemiological studies of ski resort injuries have found that terrain parks, and especially jumps, present a significantly greater injury risk to skiers and snowboarders than other more typical skiing activities. It has also been shown that the severity of impact risk can be characterized by equivalent fall height (EFH), a measure of jumper impact velocity normal to the slope, and that design algorithms exist to calculate landing surface shapes that limit EFH to arbitrarily low values. Although proposals have been made recently to introduce design, the skiing industry and other authors have objected that analysis, design, and standards are impossible because of various uncontrollable factors that allegedly make the problem intractable to analytical design techniques. We consider the list of uncontrollable factors one by one and show that, to the contrary, each is either 1) irrelevant to design, 2) has negligible effect, if any, on designed EFH, or 3) can be directly incorporated into the design process. I. INTRODUCTION The skiing industry has evolved dramatically over the last two decades. The number of snowboarders has increased substantially and now approaches parity with skiers [1]. Snow terrain parks have been introduced which include jumps and other airborne features. These have led to increases in flight maneuvers and associated landings, and some studies (e.g. [2]) have found evidence that the type of activities performed in terrain parks may increase the risk of sustaining a severe injury. On the other hand, Shealy, et al. [3] recently noted that over a period from 1990 to the present when snow terrain park use has risen, overall injury rates at resorts have actually fallen [3]. It is difficult to reconcile these apparently contradictory results. Although NSAA has released at the end of each season the number of resort related fatalities and catastrophic injuries along with the total number of resort visits, the raw data on which the study [3] was based is not publicly available. Yet many studies focusing on actual terrain park injuries clearly show that terrain parks present a special hazard to riders. Early research of Sutherland [4] recognized that snowboard injury patterns differed from those of skiers, and Dohjima [5] found that snowboard injury rates could be as much as six times higher than those of skiers. The snowboard injury rate itself was discovered to have doubled between 1990 and 2000 from 3.37 to 6.97 per 1000 participant days [6]. In addition, jumping has been suggested as the most important cause of injury [5]. Increased risk of head injury at terrain parks as compared to ski runs has continued through the end of the last decade [7]. A recent study of snowboard injury rates specifically in terrain parks found that the risk of injury on jumps was highest of all terrain park features [8],[9]. 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 [10]. But focusing on the number of injuries somewhat misses the point. Many snow-related head, neck and back injuries are very serious. A succinct summary provided by Meyers [11] explains: Spinal cord injuries (SCI) are among the worst ski outcomes. Krause et al. [12], in a broad review of SCI epidemiological trends, found an increase in SCI caused by snow (as opposed to water) skiing, and that snow skiing (presumably including snowboarding) had replaced football as the second leading cause of SCI in the U.S. Ackery, et al. [13], in a review of 24 articles between from 10 countries, found evidence of an increasing incidence of traumatic brain injury and spinal cord injury in alpine skiing and snowboarding worldwide and that this increase coincided with development and acceptance of acrobatic and high-speed activities on the mountain. As early as the end of the 1990s, Tarazi, et al. [14] had found that the incidence of SCI in snowboarders was four times that in skiers and that jumping was the primary cause of injury with 77% of snowboarder SCIs occurring from jumping. In a review by Seino, et al. [15] of six cases of traumatic paraplegia SCIs resulting from snowboard accidents at a single institution over 3 years, it was 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 these published studies may seem compelling, today it is still difficult to get a precise snapshot of overall terrain park injury statistics nation- or world-wide. In the U.S., although the NSAA collects skiing and snowboarding

2 injury data, these data are not made publicly available. What is clear is that these most serious SCIs (resulting in paraplegia or quadriplegia) exact a very large societal penalty. That they happen uniformly to young people and are permanently debilitating means that there is an enormous economic and social cost (see Devivo, et al. [16] and Dijkers, et al. [17]). 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. Several recent papers [18 21] have proposed analytical methods for increasing safety of jumps by using a formal design process to create the landing surface to prevent large landing impacts. But most jump landing surfaces are not designed at all. Rather they are built without design by somewhat experienced terrain park personnel, who often are skillful machine operators and former jumpers themselves. Virtually never is there any quantitative analysis or engineering design of the jumps by staff with formal training in engineering analysis of the designs. Although these jumps are usually tested before being opened to the skiing public, they have been found [22] to cause large impulses on landing impact that are severe enough to contribute substantially to the major injuries noted above. This approach appears to stem from a reluctance on the part of the ski areas, and their counsels, technical advisors, and trade organizations to adopt standards and a rigorous design process for jump landing safety. This reluctance can be traced to ski resorts risk management strategy [23], but the industry also claims [24] that the performance of the design is too susceptible to changes in snow conditions, including snow melt and accumulation, and other uncontrollable factors (e.g. variations in jumper aerodynamic drag) that affect jump behavior either in the design, in maintenance, or during actual use. Specifically, the NSAA asserts that, due to rider and snow variability, terrain park jump standards are impossible [24]. Although not stated explicitly, it is implied that analysis and design of such features are similarly made impossible. While it is true that the virtually... infinite number of ways that a given feature may be used by an individual... varying speed, pop, body movement, takeoff stance, angles of approach, the attempting of different kinds of maneuvers, landing stance, and the type of equipment being used (skis or snowboard)... create a wide variety of experiences for users [24], none of these in fact precludes analysis or design. Papers by other authors make similar arguments against analysis and design. Shealy and Stone [25] attempted to explain why understanding jumper pop (jumping at takeoff) was essential to make analysis possible and surprisingly concluded that [e]lementary Newtonian physical laws for projectile motion cannot be used to predict the location that a skier/snowboarder will land based strictly on the speed of the skier/snowboarder at... takeoff and the geometry of the feature, while not realizing the elementary Newtonian physical laws do not constrain the takeoff velocity to be parallel to the takeoff ramp. Later review of this work uncovered errors corrected in an erratum [26]. A later paper by some of the same authors [27] again attempted to discredit theoretical analysis and concluded (incorrectly) that theoretical models of a terrain park jumper are not developed sufficiently to model the kinematics of an individual jumper and to determine his or her landing spot [27]. This paper also concluded that there was no statistically significant relationship between takeoff speed and distance traveled, again implicitly questioning the ability of Newton s laws to predict the motion of jumpers and impugning the efficacy of design methods [18] that use Newton s laws to shape the landing surface to limit impact exposure. In spite of the previous reluctance of the skiing industry to employ design, matters appear to be changing. Committee F-27 on Snow Skiing of ASTM (previously the American Society for Testing and Materials) has voted to bring recreational winter terrain park jumps within its purview and has created a Terrain Park Task Group. It therefore appears probable that engineering design approaches will soon be applied to winter terrain park jumps. Physical modeling of jumper motion will then become an important, if not essential, component of this design process and it will be important to understand, in fact, exactly what factors are important and which are not. This paper studies comprehensively and systematically the claims that design is rendered impossible by uncontrollable factors raised by the NSAA Freestyle Notebook [24]. Using simulation methods which are extensions of those discussed in [18] and [20] it examines the effects of several such phenomena; snow coefficient of friction, snow accumulation, snow melt, changes in jumper aerodynamic drag coefficient C d due to modifications in body position in flight, wind speed and direction, variations in landing impulse, and others, in an attempt to learn whether these claims are valid. Even though jumper pop has already been explicitly treated in [28], it is included in the present paper for completeness. The main results show how a measure of jump safety at impact, here the equivalent fall height function h(x) as a function of position on the landing surface x, varies as the hypothesized uncontrolled phenomena occur. It is recognized that the potential for injury on a given jump is a function of more factors than equivalent fall height alone, including, for example, loss of control of body orientation at takeoff. Although similar design processes can address these other factors (e.g. by ensuring that the takeoff ramp is straight so as not to induce undesired rotations at takeoff [29]), these other factors are not treated explicitly or in detail in this paper. Instead it concentrates on control of equivalent fall height by shaping the landing surface. 2

3 3 II. LANDING SURFACE DESIGN A. Equivalent Fall Height Probably the most important factor in considering the relative safety of a terrain park (or any) jump is the energy that must be absorbed by the jumper at impact. A convenient way to quantify this impact energy uses the concept of equivalent fall height. When an object falls vertically onto a horizontal surface, the (true) fall height h is related to the speed v at impact through the equation describing conversion of initial potential energy to terminal kinetic energy, mgh = mv 2 /2. However, on a sloped landing surface the velocity parallel to the surface need not vanish, and the relevant energy relation becomes mgh = mv 2 /2, where v is the component of jumper velocity perpendicular to the landing surface at impact. This is the energy that must be absorbed and that therefore characterizes the severity of the landing impact. The equivalent fall height (EFH) h = v 2 /2g thus depends only on the component of the velocity normal to the landing surface. A small EFH ( say 1 meter) implies a softer landing and reduced chance or severity of injury, while larger equivalent fall heights are more likely to result in injuries, or more severe injuries, on landing. The normal component of the landing velocity v is the product of the landing speed v and the sine of the difference between the landing slope angle θ L and the jumper s flight path angle θ J ; i.e. v = v J sin(θ J θ L ), where v J is the jumper s landing speed, and thus EFH is given by h = v2 J sin2 (θ J θ L ), (1) 2g where g is the acceleration of gravity. Softer landings (corresponding to small h) are obviously characterized by a small difference between θ J and θ L. By choosing the angle of the landing surface nearly equal to the angle of the jumper flight path, the EFH can be made arbitrarily small regardless of the magnitude of the landing speed. To calculate the EFH for an arbitrary jump shape, one must solve the equations of motion for the jumper. The equations governing center-of-mass motion in flight include drag and lift, but these aerodynamic forces are small compared to ( 10% of) the weight mg and can be neglected to first order [18],[28]. In this case the jumper flight path can be shown to be approximated by the classic parabolic flight path y(x) y(x) = x tan θ T g 2v 2 0 cos2 θ T x 2. (2) Solving Eq. 2 for v 0 and using the additional energy conservation relation, vj 2 = v2 0 2gy, jumper speed v J can be obtained as a function of x and y components of position (in a coordinate system with origin at the takeoff point and x horizontal), x v J (x) = 2 g 2(x tan θ T y) cos 2 2gy. (3) θ T The slope of the jumper s trajectory is the derivative of Eq. 2: tan θ J (x) = y J(x) = tan θ T g v 2 0 cos2 θ T x. (4) Using Eq. 2 to eliminate v 0 from Eq. 4, the flight path angle θ J can be expressed in terms of x and y as ( ) 2y θ J (x) = tan 1 x tan θ T. (5) If y L (x) is the shape of the landing surface, then the landing condition is y(x) = y L (x). Using Eqs. 3 and 5 in Eq. 1 results in the following general expression [30] for the equivalent fall height as a function only of landing position x, [ x 2 ] [ ( ) ] h(x) = 4 cos 2 θ T (x tan θ T y L (x)) y L(x) sin 2 tan 1 2yL (x) tan θ T tan 1 y x L(x). (6) where θ L (x) = tan 1 y L (x) has been used. (Note that usually both θ J and θ L are negative at landing.) It is important to emphasize that Eq. 6 is completely general. It applies to every landing on any jump landing surface y L (x) whatsoever at any point x, y L (x) with landing surface slope y L (x) and from takeoff angle θ 0. In Eq. 6

4 4 TABLE I: Physical Parameters. Parameter Symbol Units Value Range Acceleration of gravity g m/s Mass of jumper m kg 75 Drag coefficient times jumper frontal area C d A m Density of air ρ kg/m Coefficient of kinetic friction µ dimensionless Lift to drag ratio ρ ld dimensionless h is a function of x alone because, for a given landing surface, both y L (x) and y L (x) are functions only of x. Eq. 6 makes plain that every jump, whether real or imagined, whether carefully quantitatively designed or not designed at all, has a function h(x) that characterizes the severity of impact at every point on its landing surface. For a given takeoff angle, the shape of the landing surface y L (x) determines the relative safety of the jump through the function h(x). This idea is used repeatedly below. One possible use of Eq. 6 for h(x) is to test the relative safety and hence adequacy of a proposed landing surface shape y L (x). Another is as an indicator of the relative safety of an already fabricated shape. In each case the landing surface shape y L (x) must be specified or measured (and its derivative y L (x) calculated). For a jump to be considered relatively safe, h(x) should be relatively small everywhere (at all possible landing positions x). Only when drag and lift are neglected, can simple analytic expressions such as Eq. 6 for the EFH be obtained [18],[20]; however it is straightforward to solve the full equations numerically including drag and lift effects (see below) to obtain the same function h(x) (even under windy conditions[20]). B. Designing Surface Shape to Limit EFH We have shown above that 1) the impact risk associated with landing is naturally quantified by EFH, 2) every landing surface shape has an associated EFH function h(x) that characterizes its relative impact safety everywhere, and 3) a soft landing arising from small EFH is possible if the jumper path and landing surface have nearly the same angle at the point of impact. We now explore the possibility of curved landing surfaces that, by design, provide acceptable EFH independent of the takeoff speed. A more complete discussion of the theory behind calculation of safer jump landing surface shapes is given by Hubbard [18] and McNeil et al.[20]. We begin with Eq. 6 as a condition on the surface shape y L (x) such that the EFH is limited to a specific value h at all values of x. In its original form Eq. 6 was used to quantify h(x), given y L (x) and y L (x). Instead, we solve for y L (x) from Eq. 6 to write an expression for the derivative of the landing surface, now interpreted as a differential equation for this surface, but assuming we know the desired value of h(x) = h (here assumed constant), y L(x) = tan [ tan 1 ( 2y L(x) x tan θ T ) + arcsin h x 2 4(x tan θ T y L (x)) cos 2 θ T y L (x) ]. (7) In Eq. 7, which has been called the safe slope differential equation [18], the first derivative of the landing surface function is itself a function of two variables (x and y L (x)) and three constant parameters (g, θ T and h). Since no jump can be definitively called safe, here we refer to such surfaces as constant EFH surfaces. Any surface y L (x) that satisfies this differential equation exposes an impacting jumper to a value of EFH equal to h no matter what the takeoff speed v 0 and consequent landing position. This surface has had its impact safety (or lack of it) designed into it through the specification of a particular value h of EFH that parametrizes the surface. The fact that the surface shape is insensitive to takeoff velocity v 0 is obvious from the absence of v 0 in Eq. 7. To find specific instances of constant EFH surface shapes without drag we would need to solve Eq. 7 by numerical integration. But first, one must specify the values of the parameters θ T and h, and a specific boundary condition y L (x F ) at some value of x F. For technical reasons [21] related to the behavior of the equation at small values of x, it is necessary to integrate Eq. 7 backward, rather than forward, in x, so we take x F 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 h parametrized by y L (x F ).

5 y (m) x (m) FIG. 1: An example constant EFH landing surface (solid line) calculated including drag and estimated maximal jumper pop, with a family of jumper paths (dashed line) terminating at the surface. All jumper trajectories intersecting this surface will incur the same EFH of 1 m, no matter what the takeoff velocity and consequent landing point. In the rest of the paper, perturbations are made to the parameters used in its design and the effects on EFH examined. To calculate the constant EFH landing surface including drag, one must solve the exact equations of motion for the jumper. We now address these flight equations of motion. The nominal values of physical parameters assumed here are given in Table I. To account for wind, the drag force depends on the rider-air relative velocity, v r = v w, where v and w are the jumper and wind velocity vectors, respectively. The aerodynamic drag force is given by [31], F drag = C daρ vr 2 ˆv r = mηvr 2 ˆv r, (8) 2 where A is the cross-sectional area of the rider perpendicular to the direction of travel, ρ is air density, C d is the drag coefficient, and ˆv r is a unit vector in the direction of v r. Hoerner [32] tabulates the drag area C d A for humans in various body orientations: standing facing forward (0.836 m 2 ), standing sideways (0.557 m 2 ), and tucked facing forward (0.279 m 2 ). Air density ρ depends on elevation Y and absolute temperature T according to the approximate relation [33], T 0 T 0 Y ρ(t, Y ) = ρ 0 T e T Y 0, (9) where Y is the altitude above sea level in meters, T 0 is the reference temperature ( K), T is the absolute temperature (T = T C , where T C is the temperature in Celsius), ρ 0 = kg/m 3, and Y 0 = m. For example, this relation gives the air density at an elevation of 3370 m as ρ=0.885 kg/m 3 at T C = -10 C. In flight, the equations of motion governing the center-of-mass motion including drag and lift are given by[20]: d 2 r J (t) dt 2 = gŷ ηv r ( v r ρ ld ẑ v r ), (10) where v r is the jumper velocity vector relative to the wind, ρ ld is the lift to drag ratio, ẑ is the unit vector normal to the plane containing the trajectory of the jumper, g is gravitational acceleration, r = (x, y) is the two-dimensional

6 position vector of the rider ignoring the transverse (z) motion, ˆv is the unit velocity vector, and η = C d Aρ/2m is the drag parameter defined in Eq. 8. Eq. 7 assumes no drag. It is important to emphasize that similar constant EFH surfaces incorporating air drag can be calculated and that these differ only marginally from those that satisfy Eq. 7. However, in the presence of drag it is not possible to show the analytic differential equation for the surface analogous to Eq. 7. Also, during the integration of the constant EFH landing surface backward, because of drag there is no analytic expression for the takeoff velocity required to pass through the present point x, y on the surface, nor for the impact velocity v J at that point. Instead, these velocities must be calculated numerically with shooting techniques, making the calculations more laborious but no less accurate. Shown in Fig. 1 is a constant EFH landing surface with takeoff angle θ T = 25. It was calculated in the above manner (i.e. accounting for drag by integrating Eq. 10 and using shooting techniques rather than neglecting drag and merely integrating Eq. 7) using nominal values of parameters. It assumes zero wind and a conservatively large amount of jumper pop velocity v p = 2.25 m/s. The surface doesn t look much like the conventional tabletop jump, but it protects against impact much more effectively than the tabletop does [21, 30]. It is one of an infinite family parameterized by the value of y L at x = 40 m. Every point on this surface has the same value of EFH (1 m in this case) and the relative safety of the surface designed into it by specifying EFH is totally insensitive to takeoff velocity. Therefore no uncontrollable factor whose sole effect is to change jumper takeoff velocity can have any effect whatever on the resulting EFH. As long as conditions (essentially η) remain at the nominal values assumed in the design, this surface imparts exactly the value of EFH it was designed to supply. If any parameters change (say at some takeoff speed), the forces in flight will change correspondingly, modifying the jumper path and the consequent landing point, as well as the jumper velocity vector and its perpendicular component at that point. This changes the EFH at that landing point. Fig. 1 implicitly assumes 1) no wind, and 2) maximum jumper pop. We below relax these assumptions and calculate whether these changes affect the ability of the design to provide constant EFH. In addition we study the other factors claimed to preclude design, i.e. to make the design unable to perform effectively. 6 III. EFFECTS OF UNCONTROLLABLE FACTORS [24] ON CONTROLLED EFH DESIGN PERFORMANCE In the Subsections that follow below we investigate, one by one, the factors suggested by the NSAA [24] to make standards and, by implication, design impossible. We calculate the EFH function h(x) in response to perturbations in the various parameters and compare it to the EFH h(x) = 1 m for the nominal design shown in Fig. 1, holding all other parameters at their nominal values. This is not to say that in actual practice simultaneous variations in more than one design parameter cannot occur, but rather only that, in order to understand their effects, it is necessary to think about them one by one. A. Varying Speed [24] The controlled EFH surface shown in Fig. 1 is completely insensitive to takeoff speed parallel to the takeoff ramp at angle θ T = 25 deg. Variations in takeoff by different jumpers have no effect on EFH. The landing surface EFH remains at h = 1 m at all speeds. B. Snow Friction The coefficient of kinetic friction µ between snow and skis or snowboards depends on many variables [34, 35], among which are temperature and melt water film thickness. It is reported [34] to lie roughly in the range 0.04 < µ < 0.12, a substantial range. Yet this relatively sizable variation in friction affects the forces on the jumper only while she is in contact with the snow surface before takeoff. It will play an important role in the ability of the jumper to accumulate velocity during the in-run/approach and can therefore affect the takeoff velocity substantially. But since constant EFH landing surfaces similar to that in Fig. 1 yield the same value of EFH independent of takeoff speed, variations in the coefficient of snow-ski friction have no effect whatever on the actually experienced EFH on an already-designed constant EFH landing surface. It is important to realize that some of the serious spinal cord injuries incurred on terrain park jumps occur as a result of over-jumping the intended landing region. Thus, perhaps the most critical factor is to insure that over-jumping cannot occur. A decrease in the coefficient of friction or drag, or both, can increase both the takeoff velocity achieved

7 when beginning from a fixed starting point and the distance jumped. Thus it is important to have the designed landing surface accommodate the maximum takeoff speed possible, which will occur at the minimum combined friction and drag combination. This is discussed more completely in [20]. The designed 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 intersection with the maximum takeoff speed jumper path is employed. Assuming that this happens, however, then the EFH experienced by a jumper is completely insensitive to the coefficient of snow-ski friction µ, whether the surface was designed with safety in mind or not. 7 C. Aerodynamic Forces Note that apart from the gravity term g, the only other term (the aerodynamic force) in Eq. 10 has as a coefficient the aerodynamic drag factor η. Because η is itself a composition of four quantities (ρ, C d, A and m), perturbations in these four parameters are indistinguishable from one another in their effect on the flight path and the consequent changes in EFH. A 1% increase in the density ρ (or the flight drag coefficient C d or the frontal area A) is essentially identical in its effect on EFH to that of a 1% decrease in mass m, since all four affect the aerodynamic parameter η in essentially the same way. Next we show the effect of perturbing (halving) drag on the EFH for surfaces calculated using nominal drag. That is, how does the surface calculated assuming full nominal drag and shown in Fig. 1 actually perform when there is only half as much drag. Fig. 2 compares h(x) = 1, the design value when η = (corresponding to nominal values for its four constituent parameters) to the case when the drag is halved and η = Decreased drag causes the highest speed landing to occur at a longer jump distance x = 33.8 m and the EFH to decrease slightly at this value of x to h = 0.88 m. Increasing the drag by a factor of two has the opposite effect, causing impact at a shorter jump length of x = 24.7 m and the EFH to increase slightly to h = 1.12 m. In practice the lift effect in terrain park jumps is extremely small. In Nordic competitive ski jumping success is based almost entirely on maximizing lift and minimizing drag, and the jumpers essentially lie down on the relative wind. Although the lift-to-drag ratio can be as large as 2 or 3, the drag and the lift are substantial because typical speeds are twice those in terrain parks and body orientation is controlled to maximize lift. But for upright low-speed terrain park jumpers, lift is of the order of only 1 2% of the weight and it depends on the orientation and configuration of the jumper in the air. This lift term is therefore neglected. In general, the flight equations (Eq. 10) must be solved numerically but, as shown by McNeil [28], for small and medium sized jumps (less than 12 m) the drag can be ignored at about the 10% level (the effects of lift are much smaller) allowing for a closed form analytic solution to the flight path that is accurate to that level. D. Wind Variations in wind cause relatively large changes in drag forces. Because the aerodynamic force is quadratic in the jumper s velocity relative to the wind, a change in wind speed of about 10% of the nominal jumper speed (say at takeoff) causes roughly a 20% change in the aerodynamic force. This variation in drag force would require more than 20% change in drag coefficient (or in the other three component variables affecting η). Although the landing surface shape may have been designed to account for the drag forces over the nominal range of jumper speeds appropriate to the jump at a nominal wind speed (presumably zero), the design cannot account for the force perturbations due to other wind speeds. Typical terrain park jump absolute maximum jumper speeds at takeoff are of the order of 18 m/s (40.3 miles per hour). Even with a low value of µ = 0.04, the approach distance required to achieve this maximum speed from rest on a 15 slope is more than 100 m [20]. Shown in Fig. 3 is a comparison of the EFH perturbations due to a positive horizontal (tail) wind of 6 m/s, 33% of the maximum jumper takeoff speed(18 m/s). At the maximum jumper speed this tail wind decreases the drag force at takeoff to only 44% of nominal, and by varying but similarly large amounts throughout flight. Drag slows the jumpers less than in the nominal case because the relative wind speed is decreased by 33% and the eventual impact point and distance jumped are increased substantially by about 3.52 m. Yet the EFH experienced at the perturbed impact point is actually decreased by 0.13 m to h = 0.87 m, because the design of the surface insures that it continues to slope downward more as distance increases, making it inherently insensitive. A 6 m/s headwind has the opposite effect, shortening the jump distance to x = 25.1 m and increasing EFH by a similar amount to h = 1.13 m. These are very small changes in EFH.

8 8 equivalent fall height (m) nominal half drag jump length (m) FIG. 2: Effect of aerodynamic drag parameter η on EFH. When the constant EFH surface is designed with η = corresponding to nominal values for its four constituent parameters ρ, C d, A and m, the EFH is the design value h = 1 m. When η = , the halved drag causes the high speed landing to occur at a longer jump distance x = 33.8 m and the EFH to decrease slightly to h = 0.88 m. Doubling the drag causes opposite effects: x = 24.7 m and h = 1.12 m. equivalent fall height (m) nominal 6 m/s wind jump length (m) FIG. 3: Effect of w = 6 m/s wind speed on EFH when η = and the constant EFH surface is designed including drag. This relatively large tailwind actually reverses the drag force for takeoff speeds less than 6 m/s (the first two data points in the figure) and significantly increases the landing distance to 34.4 m at the highest takeoff speed of 18 m/s. It has a small effect on EFH, however, comparable to that pruduced from halving the drag with decreases in η. A w = 6 m/s headwind reduces the distance jumped to x = 25.1 m and increases EFH to h = 1.13 m.

9 9 equivalent fall height (m) nominal no pop jump length (m) FIG. 4: Effect of complete lack of jumper pop on EFH, Since the nominal jump in Fig. 1 was designed for maximum jumper pop, absence of pop decreases the EFH substantially to 0.32 m for large jumps. E. Jumper Pop We return briefly to the issue of rider variability at takeoff. Probably the most common rider variable is the socalled pop, or jump just before takeoff. This phenomenon can be treated in design by altering the initial conditions as described in [18] and was examined experimentally by Shealy, et al. [27] who provided the raw data used by McNeil [28]. Of course, the EFH for the nominally designed surface is constant only if the rider leaves the takeoff at the same angle as the design assumes. Because adding pop alters the takeoff angle, the EFH for the resulting landings will be different. Since the relatively safe landing surface shown in Fig. 1 was designed assuming the maximum conceivable pop velocity of v p = 2.25 m/s, only negative perturbations in v p are considered here. Fig. 4 shows a comparison of the EFH perturbations due to a 100% decrease (total absence) of the maximum jumper pop speed(2.25 m/s). This figure shows that pop is probably the most sensitive design parameter and that the nominal design should include an assumption for a substantial amount of positive jumper pop. F. Snowfall and Melt A constant EFH landing surface similar to that shown in Fig. 1 must be able maintain its original design shape since it is this shape that accounts for the ability of the surface to provide impact protection by limiting EFH. Addition and subtraction of snow to and from the landing surface through snowfall and melt might cause the surface shape to change. It is a legitimate question whether these processes can affect the ability of the modified surface to continue to provide the constant value of EFH specified in the original design. To understand this shape change we must have models for the accretion and melting processes. Because snow falls essentially purely vertically at constant density per unit horizontal area, then every point on the landing surface grows in the vertical direction at the same rate (equal to the snowfall rate) and the shapes of the takeoff ramp and landing surfaces will remain unchanged. Other horizontal transport processes (e.g. wind) could be

10 10 equivalent fall height (m) nominal 1 meter snow melt jump length (m) FIG. 5: Comparison of EFH before and after a 1 m snow melt together with a vertical adjustment of the takeoff downward by 1.2 m. Melt increases EFH for small jumps to h = 1.13 m and decreases it to h = 0.86 m for larger ones. Because snowfall occurs vertically and changes neither the takeoff ramp nor the landing surface shape, EFH is not affected by snowfall. imagined to account for inhomogeneities in the mass density of deposition with x that might modify this assumption but these are neglected here. The location of the landing surface relative to the changing takeoff point would also remain constant, and the consequent equivalent fall height function h(x) would therefore remain unchanged as well. Thus snowfall, as long as the landing surface and takeoff ramp are packed similarly, does not change the EFH of a previously designed and built jump landing surface. This is not true for melting snow however. One conceivable model for the snow melt process is that snow is subtracted from the surface perpendicular to the surface. This process occurs much as burn of a solid rocket propellant does, in the inward direction normal to the surface. We here adopt a snow melt model that diminution of the surface occurs at constant rate normal to the surface. A unit vector in the normal direction to the snow surface is given by ) ˆn = ( y L (x) 1 / 1 + y L2 (x). The surface shown in Fig. 1 was shrunk in the normal direction at every point on its surface by allowing an S m snow melt using ( ) ( ) x s y Ls (x) = x y L (x) Sˆn (11) where x s and y Ls (x) are coordinates of the point on the new snow surface and S is the depth of the melt. Fig. 5 shows a comparison of the EFH on the melted landing surface after a snow melt of 1 m and a corresponding adjustment of the original takeoff point directly downward by δy =1.2 m, compared to that on the original constant EFH surface (Fig. 1). Because the radius of curvature of the initial surface is very large (of the order of 40 m in Fig. 1), snow melt of the order of a meter or two changes the curvature very little and results in relatively small (positive for small

11 11 TABLE II: Summary of Effects of Uncontrollable Factors - Maximum Change in EFH from Design Value of 1 m on 40 m jump Factor Variation δh(x) max magnitude (meters) Takeoff speed v 0 any 0 Friction µ any 0 Drag η -50% to +100% to Tail wind w m/s -6 to to Snow melt S m to Pop v p m/s jumps but negative for longer ones) changes in EFH of magnitude less than 0.16 m. Fig. 5 illustrates that a daily melt cannot destroy the relative safety of a constant EFH jump, but suggests it is likely that larger changes in snow depth over the course of a season may require maintenance of the surface shape. G. Variation in Landing Impulse We are not sure what is meant [24] by this uncontrollable factor. According to the time integral of Newton s second law, Impulse equals change in momentum. To annul the momentum the jumper has perpendicular to the snow surface at landing requires an impulse, defined as the integral with respect to time of the contact force from the snow on the jumper, and equal to the momentum mv before impact. The jumper has some (actually very little) control over how (in time) this impulse is absorbed but no authority to vary the impulse itself; it is by definition mv. The US Terrain Park Council [36] has recently adopted a 1.5 m maximum EFH guideline based on recent experiments showing that this value of EFH is the beginning of the range in which maintaining control of knee flex is lost by the jumper on impact. This value of EFH = 1.5 m corresponds very closely to the theoretical and experimental maximum values of absorbable fall height by elite ski athletes in a recent study by Minetti [37]. IV. SUMMARY AND CONCLUSIONS A. Comparison of Uncontrollable Effects on EFH Using a constant EFH surface designed to have h = 1 m assuming a nominal set of parameters, we perturbed each parameter in turn, holding all other parameters constant at their nominal values, and showed how the resulting EFH varied from the design value. The results are summarized in Table II. We have seen above that the effects of so-called uncontrollable factors on designed EFH are varied. They may be collected into three groups: 1) those for which there is zero sensitivity, i.e. an uncontrollable factor that makes no difference in the ability of the designed jump to deliver the designed EFH; 2) those for which fairly large parameter variations cause only insignificant maximum deviations in EFH δh(x) max, and 3) those for which the factor can be taken into account in the design process itself and its larger effect on δh(x) max completely eliminated in the unsafe direction. The striking fact in Table II is that, for very large changes in the nominal design parameters (sometimes of the order of 100% or more), small changes in the EFH are provided by the original surface shape at the perturbed conditions. Previous research [21, 30] has documented that conventional large tabletop jumps can subject jumpers to EFHs of the order of 4-6 m. Additional previous investigations (e.g. [22]) of actual terrain park accidents have uncovered values of h of 5 m, and some as large as 10 m! The small perturbations in EFH from changes in design parameters or uncontrollable factors and shown in Table II, pale in comparison to these large, potentially unsafe values of EFH, occasionally built into present day landing surfaces. The conclusion from examination of Table II is inescapable. The allegation that design of constant EFH surfaces is prevented, by the complexity of the problem and by the large number and types of parameter variations away from nominal, is false. Rather, the overwhelming impression is that these surfaces, by dint of the basic idea that the surface should ameliorate impact everywhere, are inherently insensitive to parameter perturbations. Further research should continue on the application of these ideas to terrain park jump landing surface shape design along the lines of [20].

12 12 Acknowledgments The authors acknowledge useful discussions with, and helpful suggestions from, James McNeil. We also acknowledge the help of one reviewer who presented compelling experimental evidence of the purely vertical nature of the snowfall process. [1] National Ski Areas Association, (accessed September 2011). [2] Goulet,C., Hagel, B., Hamel, D. and Legare, G., Risk factors associated with serious ski patrol-reported injuries sustained by skiers and snowboarders in snow-parks and on other slopes, Can. J. Public Health, Vol. 98, No. 5, 2007, pp [3] Shealy, J. E., Scher, I. and Johnson, R. J., Jumping Features at Ski Resorts: Good Risk Management or Not?, In: Skiing Trauma and Safety: Nineteenth Volume (ASTM STP 1553). West Conshohocken (PA): ASTM International; Forthcoming [4] Sutherland, A.G., Holmes, J.D. and Meyers, S., Differing injury patterns in snowboarding and alpine skiing, Injury, Vol. 27, No. 6, 1996, pp [5] Dohjima, T. and Sumi, Y. and Ohno, T. and Sumi, H. and Shimizu, K., The dangers of snowboarding: A 9-year prospective comparison of snowboarding and skiing injuries, Acta Orthopaedica Scandinavica, Vol. 72, No. 6, 2001, pp [6] Shealy, J. E., Ettlinger, C.F. and Johnson, R. J., Rates and modalities of death in the U.S.: snowboarding and skiing differences 1991/92 through 1998/99, STP 1397, Skiing Trauma and Safety: Thirteenth Volume, J. ASTM Intl., West Conshohocken, PA, 2000, pp [7] Greve, M.W., Young, D.J., Goss, A.L. and Degutis, L.C., Skiing and snowboarding head injuries in 2 areas of the United States, Wilderness and Environmental Medicine, Vol. 20, 2009, pp [8] Russell, K., Meeuwisse, W., Nettel-Aguirre, A., Emery, C. A., Wishart, J., Rowe, B. H., Goulet, C. and Hagel, B. E., Snowboarding Injury Rates and Characteristics Sustained on Terrain Park Features, presentation at ISSS Congress, Keystone, Co., USA, May 1-7, [9] Russell, K., The relationship between injuries and terrain park feature use among snowboarders in Alberta, Ph.D. thesis, University of Calgary, Alberta, Canada, [10] Henrie, M., Petron, D., Chen, Q., Powell, A., Shaskey, D., Willick, S., Comparison of Ski and Snowboarding Injuries that Occur Inside versus Outside Terrain Parks, presentation at ISSS Congress, Keystone, Co., USA, May 1-7, [11] Meyers, A.R. and Misra, B., Alpine skiing and spinal cord injuries: View from a national database, STP 1345, Skiing Trauma and Safety: Twelfth Volume, J. ASTM Intl., West Conshohocken, PA,1999. [12] Krause, J. S., DeVivo, M. J. and Jackson, A. B., Health status, community integration and economic risk factors for mortality after spinal cord injury, Archives of Physical Medicine and Rehabilitation, Vol. 85, No. 11, 2004, pp [13] Ackery A., Hagel B. E., Provvidenza C., and Tator C. H., An international review of head and spinal cord injuries in alpine skiing and snowboarding. Injury Prevention 2007;13: [14] Tarazi, F., Dvorak, M. F. S., and Wing, P. C., Spinal Injuries in Skiers and Snowboarders, Am. J. Sports Med., Vol. 27, No. 2, [15] Seino, H., Kawaguchi, S., Sekine, M., Murakami, T. and Yamashita, T., Traumatic paraplegia in snowboarders, Spine, Vol. 26, No. 11, 2001, pp [16] DeVivo, M., Whiteneck, G. and Charles, E., The economic impact of spinal cord injury, in Spinal Cord Injury: Clinical Outcomes from the Model Systems, S. L. Stover, J. A. DeLisa, and G. G. Whiteneck, eds., Aspen Publishers, Gaithersburg, MD, 1995, pp [17] Dijkers, M., Abela, M., Gans, B. and Gordon, W., The aftermath of spinal cord injury, in Spinal Cord Injury: Clinical Outcomes from the Model Systems, S. L. Stover, J. A. DeLisa, and G. G. Whiteneck, eds., Aspen Publishers, Gaithersburg, MD, 1995, pp [18] Hubbard, M., Safer Ski Jump Landing Surface Design Limits Normal Velocity at Impact, Skiing Trauma and Safety, 17th Volume, ASTM STP 1510, R. J. Johnson, J. E. Shealy, and M. Langren, Eds. ASTM International, West Conshohocken, PA, 2009, Vol. 17. [19] McNeil, J. A., and McNeil, J. B., Dynamical analysis of winter terrain park jumps, Sports Engineering, Vol. 11, No. 3, 2009, pp [20] McNeil, J. A., Hubbard, M. and Swedberg, A. D., Designing tomorrow s snow park jump, Sports Engineering, Vol. 15, No. 1, 2012, pp [21] Swedberg, A., Safer Ski Jumps: Design of Landing Surfaces and Clothoidal In-Run Transitions, M.S. Thesis, Naval Postgraduate School, [22] Salvini v. Ski Lifts, Inc., King County Superior Court, Case SEA, Seattle, WA, April 6, [23] Rietz, P., Ski Area Risk Management in US, presentation at ISSS Congress, Keystone, Co., USA, May 2, [24] National Ski Areas Association, Freestyle Terrain Park Notebook, [25] Shealy, J. E. and Stone, F., Tabletop Jumping: Engineering Analysis of Trajectory and Landing Impact, J. ASTM Intl., Vol. 5(6), [26] Shealy, J. E. and Stone, F., Erratum: Tabletop Jumping: Engineering Analysis of Trajectory and Landing Impact, J. ASTM Intl., Vol.?(?), 2012.

13 [27] Shealy, J., Scher, I, Stepan, L, and Harley, E., Jumper Kinematics on Terrain Park Jumps: Relationship between Takeoff Speed and Distance Traveled, J. ASTM Intl., Vol. 7, No. 10, doi: /jai [28] McNeil, J. A., Modelling the Pop in Winter Terrain Park Jumps, ISSS Congress, Keystone, CO, USA, May 1-7, 2011, and In: Skiing Trauma and Safety: Nineteenth Volume (ASTM STP 1553). West Conshohocken (PA): ASTM; Forthcoming [29] McNeil, J. A., The Inverting Effect of Curvature in Winter Terrain Park Jump Takeoffs, ISSS Congress, Keystone, Co., USA, May 1-7, submitted to J. ASTM Intl, In: Skiing Trauma and Safety: Nineteenth Volume (ASTM STP 1553). West Conshohocken (PA): ASTM; Forthcoming [30] Swedberg, A. and Hubbard, M., Modeling Terrain Park Jumps: Linear Tabletop Geometry May Not Limit Equivalent Fall Height, In: Skiing Trauma and Safety: Nineteenth Volume (ASTM STP 1553). West Conshohocken (PA): ASTM; Forthcoming [31] Streeter, V. L., Wylie, E. B. and Bedford, K. W. (1998) Fluid Mechanics, 9 th ed., McGraw-Hill, Boston, MA, pp [32] Hoerner, S. F., Fluid Dynamic Drag, published by the author, Bakersfield, CA, 1965, Lib. of Congress Card No [33] Kittel, C. and Kroemer, H.(1980) Thermal Physics, 2 nd ed., W. H. Freeman, San Francisco, CA. [34] Lind D. and Sanders S. (2003) The Physics of Skiing, 2 nd ed., Springer, New York, N.Y., 179. [35] Colbeck, SC. CRREL Monograph 92-2: A Review of the Processes that Control Snow Friction. Hanover (NH): U.S. Army Corps of Engineers, Cold Regions Research and Engineering Laboratory; [36] US Terrain Park Council, (accessed September 2011). [37] Minetti, A. E., Ardigo, L.P., Susta, D., and Cotelli, F. Using leg muscles as shock absorbers: theoretical predictions and experimental results of drop landing performance, Ergonomics, Vol. 41, No. 12, 2010, pp