美赛优秀论文

Team # 11840 Page 1 of 23
The Design of Snowboard Halfpipe
Abstract: Based on the snowboard movement theory, the flight height
depends on the out- velocity. We take the technical parameters of four
sites and five excellent snowboarders for statistical analysis. As results
show that the size of halfpipe (length, width and depth, halfpipe slope)
influence the in- velocity and out- velocity. Help ramp, the angle between
the snowboard’s direction and speed affect velocity ’s loss.
For the halfpipe, we established the differential equation model, based on
weight, friction, air density, resistance coefficient, the area of resistance,
and other factors and the law of energy conservation. the model’s results
show that the snowboarders’ energy lose from four aspects
(1) the angle between the direction of snowboard and the speed, which
formed because of the existing halfpipe
(2) The friction between snowboard and the surface
(3) the air barrier
(4) the collision with the wall for getting vertical speed before sliping out
of halfpipe.
Therefore, we put forward an improving model called L-halfpipe，so
as to eliminate or reduce the angle between the snowboard and the
speed .Smaller radius can also reduce the energy absorption by the wall.
At last, we put forward some conception to optimize the design of the
halfpipe in the perspective of safety and producing torsion.
Key words:

snowboard; halfpipe; differential equation model;L-halfpipe

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Contents
1. Introduction..... .................................................. .................................................. .... 3
简介
1.1 the origin of the snowboard course problems................................... ...................3
滑雪课程的起源问题。

1.2 the background.................................... .................................................. ..............3
背景
2. The Description of P roblem............................................ ........................................3
问题的描述
2.1 Practical halfpipe’s requirement s................................................. ........................3
实用halfpipe的需求

2.1.1 the maximum vertical and the largest body twist...........................................3
最大垂直和最大的身体扭曲

2.1.2
Speed analysis< br>............................................... .................................................. ...3
速度分析
2.2 Halfpipe’s own conditions... .................................................. ..............................4
Halfpipe自身的条件

2.2.1 Friction.......... .................................................. ................................................4
摩擦
2.2.2 the size of halfpipe............. .................................................. ...........................4
halfpipe的大小
3. Model...................................... .................................................. .................................4
模型
3.1 Definitions and Symbols........................... .................................................. .........4
定义和符号
3.2 Assumptions.......... .................................................. .............................................5
假设
3.3 the simple analysis of gravity and friction when sliding in the halfpipe.............5
简单的分析重力和摩擦力的halfpipe时滑动

3.4 in- velocity of factors............................... .................................................. ...........6
速度的因素
3.4.1 the snowboarder’ angle when in and the speed loss.......................................6
滑雪在角和速度上的损失
3.5 out-velocity of factors... .................................................. .....................................8
初速度的因素
3.5.1 Help ramp............................. .................................................. .........................8
帮助坡道
3.5.2 the force point and the plate angle when out...... .............................................9
力的点和板角

3.5.3 the snowboarder’ angle when out and the speed loss......................................9
滑雪在角和速度上的损失
3.5.4 Halfpipe’s Radius....... .................................................. .................................11
Halfpipe的半径

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3.6 the in-velocity comparison with the out- velocity................. .............................14
速度与速率的对比

3.7 Snowboarder’s position impact on the speed .................................................. ...14
滑雪的位置影响速度
3.8 the entire movement of the energychange in the halfpipe..................................15
在halfpipe中整个运动的能量变化
3.9 the balance of speed after considering the air resistance....................................18
后速度的平衡考虑空气阻力

3.10 L-halfpipe.......... .................................................. .............................................19
左halfpipe
3.11 Solution and Result........ .................................................. ................................20
解决方案和结果
4. Conclusions................................ .................................................. ...........................21
总结
4.1 Conclusions of the problem…………………………………………...………21
结论的问题
5. Future Work...................... .................................................. ...................................21
工作展望
5.1 other models………………………………………………………….....……..21
其他模型
5.1.1 Halfpipe’s location outdoor………………………………………………..22
Halfpipe位置的户外

5.1.2 Halfpipe’s material………………………………………….....…………...22
Halfpipe的材料

6. References............. .................................................. ................................................22
参考文献

Team # 11840 Page 4 of 23
1. Introduction

In order to indicate the origin of the snowboard course problems, the following
background is worth mentioning.
1.1 The origin of the snowboard course problems
In the past, a significant amount of half pipe anxiety was due to the learning curve
of a new sport, and educating resorts and pipe construction person nelson how to
prepare the best shapes with basic resort equipment. This mode of operation is
changing with the advent of new snowboard specific technology both in machine and
hand tools. As technology has made half pipes better, the standards have also been
proved. Most half pipe riders have a vision of what an ideal pipe should look like, but
shifting that vision into reality seems to be a quantum leap.
1.2 The background
The problem lies in the fact that too many people who control the decision making
process view of the half pipe as a fixed and static feature, and that once built, a pipe is
left to the forces of nature. A severe change of opinions needed, as the half pipe needs
to be thought of as an elastic form (almost lifelike) that changes daily and which
needs continual maintenance. Another huge factor in developing consistent half pipes
is a set of standards. Over the years, the NASBA, OP, USASA, USSA, ISF, and FIS
have given differing pipe dimensions to resorts. All this help from various
organizations has left pipe building more of an art than a science. Both the ISF and
the FIS are now promoting similar versions of half pipe dimensions. So we need to
redesign the shape of a snowboard course to maximize the production of vertical air
by a skilled snowboarder.
2. The Description of the Problem
2.1 Practical halfpipe’s requirements
2.1.1 the maximum vertical and the largest body twist
Snowboarders’ greatest height, the number of rotations (the largest body twist) and
the beautiful action will affect the athlete's score. the longer the spare time left, the
more rotations to do for snowboarders. The basic physics principle at work here is the
conservation of angular momentum. The angular momentum of the snowboarder is
determined at takeoff, and cannot be changed once the snowboarder is airborne. So to
make turns in the air the snowboarder must give himself initial rotation upon takeoff.
In order to reach the maximum height, the maximum out-velocity would be
we analyzed the in- velocity and the out- velocity, and the shape of space
(length, width, depth, field gradient) affect the in- velocity and the out-velocity
the height can not be too high, because too high speed would be a big
threat to the safety of snowboarders. Therefore, in order to control the maximum
speed, we need to redesign the halfpipe.
2.1.2 Speed analysis
Whether to reach the maximum vertical height or to produce the largest body twist
speed is is a reflection of practical indicators to the halfpipe composition
of the factors in the ing the fly height, difficulty, diversity, quality

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completion of the action, Site use and landing conditions and so on because the height
have an limit effect on difficulty, diversity, quality of action completement, so the fly
height is the core elements of many conclude,no height,no no flight time
and no flight time,no difficult action.
As the free fall shows:
V
y
?2gh
.The height snowboarders can reach have a very
close relationship with the speed.
2.2 Half pipe’s own conditions
2.2.1 Friction
Friction, including friction between the board and the snow as well as air
dynamic friction coefficient between Snow and the board changes from
0.03 to 0.2 for example, the maximum friction coefficient and the full effect
of body weight to calculate the vertical friction
f?0.2W
, that the acceleration less
due to friction is generated to accelerate the role of body weight 0.2 times, much
smaller than resulting in the acceleration of gravity effect. Air friction
f
a
=0.5r
a
C
d
Av
2
, in our model, we do not consider the influence of air friction.
2.2.2 the size of halfpipe
Under certain circumstances,as the length, depth, tilt angle increases, the speed will
be. In view of snowboard safety, speed can not be infinite, which has some of the
value of the constraints.
3. Models
3.1 Definitions and Symbols
Flat
：
the bottom ground of U groove
Transitions
：
the transition zone between Horizontal and vertical groove bottom wall
Verticals
：
the vertical parts of the walls between the Lip and the Transitions
Platform
：
the level platform on the snow wall surface
Entry Ramp
：
the slippery position of U-shaped slot
m
：
Athlete's quality
g
：Gravity acceleration
V
1
：Athletes’ speed when first enter u-shaped slot
V
t
：Athletes’ speed when last sliding out u-shaped slot

l
1
：under side rectangular width of U-shaped slot

l
2
：the length of U-shaped slot
R
：the deep of U-shaped slot
n
：Athletes emptied times

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?
：Angle between Athletes’ speed and slot edge horizontal when first enter
u-shaped slot

u
：
the frictional factor between Skateboarding and snow
A
f
：how much work friction do when Athletes vertically into a u-shaped slot in arc
C
d
：Air resistance coefficient
?
a
：Air density
A
：
Corresponding to the projective area of
v

3.2 Assumptions
ng frictional factor is a constant when athletes are in taxiing process
ng no melting snow when athletes are in taxiing process)
ng the maximizing friction is gravity, frictional factor as the biggest 0.2,
when compared friction work and gravity work
ng the loss of speed is 2 meters per second because of the Angle between
the speed and direction of existence with blade when athletes come into (out) the
slots every time
3.3 the simple analysis of gravity and friction when sliding in the
halfpipe
If the athlete slip into the half pipe with a certain speed. Athletes in motion of
constantly falling in vertical direction Increasing gravitational potential energy. The
process in motion need to overcome the frictional resistance acting between the skate
and snow acting must also overcome the air resistance acting. We use all ski areas in
China to analyze the data[1] as follows in Table1:
Table 1 “Board rules

Site slope
??

o
?
：Athletes’ speed relative air movement
Minimum
14
100
14
3
Recommended
16
120
16
3.5
Maximum
18
140
18
4.5
Site length
?
m
?

U-width
?
m
?

U-depth
?
m
?

Table 2 National snowboard half pipe skiing skill to the situation Championship
Series


the mountain Bai Qing Zhai Harbin Ya Bu
The mountain
of Shenyang Li
of Maor in
Yun Fo in

Team # 11840 Page 7 of 23
Beijing
halfpipe slope
Site length
Harbin
16.0
110.0
30.2
27.6
15
3.0
18.0
110.0
34
30.9
15
3.5
17.0
150.0
43.8
29.2
18
5.0
??

o
10.0
110.0
19.1
17.4
15
2.8
?
m
?

Altitude gap
?
m
?

Gap(%)
U-width
?
m
?

U-depth
?
m
?

From the Table 2, The data can be seen through the site, The competition in the
17 ° slope of more than 100 meters along the length of glide in the groove The
competition is in the17 ° slope and along the length of more than 100 meters slide in
the grooves and do all kinds of flip, twist, grasp the difficulty of board action, the
action is completed in a certain vertical height of drop. The standards of international
competition venues, can be obtained by calculating the U-groove vertical drop
h?150*sin17
?
.Those athletes complete the maneuver in the vertical direction to
produce the height of 40 meters gap. A gap of more than 40 meters in the vertical
direction athletes can have a very substantial increase in the rate. A gap of more than
40 meters in the vertical direction athletes can have a very substantial increase in the
rate. In terms of free fall calculations
V
y
?2hg?2?10?40?20
m s, However the
snow and the board’s dynamic friction coefficient between 0.03 to 0.2, the maximum
friction coefficient and the full body weight to calculate the friction force acting
perpendicular
t
of
?0.2W
.That the speed less is due to the friction resistance, it is
weight generated to accelerate the role of body weight 0.2 times, far less than the
acceleration of gravity produces results. Therefore, venue’s height of fall is an
important way for athletes obtained the vertical velocity. Athletes can complete the
vertical velocity and level velocity conversion with a reasonable technology, So that
Athletes most likely to get to the maximum vacate height at the last vacate.
3.4 in- velocity of factors[1]
3.4.1 the snowboarder’ angle when in and the speed loss
Players control the skis taxiing around the edge of the board into the slot，both
the before and the after of snowboard have the effect of braking, so in order to reduce
the loss of speed, so that，the speed of the body center of gravity in the same direction
with the board's longitudinal axis as far as possible，to reduce the braking effect when
the snowboard have instant contact with the snow, and homeopathic slide, taking full

Team # 11840 Page 8 of 23
advantage of wall height difference obtained acceleration. It can be seen the speed of
full contact is less than the speed of front panel from Table 3, indicating that the
human body has a loss of speed when completely into the slot, Since the existence of
wall resistance, the speed loss is normal. However, if the speed of body center of
gravity has the same direction with the blade, the speed of the losses will be reduced.
As can be seen from Table 3, the athlete’ gravity speed direction has an angle with
direction of blade center, the minimum is
1.2
?
, and the maximum is
5.4
?
, the speed of
direction and the direction with the blade did not reach exactly the same. Decrease the
maximum rate reached 27.5%, a minimum rate of 6.8%.
Table 3 board Kinematic Parameters of some outstanding players when into half pipe.
name action The speed of
gravity when part
of snowboard
contact with half
pipe
(ms)

The speed of
gravity when
the whole
snowboard
contact with
half pipe
(ms)

Shi Wan
Cheng
Huang
Shi Ying
Zen Xiao
Hua
Liu Jia
Yu
Pan Lei
anti-front
720

o
The loss
speed
(ms)

The rate of
loss speed
（％）
The angle
between
speed and
bald
??

o
11.06 9.04 2.02 18.3 5.4
anti-front
720

o
13.68 12.51 1.17 8.5 2.6
front
720
o

11.27 10.50 0.77 6.8 3.7
behind
540
o

behind
540
o


12.08
14.93
10.62
10.82
1.46
4.11
12.0
27.5
1.2
2.4

Team # 11840 Page 9 of 23

Figure 1 the angle between the rate of speed loss and direction with the blade when
into the slot
It can be seen that the speed loss rate and direction with the blade angle has not
exactly the same trend from Figure 1, there may be several reasons as follows:
(1)players is not very skilled when sliding into the slot, the ability of controlling board
is not strong
(2) It may require different sliding speed for the different air movement in the next
time, resulting in players want to control taxi speed on purpose
(3)the center of gravity is too forward, the gravity torque is too large, have Side effect,
So the technology will have a major impact in speed.
3.5 out-velocity of factors [1]
3.5.1 help ramp
Athletes for the first time into the slot before sliding into the slot with help,
Athletes should be actively obtained the speed of access to controlled, If the
snowboarder into the slot before , after slide a certain distance at the edge of the slot,
Obtain a certain speed. and before leaping into the slot and in a certain height
E
0
,
you'll get some initial energy reserves
E
0
=E
动0
+E
势0< br>(
E
0
Representative athlete of
the initial energy,
E
动0
representing athletes initial kinetic energy,
E
势0
.
Representing athletes Initial potential) With the completion of the action into the
groove, getting smaller and smaller potential energy athletes to complete, in the case
of gravity does positive work, the potential energy of the players is corresponding

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increase, that the athletes will get the vertical speed by energy transfer. After get some
of the vertical velocity into the tank, the athletes have a certain amount of kinetic
energy reserves; athletes using the kinetic energy reserves, transformation to the
potential when out the half pipe, it can achieve the purpose of improving flight
altitude; flight altitude do reserve for potential of the next action into the half pipe for
the next action to provide time and space to ensure the successful completion
However, athletes in the kinetic and potential energy conversion, to achieve the speed
must be controllable. If the speed is not controllable, it will affect the athlete’s
performance, Otherwise it will lead to serious accidents. From Table 4, it can be seen
that the athletes Lei Pan rear positive blade rate of
540
?
movement into the tank the
largest; is
14.93ms
, the minimum Shi wan Cheng's anti-blade rate of
720
o
front foot
movement, is
11.06ms
. The actions are successful action, but also a national athlete,
so you can give a preliminary conclusion: the speed of athletes in the following speed
control 15 meters per second.

3.5.2 the force point and the plate angle when out
In the trench wall of the moment, because of losing the support of the front skis,
then, the stress point should be to leave the center of board, and gradually transition
back to the board, so that the stress point is always forcing plate wall, front foot
homeopathic slide, back foot should be gradually forced pedal. When reaction force in
sufficient, maintain parabolic path smooth, increasing the speed, and maintain a
reasonable angle of the slot. At the same time of achieving the goal of increasing
height highly effective, also get into the appropriate slot speed and angle of twist.
Reaching movements while floating high, reducing the level of speed and the effect of
resistance into the half pipe, reasonably come into the groove; do energy reserves for
the next the action.
3.5.3 the snowboarder’ angle when out and the speed loss
Table 4 some snowboarders’ Kinematics parameters

the speed of
snowboard
when part of
snowboard
contact with
half pipe
(ms)

the speed of
snowboard
The angle
between
The rate
of loss
speed
（％）
name

action

when on part speed and
of snowboard
bald

contact with
half
pipe
(ms)

??

o
Shi
Wan
Anti-front
720

o
11.39 7.73
The board
have turned
32.1

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Cheng

Sun Zhi
Feng

Huang
Shi
Ying

Zen
Xiao
Hua

Liu Jia
Yu

behind
540

o
（20）
front
720

o
10.20 8.24 4.0 19.2
Anti-front
720

o
13.73 12.09 3.9 11.9
front
720

o
11.65
11．55
0.3 0.9
11.20

12.00
9.82 4.1 12.3
Pan Lei

behind
540
o

9.11 3.0 24.0
Table 4 is part of the elite athlete’s slotting board kinematic parameters. By
comparing the data in Table 4, we can find what the speed of completely clear out the
slot is less than the speed of the front panel instantaneous slip out the slot. It can be
seen that five players’ speed and the direction of blade angle have positively
correlated with the loss rate in Figure 2, indicating that the greater of angle between
speed and direction with the blade, the greater of loss speed, so you need to control
the sliding board direction, letting the long axis have the same direction with the
speed of human body.

Figure 2 the angle between the rate of speed loss and direction with the snowboard

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when out of the halfpipe
3.5.4 Halfpipe’s Radius
Appropriate reduced orbit radius can increase the speed when athletes slip out half
pipe, and favor the athletes to make various actions in the air. Sides rail identifiable
by two
14
arcs, we can deduces the formula
tf

Then taking orbit design into consideration, the optimal speedup method is to
reduce the rail depth (by our hypothesis know depth and arc radius is equal), namely
ti
w
f
?(I
oi
w
i
?
?
?
M
o
dt)I
of
decreases
r
of
, and so can reduce
I
of
, effectively increase
w
f
. But, taking the athlete's
safety into consideration, the athletes' speed may not excessive, namely orbit radius
cannot be too small. General provisions half pipe orbit radius scope for 3-4.5m,
guarantee the slot speed are not more than 15, also ensures the athlete's safety.
The basic snowboarding physics behind this phenomenon can be understood by
applying the principle of angular impulse and momentum.
The schematic of the physics of snowboarding in this analysis is given below.

Figure 3 the analysis of force
Where:

Team # 11840 Page 13 of 23
w
i
is the initial angular velocity of the body (consisting of snowboarder plus
board), at position (1)
w
f
is the final angular velocity of the body, at position (2), which is the point at
which the snowboarder exits the half-pipe
V
i
is the initial velocity of the center of mass G of the body, at position (1)
V
f
is the final velocity of the center of mass G of the body, at position (2)
r
i
is the initial distance from the center of rotation o to the body's center of mass
G, at position (1)
r
f
is the final distance from the center of rotation o to the body's center of mass
G
, at position (2)
g
is the acceleration due to gravity
N
is the normal force acting on the snowboard, as shown
F
is the friction force acting on the snowboard, as shown
It is assumed that the half-pipe is a perfect circle with center at o. The physics of
snowboarding in this analysis can be treated as a two- dimensional problem. Now,
apply the equation for angular impulse and momentum to the system (consisting of
snowboarder plus board):
tf
I
oi
w
i
?
?
?
M
o
dt?I
of
w
f
ti

Where:
I
oi
is the initial moment of inertia of the body (consisting of snowboarder plus
board) about an axis passing through point o and pointing out of the page, at position
(1)
I
of
is the final moment of inertia of the body (consisting of snowboarder plus
board) about an axis passing through point o and pointing out of the page, at position
(2)
?
M
o
is the sum of the moments about point o. These moments are integrated
between an initial time
t
i
(at position 1) and a final time
t
f
(at position 2)
Here we are assuming that the body can be treated as rigid at positions (1) and (2),
even though the snowboarder does in fact change his moment of inertia between these

Team # 11840 Page 14 of 23
two positions. But as it turns out, when using this equation we only need to know the
initial and final values of the moment of inertia of the body.
The line of action of the normal force N passes through point o, so it does not
exert a moment on the body about point o. The friction force
F
is small so it can be
neglected in terms of its moment contribution. This leaves only the gravitational force
which exerts a moment on the body about point o. (Note that the gravitational force
acts through the center of mass of the body, consisting of snowboarder plus board). In
the above equation isolate
w
f
. Thus,
tf
w
f
?
I
oi
w
i
?
?
?
M
o
dt
ti
I
of

Now,
I
oi
?I
Gi
?mr
i
2
I
of
?I
Gf
?mr
f
2

Where:
I
Gi
is the initial moment of inertia of the body about an axis passing through
point G and pointing out of the page, at position (1)
I
Gf
is the final moment of inertia of the body about an axis passing through point
G and pointing out of the page, at position (2)
m
is the mass of the body
In the above equation for
w
f
, if we decrease
I
of
the angular velocity
w
f
will
increase beyond the value it would be if we did not decrease
I
of
. In practice this can
be accomplished by sufficiently reducing the distance from the center of mass of the
body G to the point o. In other words, make
r
f
small enough and
w
f
will increase.
Note also that the terms
I
Gf
and
?
M
o
may also change somewhat. But the
dominant effect will be that of reducing
r
f
.
At positions (1) and (2), the velocity of the center of mass G is given by
V
i
?w
i
ri
V
f
?w
f
r
f

These two velocities are parallel to the half-pipe since the body is rigid at
positions (1) and (2).

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Looking at the above equations for velocity, if we makes
r
f
small appropriate,
the snowboarder will significantly increase
w
f
. This in turn will result in his velocity
exiting the pipe (
V
f
) being greater than otherwise.
3.6 The in-velocity comparison with the out- velocity [1]
It can be seen that the speed of athletes when athletes slip out half
pipe is less than the speed of athletes when athletes slip out half pipe
from Figure 4. The biggest difference between the two is the Shi wan Cheng,
the smallest difference between the two is that Zen Xiao Ye. The average
speed is
11.69ms
when slip into half pipe, the average down is
1.94ms
,the
speed decline will lead to altitude declining when slip out half pipe,
having effect on the speed of slipping into half pipe next time, which
restricts movements of athletes and sports techniques to improve the
difficulty level of play, but also make the action quality greatly reduced,
so the players should pay attention to the completion of a continuous
action of the hair lower limb muscle strength.


Figure 4 the chart of comparison about speed change when into (out of)half pipe
3.7 Snowboarder’s position impact on the speed
Pumping on a half-pipe is used by snowboarders to increase their vertical take-off
speed when they exit the pipe. This enables them to reach greater height and perform

Team # 11840 Page 16 of 23
more aerial tricks, while airborne. The principle is exactly the same as for
skateboarders pumping on a half-pipe.
The snowboarder is able to increase his speed on the half-pipe with his feet
remaining firmly on the board. This begs the question, what is the physics of
snowboarding taking place that enables the snowboarder to increase his speed on the
half-pipe?
To increase his speed, the snowboarder crouches down in the straight part of the
half-pipe. Then when he enters the curved portion of the half-pipe he lifts his body
and arms up, which results in him exiting the pipe at greater speed than he would
otherwise.
Looking at the above equations for velocity, if the snowboarder makes
r
f
small
enough (by lifting his body and arms up), he will significantly increase
w
f
. This in
turn will result in his velocity exiting the pipe (
V
f
being greater than if he did not lift
his body and arms up.
By continually pumping his body (by crouching down and lifting his body and
arms up in the curved portion of the half-pipe), the snowboarder is able to continually
increase his velocity, eventually allowing sufficient height to be reached (upon exiting
the half-pipe) to perform a variety of mid-air tricks.
A more intuitive (non-mathematical) explanation of the physics of snowboarding
taking place here is that pumping adds energy to the system in the same way that a
child pumping on a swing adds energy, and results in him swinging higher. Therefore,
the physics of snowboarding related to pumping on a half-pipe is similar to pumping
on a swing.
As a snowboarder lifts his arms and body up he feels resistance due to the force of
centripetal acceleration which tends to push his body away from the center of rotation
o. This resistance is proof that work is being done, and therefore energy is being
added to the system.
3.8 the entire movement of the energy change in the halfpipe

How the energy change during the Athletes’ entire movement in the half pipe.

Team # 11840 Page 17 of 23

Figure 5 3-D half pipe


?



l
2



R


?

l
1



Figure 6 halfpipe’s cross-section


From Figure 6, we can know both sides of the curved part is the 1 4 cylinder in
the side, the middle is rectangle.
As shown, we assume that the depth of half pipe is R, the middle part length is
l
1
,
the width of half pipe is
l
1
+2R
, the half pipe’s length is
l
2
, half pipe’s inclination
angle is
?
.
When the athletes straight down into the tank by the vertical speed, we analysis
the friction’s work in this process.
When the athletes straight down into the tank where has friction, the friction’
work can be applied to functional principle, considering the given state can find out
friction’s work, But this does not consider the specific forms of friction force. By the
analysis of analytical solution, we can describe its distribution characteristics.[2]

Team # 11840 Page 18 of 23
As shown in Fig 7, Objects satisfied Newton equations, the tangent of the form
and normal directions form is (considered
f?uN
),
mgcos
?
?uN?m
dv

dt
，————————————————（1）
N?mgsin
?
?mv
2
R
，————————————————（2）


R
?


N
f






mg
Figure 7 objects in circular orbit force
Pray for (2) a derivative time
dNd
?
mdvd
?
and (1) David into the type
?mgcos
?
?2v
，
v?R
dtdtRdtdt
dNd
?
d
?
， ?mgcos
?
?(2mgcos
?
?2uN)
dtdtdt< br>dN
?2uN?3mgcos
?
—————————————————— (3)
d
?
Solving (3) type is the key to solve the solution of friction

?
?
?
3mgcos
?
*exp
?
?
2ud
?
?
*d
?
?c
?
——————————（4）
?exp (?2u
?
)
?
?
3mgcos
?
*exp
?
2u
?
?
*d
?
?c
?
N?exp?
?
2ud
?
?
cos
?
sin
?
exp(2u
?
)?
?
exp
?
2u
?
?
d
?
2u2u
Among them;
cos
?
sin
?
1
?exp
?
2u
?
?
?exp
?
2u
?
?
?
2
?
cos
?*exp
?
2u
?
?
*d
?
2
2u< br>4u4u
?
cos
?
*exp
?
2u
??
d
?
?
namely
?
cos
?
* exp
?
2u
?
?
*d
?
?
So:
N?
exp(2u
?
)
(2ucos
?
?sin
?
)
，
2
4u?1
3mg
(2ucos
?
?sin
?
)?Cexp(?2u
?
)
，
2
1?4u

Team # 11840 Page 19 of 23
Because of:
?
?0,N?0
， will,
C??2u
So:
N?
3mg
.
1?4U2
3mg
(2ucos
?
?sin
?
?2uexp(? 2u
?
))
———————————（5）
2
1?4u
3 mg
And:
f?uN?(2ucos
?
?sin
?
?2u *exp(?2u
?
))

2
1?4u
?
A
f
?
?
?uNds?
?
?uNRd
?
??
?
??
3umgR
?2u
?
2ucos
?
?si n
?
?2ued
?
2
0
1?4u
??
3u mgR
(2usin
?
?cos
?
?e
?2u
?< br>)
2
1?4u

3.9 the balance of speed after considering the air resistance [3]
If in the process of straight downhill snow is flat and snowboard does not leave
the ground can be approximately described by plane hinged to the relationship
between ski and snow we watch skis and skiers as a whole force people ski and
snowboard in the force of both concentration and reduced to a couple Torques’s’. At
this point slide in the snow is equivalent to a single degree of freedom motion system
as Figure 8
When the system is in static equilibrium with
?
mgsin
?
?(f
a
?f
r
)?0< br>
?
f?mgcos
?
?0
?
s
Which
f
a
?0.5
?
a
C
d
Av
2< br>,
f
r
?uf
s
.Joint Solution available
：
f
r
?umgcos
?
Can be seen, friction and gravity components is balance at the balance.


Figure8 single degree of freedom motion system
Into the above equation can be obtained:
mgsin
?
?(0.5< br>?
a
C
d
Av
2
?umgcos
?
)?0

v?
2mg(sin
?
?ucos
?
)

?
a
C
d
A
1?u
2
)
，
suppose
?
?sin
?1
(u

Team # 11840 Page 20 of 23
Simplification
：

v?2mg1?u
2
sin(
?
?
?
)
?
a
C
d
A
2mg1?u
2

?
a
C
d
A
when
?
?
?
?90
， Has a maximum value：
v?
o
Visible: the speed of downhill factors: weight, friction, air density, drag coefficient,
drag area the greater weight and the tilt angle slopes, the greater the equilibrium rate,
whereas the smaller ,the greater the friction coefficient, air density, drag
coefficient, ,and the resistance area , the smaller the equilibrium rate, whereas the
larger. Access parameters
m=68kg
，< br>g=9.8
，
u=0.03
，
r
a
=1.2kgm< br>3
，
C
d
=0.45
A=0.3m
3
Research balancing speed in different snowboard.
3.10 L -halfpipe
Existing assistant slide, as a result of the movement plane when into the orbit is
different from the half pipe; there is a certain angle between them. Snowboarder into
the orbit each time, the speed direction is needed to change, so when the athletes
started going to tank with energy loss.
Obtained from the above analysis, Athletes of the average loss rate into the tank
is2ms .To reduce energy loss, we have removed the original slide boost, the half pipe
is designed to
small piece of material,
tf

I
o i
w
i
?
?
?
M
o
dt
tiw
f
?
I
of
We can see from the above formula. L type reduced the radius of the B side arc,
reducing the radius of the tank can improve the speed and benefit snowboarder do all
sorts of trick. In addition, we can see from the theorem of energy conservation: As
both sides have a certain height difference between two wall and wall A and wall B
gap difference can reserve some gravitational potential energy .Gap width of the
design. Formal competition, the track at some distance will draw a color line; here we
are tentatively scheduled for the red, hint the Figure and athletes will be completed
within the three red cells on both sides wall once therefore, the gap width should not
greater than the width of 3 red cells, and the snowboarder can not climb more than
twice in the gap at the slope. Notch depth of the design, we can see from the energy
conservation theorem:
E
A势
=E
B动
+E
B势
+E
A?B损

In summary, from Figure 9,snowboarder with the initial velocity of zero from the A
point of departure, there is a gap in the B point, the mechanical analysis, the gap
width is not greater than 3 red cells, the notch depth according to the speed required

Team # 11840 Page 21 of 23
by snowboarder so the design of ’L’-half pipe help increase the speed and promote
their athletes play. Snowboard does not have to worry about Energy reduction with
Assistant slide.
3.11 Solution and Result
Athletes can easy achieve maximum post-flight height in the last time by our
analyzing; we obtain the biggest speed when athletes in the air through mechanical
analysis for the system, namely balance speed. In order to protect athletes, we
analyzed know what athletes can control maximum speed for 15 meters per second by
himself, when analysis the process of athletes in (out) a groove, because of player's
speed direction and using blade direction have angle, athletes has certain existing
energy loss Finally, because of the athletes’ energy conservation in the whole
movement, we write equations contains four variables:
?
,l
1
,l
2
,R
, Then using control
variable method, first control three variables, then optimized another variable, finally
get the optimal solution of the four variables, Due to the athletes state to swivel action
in the air when leave the U-shaped slot. According to the laws of momentum
conservation, we analysis the relationship between halfpipe and post-flight height.









l
2


A
b

B
R

c

l
1

Figure 9 the design of L-halfpipe
When athletes in a certain initial speed enter half pipe, the direction of speed with
the u-shaped slot edge horizontal direction existing angle
?
, Hypothesis athletes
vacate total
n
times in entire movement process in half pipe, Then in the whole
movement process, friction’s work is
2*n*A
f
sin
?
when athletes Movement in slope ,
while friction’s work is
n*umgl
1
when athletes Movement in flat.
sin
?

Team # 11840 Page 22 of 23
Of the above analysis, we assume that the loss of speed for 2ms when the athlete
slip into (or out) the half pipe because discrepancy slot speed direction and direction
1
with blade, then energy loss is
2n*mV
0
2
?4nm
in the whole sports.
2
Hypothesis athletes first with speed
V
1
enter half pipe, the last time with speed
V

sliding out half pipe, half pipe tilt for
?
, then in the whole process, gravity work
for
mgl
2
*sin
?
, energy conservation formula is:
2*n*A
f
n*umgl
1
11
mgl
2
*sin
?
?mV
12
?mV
t
2
???4nm

22sin
?
sin
?
During the optimization process, we establish
?
for certain value, the value
u

scope to 0.2 between in 0.03, the goal is to get the maximum, then by using lingo
programming optimization, get the optimization results are as follows:

4. Conclusions
4.1 Conclusions of the problem
?
The smaller the snowboarder’s angle when in is,the smaller snowboarders’ energy
lose. So, we put forward a L-halfpipe regardless of the assistant slope.
?
The
smaller Halfpipe’s radius is, the greater the speed can reach. So, for
L-halfpipe one side’s radius is big, while the other is small. And the gap is the
main way to gain energy, L-halfpipe is conducive to the savings of some Initial
energy.

?
The greater friction is, the greater energy ore, in order to get enough
speed, we should reduce the friction, which requires a certain smoothness
halfpipe.
? We require speed the bigger the better. But, in view of snowboarder’s safety,
speed limits will generally require less than 15
m
s
.
5. Future Work
5.1 other models
5.1.1 Halfpipe’s location outdoor
We make an analogy, just as Einstein's theory of relativity explained (story of
standing next to the beauty and fire.). Because of the direction of rotation of the Earth
from west to east, so the sun always rises in the east, down the west. Assuming the
direction football is east to west , if competition in the morning, sunrise, the entire
morning in the east direction of the sun, the sun will direct players to attack the east,
the athlete's eyes shone dazzling; if the game in the afternoon, the setting sun, the sun
in the afternoon west direction, the sun will be the offensive player is on the west, the
sparks fly from the athlete's eyes tan, athletes would not dare look up, which of course
will affect the athletes. Football is a north-south, the sun was just coming from the

Team # 11840 Page 23 of 23
irradiated side of the athletes, and Athletes can avoid direct sunlight, and will not
affect the athletes. Similarly, outdoor half pipe is the best north-south direction, as
shown, construction of the compass pointing out the halfpipe layout situation in
Figure 5.
5.1.2 Halfpipe’s material
Under the Snow, the panel should have some flexibility, Can know by the theorem
of momentum
I=Ft
, which can make players have a sufficient buffer time, have an
active effect on snowboarders’ show. In a nutshell, half pipe panel wood structure,
steel structure supporting materials. In this way, the stability of the structure
considered half pipe, but also considers the optimal selection problem.
6. References
[1] YAN Hongguang. Liu Ping. Guo s Influencing Velocity Away from
Decks in Snowboard Half-pipe. Journal of Shenyang Sport University. Jun. 2009.
[2] Xu s along the arc rail slide friction with numerical
solution of the analytical solution. Journal of Gansu education college .Ju1．1999.
[3] Chen Li .The Biomechanical Simulation of Skiing Movement .Form page 40 to
page 1.2009.
[4] Return from The Physics Of Snowboarding to Real World Physics Problems home
page