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# 2011 MCM A 题答案

Design the half-pipe for Better Performance

Abstract
In order to design an idea model of the half-pipe course to meet different requirements, the conversion relationship between work and energy in the system consisting of snowboarders and the course comes first. Since the air drag is negligible, the work that friction does becomes the key point of solving the problem. We mainly use the differential equations and physical theories such as energy conversion and energy conservation to solve for the maximum value of the vertical air and the specifications of the course. And we build three models to gradually meet the requirement of the practical course. In the basic model, to simplify the calculation, we haven’t consider the slant angle of the course, and we design one with a semicircular transversal surface. We remove the flat bottom part of the course to minimize the work done by friction aiming at maximize the vertical air, meanwhile, the trajectory of snowboarders can be considered as a semi-circle. In the improved model, we take slant angle of the course into consideration, for it can provide the athlete with an extra acceleration in the race. Under this circumstance, the trajectory of athletes is almost a parabola. And we divide the athletes’ movement into two parts, the inclined one and the vertical one. But we draw the conclusion that without upper limit, we cannot determine the vertical air. So we reference the recommend specifications set by the FIS(Federation Internationale De Ski) and make the trad-off by limit the snowboarders’ maximum speed, thus, we can solve for the width and length of the course. In the final model, we consider the reaction time the athletes need to adjust their bodies and choose the ideal path, consequently, we add the flat bottom part into the course. In this model, we use a new integration method, meantime, to solve for the work that friction does, we divide the trajectory of athletes into three individual parts. To solve for the numerical value of specifications of the course, we use the data of some outstanding snowboarders in China, and we work it out. The final result is that the radium of the course’s transversal surface should be 3.5 meters. The length of the course should be 133.7 meters, the slant angle of it should be 16 degrees and the width of it should be 15.2 meters. Keywords:differential model, energy conversion, energy conservation,vertical air

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Contents

1

2 3

4 5 6

Introduction............................................................................................................................. ............. .....3 1.1 History of Snowboarding …………………………………………….……………………......3 1.2 half-pipe Snowboarding ………………………………………………………………………...3 1.3 What is vertical air? ……………………………………………………………………….……...5 1.4 How a skilled snowboarder will perform………………………………………………......5 Problem Description and Analysis…………………………………….………………………..6 Models……………......................................................................................................................... ...............6 3.1 Basic Model..........................................................................................................................................6 3.1.1 Model Overview………………………………..…...……………………….......................6 3.1.2 Terms Definitions and Symbols……………………………………………………….7 3.1.3 The Model ……………………………………………………….………….………….……7 3.1.4 Strength and Weakness………………………………………………….………….......11 3.2 Improved Model…………………………………….……………………………………..............11 3.2.1 Model Overview……………………………………..………………………...………………….............11 3.2.2 Extra Symbols……………………………………………….……………….…................11 3.2.3 The Model ……………………………………………………….………….………….…..12 3.2.4 Strength and Weakness………………………………………………….………….......15 3.3 Final Model…………………………………….…………………………………….........................15 3.3.1 Model Overview……………………………………..………………………...…………………............15 3.3.2 Extra Symbols……………………………………………….……………….…................15 3.3.3 The Model ……………………………………… ……………….………….…………. …15 3.3.4 Strength and Weakness………………………………………………….………….......20 Analysis of the Result.........................................................................................................................20 Future Work…………………………………….…………………………… ……….… …………....21 References …………....……………………….…… ……… ……………………....................... 21

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1 Introduction
In order to indicate the origin of snowboarding , the following background is worth mentioning.

1.1 History of Snowboarding
Snowboarding is a very recent sport and is similar to Surfing, skateboarding, and Skiing. Although it is hard to pinpoint the pioneer of Snowboarding, the History of Snowboarding tells us that it was initiated around the 1950s by a few surf and skate enthusiasts who used self-made boards to convey their skills to a new terrain: the snow. Snowboarding and Skiing are similar in several ways. On the other aspect, these two sports also have several differences. One would be that Snowboard riders frequently have to be seated or put forth energy to maintain on edge while at a stop, unlike in Skiing which makes use of poles to aid the skiers in moving uphill or downhill and to serve as support when in upright position. In 1983, less than 10 percent of United States ski areas allowed Snowboarding. But by 1997, few Resorts excluded it. Today, Snowboarding is now as accepted as Skiing in most Ski Resorts Worldwide and its popularity and fan base is growing at such a pace that the number of skiers in the US has declined by 25%. The number of snowboarders has increased by 77%, making Snowboarding the fastest growing winter sport in the US. Today, more than 3.4 million people Snowboard. This number comprises about 20% of the visitors to US ski resorts. Also, the number of people who snowboard is predicted to overtake Skiing by 2015.

1.2 half-pipe Snowboarding
Half-pipe Snowboarding traces its roots from skateboarding. It is basically an event where riders move from one side to the other of a u-shaped bowl or the half-pipe.They then make jumps and perform tricks while on mid-air. Half pipes originated in skateboarding, and the whole phenomenon has now made its way to snowboarding. Here we take a look at the Half-pipe Schematic drawn as figure 2:

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Figure 1. World Cup standard snowboard half-pipe. Kreischberg, Austria 2006.

Figure 2 ：Snowboarding Half-pipe Schematic These are the Elements of a half-pipe: Flat Is the center flat floor of the half-pipe Transitions/Trannies The curved transition between the horizontal flat and the vertical walls Verticals/Verts The vertical parts of the walls between the Lip and the Transitions Platform/Deck The horizontal flat platform on top of the wall Entry Ramp The beginning of the half-pipe where you start your run

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1.3 What is vertical air?
"Vertical air" is the maximum vertical distance that athletes fly above the edge of the half-pipe. Higher the athletes fly, longer delay time. And the player will have more time to complete the action. So you can view, flying height is the foundation of completing the difficult actions and performing more complex actions. In the statutes of half-pipe snowboarding skills development, there once had been one referee specifically designed for athletes to fly up high scores. From this we can see the height of flying is importance in half-pipe snowboarding. For half-pipe snowboarding, no height means no casting technology of space. In other words, no height strictly will lose the characteristics of athletic events of half-pipe snowboarding. You can think, flying height is the characteristics skill of half-pipe snowboarding, as well as a referee of half-pipe snowboarding requirements for high quality technical evaluation on snow skills. What is more， is the need of athletes to complete difficult and high quality technical it actions, as well as the need for quality, performance and innovation.

1.4 How a skilled snowboarder will perform
The basics of riding the half-pipe included traversing , slide turns ， jump turns. While a skilled snowboarder can accord to the specific needs, make the best possible choices, to achieve the best effect. First let we have a glance on the composition of the half-pipe snowboarding. half-pipe snowboarding in the region is divided into: jumping, arc-shaped transition area, the end of the landing area, arc-shaped transition area, landing area. One gliding skill of the players' actions should include: one release of the landing operation, decline after landing, tank bottom slide, arc transition region slide, take-off area slide, departure, in the air, take-off landing and so on as specific technologies. Departure: flying into the air should take advantage of the upper speed. A skilled snowboarder will continue to maintain natural posture flew up into the air. Actions should remain consistent ， example， action should be consistent with for arm foreleg action. In the air: After departure, athletes will fly up along the parabolic flight ， implementing the sky turning. A skilled snowboarder will curl his thighs as far as

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possible，keeping close to the chest and maintain that position. Before reaching the highest point he will keep an eye all the way to the tip above the line. Then when he feel reach the peak,his eyes turned to the observation of corner, preparing for landing. A skilled athlete in flight should always keep the body's natural, relaxed and reduce redundant action. Take-off and landing: Landing smoothly, athlete speed up at the arc transition region, then slide down. At the bottom of the first half athlete will adjust arc and body posture. At the bottom of the second half, athlete will stable glide preparing for next action. Due to the time and acceleration when players decline after landing, it is difficult for athletes to adjust body position in the arc transition region. But a skilled snowboarder can adjust taxiing blade when there is instability of the body posture. In fact, excellent athletes usually selected pick linear slide line arc adjusted.

2 Problem Description and Analysis
To solve this problem we need to build a optimized model to maximize the vertical air in the snowboard race, first we consider the snowboarders as particles and the model can be a kinematic differential model. We can neglect slant angle of the half-pipe course and air drag to build a basic model. And we can solve for the maximum of the vertical air by using differential equations and kinematic theories such as energy conversion and energy conservation. Apparently, the initial velocity of the snowboarders when they first reach the bottom of the course is vital to the vertical air, so we can provide them with an extra acceleration by building the half-pipe course on an inclined plane. Also take their reaction time into consideration, we can add the flat bottom part, so the racers could have more time to adjust their body position and choose their path. So here comes the improved model and the final model. We can solve for the vertical air and the radium of the course’s transversal surface along with the length of it by using differential equations and physical analysis.

3 Models
3.1 Basic Model
3.1.1 Model Overview
Here, we build a basic simplified differential model and analyze snowboarders’ motion in the half-pipe course which is built on a horizontal plane with a semicircular transversal surface. Under this circumstance, the trajectory of a snowboarder is almost

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a semi-circle, and it’s easy for us to understand and analyze their movement in the race. Since we neglect some complex factors, the model will be less practical, but in subsequent modeling those defects will be redeemed.

3.1.2 Terms, Definitions and Symbols
m g μ h θ r s The total mass of athlete and snowboard Gravitational acceleration Dynamic friction coefficient between the course and snowboard The maximum vertical distance that athletes fly above the edge of the half-pipe Angle between moving radius and vertical direction Curve radius of moving trace Length of moving trace

v0 Athletes’ velocity at the end of the entry ramp v Athletes’ velocity when taking off

v’ Athletes’ velocity at the highest point of the flight Fc Centripetal force acting on the athlete F E Joint forces acting on the athlete Kinetic energy of the athlete at the end of the flight

E0 Initial kinetic energy of the athlete L Length of the course

L1 The minimum distance athlete need to enter the course L2 The minimum distance athlete need to leave the course

3.1.3 The Model
? Step 1. Problem analysis: We analyze the movement of the snowboarder in the half-pipe. According to Newton's second law and the definite integration, we draw a conclusion that the friction is reciprocal to the radium of the half-pipe. Apparently, in order to maximize the vertical air, we should minimize the work that friction does to decrease the snowboarder’s kinetic energy. Thus, in the model, the radium of the half-pipe should be maximized in reasonable value range. Despite the friction coefficient between snow and snowboard can be very small, while moving on the flat bottom, the work that friction does can decrease the kinetic energy

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of the snowboarder. Therefore, we cut out the flat bottom of the half-pipe. So, in the basic model, we design the half-pipe as a course which lies on a horizontal plane with a semicircular transversal surface.

Figure 3:transversal surface of the half-pipe course We can see the detailed transversal surface of the half-pipe course in Figure 3 and the movement of the snowboarder showed in Figure 4.

? Step 2. Force analysis and solve differential equations: Through the force analysis of the system consist of snowboarder and snowboard, we can get equation set below:

Figure 4: transversal surface of the half-pipe course
? v2 Fc = m ? r ? ? F ? mg cos ? ? F c ? n ?f ? ? Fn ? ? Fa = ? f ? mg sin ? ?

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Substitute the inference result of the equation set into the formula of power, we get that instantaneous mechanical power dE equate instantaneous velocity of the snowboarder(same as the differential of snowboarder’s moving trace) times the joint force:
mv 2 ds r

dE ? F ds ? ?mg sin ? ds ? ? mg cos ? ds ? ?

According to kinetic energy formula:

1 2 mv ? E 2
And the formula of the relationship between trajectory and the direction of the speed:

r?
Substitute them into formula(2), we get:

ds d?

ds ? dE ? d? ? 2 ? E ? ? mg (sin ? ? ? cos ? ) d? ? ? 1 E ? mv 2 ? 2 ? ? 1 2 E0 ? mv0 ? ? 2

When

?=
Solve the differential equations, we get:

?
2

Ee2?? ? E0e? ?? ? ?? e2?? mg ? sin ? ? ? cos? ? ds
s 0

According to the model, the transversal surface of the course is semi-circle, so we have:

ds ? r 2 ?? ? ? ?r ' ?? ?? ds ? ?
2

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And there is always:

r ' ?? ? ? 0
Substitute them into result(4), we get equation below:

Ee2?? ? E0e?2??0 ? ?? e2?? mg ?sin ? ? ? cos? ? rd?
?0

?

Substitute equations below into the result,
1 2 1 2 mv ? E & mv0 =E 2 2

We get:
1 2 ?? 1 2 mgr mv e ? mv0 ? ? 2 ? e?? ? 1? ? 2? ? 1? 2 2 4? ? 1

When the snowboarder takes off, his trajectory can be considered as a circular motion in vertical plane. When he reach the peak, the force of gravity provide centripetal force:
v '2 h

mg ? m

During the period the snowboarder take off till he reach the peak, according to the theorem of kinetic energy, we have:

1 1 ?mgh ? mv'2 ? mv2 2 2
Solve it, and we get:
2 v0 2r h? ? ?? ? e?? ? 1? ? 2? ? 1? ?? 2 3ge 3e (4? ? 1)

Here, h refers to the height from horizontal plane to the peak. ? Step 3. Radium of the course:

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Observe the equation we get, we find that h is reciprocal to r. Thus, we should make a trade-off to decide r, in order to provide snowboarders better course, and we should also take the time they need to adjust their bodies’ balance into consideration. Here, for the time they need is proportional to the speed, so, we should maximize the radium of the half-pipe under the condition that snowboarders’ initial speed at the end of the entry ramp is limited. Here we can see that the initial speed plays a vital roll in this problem. According to previous research, if one wants to get a ideal score in the race, the initial speed at the end of the entry ramp cannot be less than 11.06 meters per second and cannot beyond 15 meters per second. So, we make the trade-off and get the most suitable radium of the half-pipe should be 3.5 meters, which is in accordance with the provisions of FIS(Federation Internationale De Ski). ? Step 4. Length of the course: The last thing we should consider is the length of the course. According to some research and the game rules, we know that snowboarders supposed to make 5 to 8 jumps and perform tricks while on mid-air. But generally, they make 7. So, we assume that every snowboarder make 7 jumps, as figure 4 shows, we can get that the length of the U-shaped course:

L ? 14h ? L1 ? L2
3.1.4 Strength and Weakness
? We considered snowboarders as particles and analyze the their twist movement during the flight, proved that the twist in the air won’t influence the vertical air. The transition part of basic model is a smooth semi-circle, and the flat bottom part of it is cut out, so as to increase the vertical, and it can simplify the compute.

?

3.2 Improved Model
3.2.1 Model Overview
Based on the basic model, we consider building the half-pipe course on an inclined plane. Since that slant angle can provide an extra acceleration, and contribute to higher vertical air, so snowboarders can get a longer hang time and perform better tricks.

3.2.2 Extra Symbols
Fc21 Centripetal force acting on the athlete in the vertical direction to the course

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Fn21 Stamina acting on the athlete in the vertical direction to the course Fa21 Joint force acting on the athlete in the vertical direction to the course f21 Friction acting on the athlete in the vertical direction to the course

3.2.3 The Model
? Step 1. Force analysis: In order to analysis the relationship between slant angle and vertical air, we assume the slant angle of the half-pipe course beα , and we analyze the forces in both direction which is parallel and vertical of the course. In the direction vertical to the course, we have equation set below:

Figure 5:longitudinal section of the half-pipe course

? v2 Fc 21 = m ? r ? ? F ? mg cos ? cos ? ? F c 21 ? n 21 ? f ? ? Fn 21 ? 21 ? Fa 21 = ? f 21 ? mg cos ? sin ? ?
? Step 2. Solve differential equations: Substitute the inference result of the equation set into the formula of power, we get that instantaneous mechanical power dE equate instantaneous velocity of the snowboarder(same as the differential of snowboarder’s moving trace) times the joint force, we get:
mv 2 ds r

dE ? F ds ? ?mg cos ? sin ? ds ? ? mg cos ? cos ? ds ? ?

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According to kinetic energy formula:

1 2 mv ? E 2
And the formula of the relationship between trajectory and the direction of the speed:

r?
Substitute them into formula(2), we get:

ds d?

ds ? dE ? d? ? 2 ? E ? ? mg (sin ? ? ? cos ? ) d? ? ? 1 E ? mv 2 ? 2 ? ? 1 2 E0 ? mv0 ? ? 2

When

?=
Solve the differential equations, we get:

?
2

Ee2?? ? E0e? ?? ? ?? e2?? mg ? sin ? ? ? cos? ? ds
s 0

According to the model, the transversal surface of the course is semi-circle, so we have:

ds ? r 2 ?? ? ? ?r ' ?? ?? ds ? ?
2

And there is always:

r ' ?? ? ? 0
Substitute them into result(4), we get equation below:

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Ee2?? ? E0e?2??0 ? ?? e2?? mg ?sin ? ? ? cos? ? rd?
?0

?

Substitute equations below into the result,
1 2 1 2 mv ? E & mv0 =E 2 2

We get:
1 2 ?? 1 2 mgr mv e ? mv0 ? ? 2 ? e?? ? 1? ? 2? ? 1? 2 2 4? ? 1

? Step 3. Vertical air: When the snowboarder take off, his vertical motion can be considered as an uniformly retarded motion. When his vertical velocity reduced to zero, he comes to the peak. According to kinematics formula, we have:
v2 2 g cos ?

h?

During the period when snowboarders take off till they reach the peak, according to the theorem of kinetic energy, we have:
? mgh cos ? ? 0 ? 1 mv 2 2

Solve it, and we get:

h?

v0 2 (4? 2 ? 1) ? r (e?? ? 1)(2? ? 1) 2 ge?? (4? 2 ? 1) cos ?

? Step 4. Length of the course: Now, we need to solve for the length of the U-shaped course. In the direction which is perpendicular to the course, we neglect the pressure on the course caused by snowboarders’ circling motion, we can solve for the time snowboarders need to take to perform 7 complete jumps by using the kinematics formula:
8(v0 ? v) 14v ? g cos ? ? g cos ? cos ?

t?

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So, the length of the U-shaped course should be:
8(v0 ? v) g sin ? 14v ( ? )2 ? L1 ? L2 2 g cos ? ? g cos ? cos ?

L?

3.2.4 Strength and Weakness
? We take the slant angle of the half-pipe course into consideration, which makes the vertical air higher, and is more convenient for snowboarders. We haven’t consider the reaction time of snowboarders, for we remove the flat bottom part of the course to decrease the work friction does. We haven’t analyze the force acting on the snowboarders every moment, because its complexity.

?

?

3.3 Final Model
3.3.1 Model Overview
Based on the improved model, we add the flat bottom part into consideration. This will provide snowboarders more time to adjust their path, in order that they will be safer and can perform better in the air. Also, the model will be more conform to reality.

3.3.2 Extra Symbols
α Slant angle of the half-pipe course θ ’ The angle between snowboarders’ initial position and present-position Wf1 Work that friction does to the snowboarders in trace AB Wf2 Work that friction does to the snowboarders in trace BC Wf3 Work that friction does to the snowboarders in trace CD vt the instantaneous velocity when the snowboarder take the jump t time that snowboarders take to reach the peak in the flight X the component part of trace AB in X’ axis Y the component part of trace AB in Y’ axis

3.3.3 The Model
Since we consider the flat bottom. The trajectory of the snowboarders is shown in figure 6. We assume that the angle between the speed and the positive direction of x’

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axis always beβ . In figure 6-a, we expend the course to a plane, so, it’s obvious that the trace of the snowboarder is A-D. In figure 6-b, we establish a space rectangular coordinate system as shown. Then we resolve the speed, friction and gravity into two planes, in order to analyze the forces and solve them.

a Figure 6

b

Take trace A-D for example, we divide it into three parts: AB(gliding down on the transition part), BC(during the flat bottom) and CD(gliding up on the transition part). So we can analyze each part individually. Trace AB and CD consist of two kinds of motion, one is the rectilinear motion along the x axis, another is the circular motion in plane Y-Z whose radium is R. Thus, we get: X = X and Y ? X tan ?

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Figure 7 First, we resolve v0 (the speed of snowboard) in x axis and plane Y-Z. Then we get the friction of the snowboard when it goes down and the component force of it on plane Y-Z( fr ). So we can get the joint force acting on the whole snowboard on plane X-Y. According to the analysis of forces, we can get the tangential and normal equation of the motion. In trace AB:

v2 ? v cos ?
mg cos ? cos ? ' ? fr ? m dv2 dt

N ? mg cos ? ? m

v2 r

Take the derivative of the equations with respect to x, we get:
dv dN d? ' ? mg cos ? cos ? ' ? 2mv2 2 dt dt dt dN d? ' d? ' ? mg cos ? cos ? ' ? 2(mg cos ? cos ? ' ? ? N sin ? ) dt dt dt

This can be reduced to:

dN ? 2? N sin ? ? 3mg cos ? cos? '

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' ' ' 3mg cos ? e(2 ? cos? sin ? ?sin? ) ? c e( ?2 ?? sin ? ) 2 2 1 ? 4? (sin ? )

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N?

Here, c is decided by the initial conditions. When the snowboarder is at point A, we have:
(v0 sin ? ) 2 N ?m R

Meanwhile,

?' ? 0
(v0 sin ? )2 3mg cos ? c1 ? m ? e2 ? sin ? 2 R 1 ? 4? (sin ? )
? ' ' ' 3mg cos ? W f 1 ? ? 2 ?[ e(2 ? cos? sin ? ?sin? ) ? c1 e( ?2 ?? sin ? ) ]d? ' 2 2 0 1 ? 4? (sin ? )

Solve it, and we get:
c2 3mg cos ? e2 ? sin ? ? e( ? ?? sin ? ) 2 2 1 ? 4? (sin ? ) 2? sin ?

Wf 1 ?

In trace BC:

Wf 2 ? ?mg cos ? SBC
In trace CD:
(vc sin ? )2 ? mg cos ? R

N ?m

? ? (v sin ? )2 3mg c2 ? e(?? sin ? ) ? mg cos ? ? m t ? 2? R 1 ? (4? sin ? ) ? ?

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Wf 3 ?

c2 3mg cos ? e2 ? sin ? ? e( ? ?? sin ? ) 2 2 1 ? 4? (sin ? ) 2? sin ?

We use the theorem of kinetic energy in the whole period of motion, and we get:
mgY cos ? ? W f 1 ? W f 2 ? W f 3 ? 1 1 mvt 2 ? mv0 2 2 2

Figure 8 Then, the trajectory of the snowboarder can be considered as a parabolic shape in plane X-Z. Thus, according to the kinematics formula, the vertical air is:
vt 2 2g

h?

h ? gt cos? '
x ? 2(vt t cos ? ? 1 g sin ? ? t 2 ) 2

The skilled snowboarders will perform 7 jumps in the race, so we get the length of the half-pipe course:

L ? L1 ? L2 ? 7 x ? 7 X
In order to calculate the specific numerical values of each part of the course, we search the data of some outstanding Chinese snowboarders, shown in table 1:

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Table 1: Partial kinematics parameters of outstanding veneer skiers in China when they are in the course velocity when the nose velocity when the Angle between body name of the board contact the board fully contact and blade(° ) course(m/s) the course(m/s) Shiying 13.68 12.51 2.6 Huang Xiaoye 11.27 10.50 3.7 Zeng Jiayu Liu 12.08 10.62 1.2 Wancheng 11.06 9.04 5.4 Shi Lei Pan 14.93 10.82 2.4 Take the provisions of FIS and trad-offs into consideration, we use the recommend data from FIS and the athletes’ data in table 1, substitute them into our final model, and we get the best architectural design of the snowboard course: The radium of the course’s transversal surface should be 3.5 meters. The length of the course should be 133.7 meters. The slant angle of the course should be 16 degrees. The width of the course should be 15.2 meters.

3.3.4 Strength and Weakness
? We add the flat bottom part into consideration so snowboarders have more buffer time to adjust their position and path. We simplify the calculation by separate the trace of the snowboarders into three individual part. We haven’t consider the condition that the snowboarders’ trajectory may be semiellipse and the continuous change of their barycenter.

?

?

4 Analysis of the Result
According to the international snow league veneer rules,the gradient of the halfpipe minimum for 14° recommend for 16° and maximum for 18° , , ,Site for the minimum length of 100 m respectively, recommend 120 m, the biggest 140 m,the wide of the halfpipe is 14m, 16m or 18m.Its depth is 3 m, 3.5 m or 4.5 m. In accordance with the date of the model that meets the international snow united veneer rules of the parameters specified requirements, so the model is successfully

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meets the needs of the athletes fly high.

5 Future Work
To optimize the model, here are some more things to consider: ? Computer simulation technology can be used to simulate the whole motion of snowboarders in the race. And we can gather more data of envionment variables such as friction coefficient of every part of the course and the air drag to make the model more conform to reality. We can analyze snowboarders’ twist movement in the flight and not consider them as particles.

?

6 References
[1] 国家体育总局冬季运动管理中心. FIS 单板滑雪竞赛规则[M].北京: 人民体育 出版社, 2006 [2] 李淑媛. 我国单板型场地滑雪项目的发展与项群训练理论的应用[ J]. 冰雪运 动, 2008(3): 46-48. [3]王葆衡 对 U 型场地单板雪上技巧项目基础滑行和飞起高度的思考 沈阳体 育学院学报 2005 212-215 [4]徐信洪 由功的定义式计算物体沿圆弧轨道下滑时摩擦力的功 舟山师 专学报（自然科学版） 1998（1） [5] 王葆衡. 第 11 届冬运会女子单板前三名技术与都灵冬奥会女子 前三名技术的比较分析[ J]. 沈阳体育学院学报, 2008( 2): 25 [6]Jason William Harding, Colin Gordon Mackintosh, David Thomas Martin, Allan Geoffrey Hahn and Daniel Arthur James, Automated scoring for elite half-pipe snowboard competition: important sporting development or techno distraction?, Sports Technol. 2008, 1 , No.6, 277–290 [7]The International Snowboard Competition Rules(ICR) ,2010

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### 2011美赛MCM赛题翻译数学中国提供

2011MCM一等奖论文WORD版 20页 5财富值 2011美赛B题解答 23页 免费 数学建模...A 题:单板滑雪场地 请设计一个单板滑雪场 (现为“半管”或“U 型池”) ...

### 2011美赛MCM赛题中英对照版

2011年美国大学生数学建模竞赛MCM竞赛试题。提供中英对照,仅供参考。隐藏>> 2011 Contest Problems MCM PROBLEMS PROBLEM A: Snowboard Course Determine the shape of...