数学建模真题训练「MCM/CUMCM」-滑雪场地设计（原创）美赛英文版

Abstract

  In this paper, we analyzed the snowboarding process and designed the half-pipe slide, which can make the skilled athletes perform vertical flight to the greatest extent.

  Firstly, according to the data, the half-pipe slide is a U-lag with a certain inclination. In order to facilitate the analysis of the problem, the slide section under the flat state is studied without considering the inclined condition of the slide. Based on the force analysis of the athletes in a certain place of the slide, the energy conservation law is used to express the energy transformation relationship when a periodic slide (from one edge of the slide to the same edge) is carried out in the section and the vertical flight is generated. Then, for the slide analysis under the inclined state, regarding the flat of U-lag bottom’s median as X-axis. A horizontal line which is perpendicular to X-axis is Y-axis. The intersection of X-axis and Y-axis is point O. The line which is perpendicular to surface XOY is Z-axis. Through force analysis, the work done by friction force in X, Y and Z directions is expressed, and substituted back into the energy conversion equation above, and the expression of vertical lift distance on each part parameter of the slide is obtained by simplification, At the same time, every parameter has its corresponding conditions, in the case of meeting these conditions, the vertical distance, when you pick up the maximum suspension parameters appear effective convergence solutions: l（slide length）=118.6m, h（height of a vertical part）=1.2m, R（radius of the circle similar to transition）=2.4m, d（flat width）=15m, H（height of slide without leaning）=3.5m, µ（dynamic friction coefficient in slide）=0.05，θ（angle of inclination）=18°, the curvature of the transition is 0.41667. The slide constructed according to these parameters is the slide meeting the requirements of the problem.

Secondly, because the athletes need to consume energy in the process of soaring, and it needs a certain angular velocity and height to complete the flipping and return to the slide when they leave the slide in a certain situation. Before in a model on the basis of further assumptions, seek of the center of mass velocity, leaving the track of air rotating angular velocity and turnover time expressions, and take into the slide speed of 15m/s, take off and landing speed loss rate was 13% and 17%, and into the edge angle velocity direction and chute (hereinafter referred to as the angle) is 90°， 87° and 70° and arbitrary. We simulate movement locus severally. Next, we found that as long as entering velocity is 15m/s and angle is 70°, athletes can complete aerial demanding actions if they slide 10~12 meters in X-axis. Meanwhile, the velocity  we used in model1 is also 15m/s and the angle is arbitrary, and length in X-axis is enough. So model1 is qualified and the shape meets the requirements.

Finally, through online search and looking through books and journals, we learned about the needs should be meet when building the snowboarding course.  After sorting out, we divided into safety and environmental aspects to describe.

Keywords: simplified model, law of conservation of energy, three dimensional coordinate system, energy consumption, moment of inertia.

1. Introduction

1.1. Background

  Snowboarding, which originated in the United States in the 1960s, is still a new sport in the word, but it has become popular all over the world with its own charm. It was designated as a winter Olympic sport by the international federation of ski and snowboarding in 1994.

  Snowboarding requires special skiing areas and halfpipe runs. In snowboarding, the skaters need to use the half-pipe slide to generate a vertical lift (“vertical lift” is the maximum vertical distance beyond the edge of the half-pipe slide.), and then complete a variety of spins and jumps in the air (mainly including leaping catch plate, jump non-grab, handstand, leaping handstand, rotate).

  For the challenging winter extreme sport, good field conditions make training more efficient. The design of half-pipe slide is particularly worth considering in the training field.

1.2. Our tasks

•  Design a snowboarding course, making skillful athletes perform “vertical lift” as much as possible.

• Optimize the snowboarding course shape, to satisfy the need of performing maximum twist. 

•  What should we consider to ensure the snowboarding course is practical?

2. Assumptions

  Due to the fact that some factors cannot be controlled artificially in the actual situation and have some influence on the problems. In order to facilitate the establishment of the model, the following assumptions are made:

• Assume that the ski resort in model 1 and 2 exists without wind direction, that is, without considering air resistance.

•  Assume that the weight of the skier and the skis are evenly distributed, and that the skier and skis are always together during the modeling process.

•  Athletes are regarded as particles in the establishment of vertical flight correlation model.

• Assume that the initial speed of a skier entering a snowboard resort is easy to control.
• Assume that the coefficient of kinetic friction is the same everywhere in the slide.
• Assume that only the gravitational potential energy is taken into account when the athlete takes off vertically.
• Assume that the angular velocity remains the same when the athlete flips in the air.

3. Symbol Descriptions

  In the process of building the model, the following symbols are defined and used. For the convenience of reading, the meanings of each symbol are summarized as follows:

Symbol		Meaning
α   β    θ  l  h  R  d   H    v0   v2  vB  vC  vD  f   rf  N  μ  Wf   vmax   G  Gmax	The slope angle of snowboarding course  The angle between the X-axis and line AD The angle between the supporting force and Y-axis on YOZ The length of snowboard resort The height of the vertical wall The radius of curvature of the slide Width of slide bottom flat The height without inclination Initial speed The value of v0 θ projected onto YOZ The instantaneous velocity at point B The instantaneous velocity at point C The take-off velocity at point D Frictional force The value of f projected onto YOZ The normal force of a skier The coefficient of friction Work done by sliding friction Maximum sliding velocity The acceleration of gravity acting on the skis Maximum gravity acceleration when balanced

 

4. Problem Analysis

4.1 Problem one

Need to establish the relationship between the vertical lift and the parameters of each part of the half-pipe slide track，to determine the shape of the half-tube slide when maximum height is reached in the air. It is necessary to establish the relationship between the vertical evacuation and the parameters of each part of the half tube slide to determine the shape of the half tube slide when the vertical empty reaches the maximum limit.

First of all, the analysis is made on the section diagram of the half tube slide. By using the law of conservation of energy, the variation of energy is expressed when the athletes slide and rotate in the section.

Secondly, the force acting on the glide is analyzed in the three-dimensional slideway diagram with slope，and the concrete expression of the friction work in each direction during the actual gliding process is obtained. Substitute back into the relation of energy transformation, and simplify to get the expression of the vertical clearance distance with respect to the parameters of each part of the slide. Under the condition that the parameter limit condition is satisfied，he parameters of the corresponding slide are the optimal condition when the vertical lifting distance reaches the maximum value.

4.2 Problem two

We need to find out the conditions that the athletes need to meet when they finish the flipping in the air, to determine whether the shape of the half tube slide path designed in Problem 1 is suitable. If appropriate, it is good; If not, optimize. In the process, the energy loss of athletes must be considered.

4.3 Problem three

Need to investigate and collect relevant information that should meet the actual conditions in the construction of ski slides. Such as: safety factors, environmental factors, etc.

5. Models

5.1. Establishment and solution of model 1

5.1.1 Preparation of model

Snowboarding is a winter sport that evolved from skateboarding and surfing. There are three main types of skiing: alpine skiing, freestyle skiing, and cross-country skiing. Snowboarders use one board and use their body and feet to steer. This paper studies the u-shaped field relative to the skilled skiers. Snowboarding is a winter sport that evolved from skateboarding and surfing. There are three main types of skiing: alpine skiing, freestyle skiing, and cross-country skiing. Snowboarders use one board and use their body and feet to steer. This paper studies the u-shaped field relative to the skilled skiers. Table 1 shows the relevant professional concepts of the "half-pipe" track: shows the relevant professional concepts of the "half-pipe" track:

 

Table 1: U-shaped slide’s relative professional concepts

Symbol	Professional concepts	Concepts explanations
d	Flat width	The flat of U-lag bottom
	Transitions	Cambered transitions between U-lag bottom and vertical parts
h	Verticals height	The vertical parts of the walls between the Lip and the Transitions
----	Platform	A platform on the snow wall
c	Entry Ramp	U-lag’s entry ramp

 

The relevant parameters of "single tube" will vary according to the needs of athletes, which is also known as the u-shaped field. The quantitative way to determine its shape is to calculate its most scientific parameters according to the needs.

5.1.2 Establishment and solution of the model

To study the problem, determine the maximum height "vertical" suspension, need to comprehensively consider the effect of circuit parameters bring players (including the transition curve, base platform width, etc.), because the track is three-dimensional structure, to facilitate understanding, model from simple to deep, from intuitive to complicated, first of all, a plane model of the vertical section, study one-way players particle motion;

Fig. 1  sectional view of half pipe slide

 

When A contestant moves from point A to point D and starts entering the track vertically with v0, then:

By the law of conservation of energy

                              （1）

be

 

where  is the internal energy generated by snowfield friction on snowboard.

The key of this conception is to find out the distance of skidding in the transition zone, where the curvature and radius of curvature are introduced【1】；

Curvature: is the bending degree of the equilibrium curve, It's denoted by the symbol.

The definition of a reference derivative, Find the mean curvature,  define AB.

The average bending of a curve，Among them  is the change Angle of the tangent line on the curve AB, is the arc length of AB。

By:                                                            

                    (2)

and

                                      (3)

obtain radius of curvature,

                               (4)

Fig .2  stress analysis of a certain position in the half pipe slide

 

First of all, decompose  in the plane，And then you get the magnitude of the friction that the player is causing as he's sliding down due to gravity, and the frictional force in the y-z plane, In the  plane of the coordinate system, we can obtain the stress analysis of the entire process of the contestant (as shown in the figure2).According to the law of conservation of energy, the initial state v0 is decomposed into a process of tangential motion:

                               (5)
 

and

                                   (6)

According to the graphic analysis,fr=μNsinβ, among them f2 is generated by gravity in the Y-Z plane, and ,

then

                         (7)

Take the partial derivative of (5)【2】，and let  and（6）,if plug in x for 5, we will know,

be            

                          (8)

As shown above，this is a first order linear inhomogeneous equation，the method first uses arbitrary constant variation, then：

   (9)

due to ,

（10）

Let the competitor's speed point at A be v0, and，so    

                          (11)

then

                          (12)

The friction force on curve AB:

 (13)

Combined with (1) :

be

The expression of h0 is denoted as:

       12mv2=mgh0-[f1σR+μmgd]            （14）

◆ This expression can be used to calculate the vertical flight height  relatively directly. However, it is difficult to see the most accurate parameter value corresponding to it, and the slope Angle of the track is not taken into account, which is not in line with the reality. The more comprehensive and scientific three-dimensional track will be analyzed as a whole below.

Fig. 3  halfpipe slide stereogram

Combined with the experience of the above model, the coordinate system as shown in figure 3 is established.

The movement status of a cycle in the sliding process is as follows: Enter the track at point A, then slide along AB on the bottom platform to point B, then slide to point C on the BC section of the platform, then slide onto the uphill track to point D. Taxi from point D to transition point D1 and empty. Back to point D' after motion again into the u-shaped pool, complete a fly. Since the D' slide downhill to C', Along the platform to B', after is uphill section to A', transition section to A1',by A1' empty curve fell after exercise to A1",By A1' 'point curve suspension movement after the fall back to A1'' to complete the second empty, The transition section have A'' to complete A cycle of movement.

Since the motion characteristics of the itd and the flying segment are consistent, the motion characteristics of the two segments are merged. Therefore, force analysis was performed ',C'B',B'A' and A'A”,then the equations of motion were established

Fig. 4  side view of u-tube slide

The establishment of the equation of motion, in the curvilinear coordinate system as shown in the figure,ax,ay,az represent the accelerations along the tangent line y, the principal normal line z,and the adjoining normal line x in curvilinear coordinates, respectively. Since it is reciprocating motion, There are differences in force analysis in different directions of motion. In order to express clearly, In order to express clearly,the first cycle can be divided into A-D'Dthe first stage of D'-A second phase of analysis.

Fig 5  force analysis diagram of the first stage

1) The force analysis in the first stage is shown in the figure. According to Newton's second law, the dynamic equations are established： part goes down

                (15)

where the sliding friction is opposite to the motion direction and has a magnitude of .

If the Angle between the total friction and the x direction is 

                             （16）

2) BC part,

                   (17)

The friction force is the same as equation (16).

 3) CD section upward

                (18)

The friction force is the same as equation (16).

4)  Parabolic segments

                          (19)

In the second stage, the force distribution diagram is shown in the figure. According to Newton's second law, The following dynamic equations were established:

Fig. 6  schematic diagram of force analysis in the second stage

 

1） section downward

                (20)

The friction force is the same as equation (16).

2）   part

                   (21)

The friction force is the same as equation (16).

3） section upward

                (22)

The friction force is the same as equation (16).

4）Parabolic segments

                          (23)

5.1.3 The model results

Since there is no analytical solution, in order to verify the numerical model, the convergent solution is adopted as the reference solution in this paper, that is, when the parameter value is small enough, the physical quantity is basically unchanged.

Selecting=15ms,, When the time step values are different, the displacement of one period is shown in Fig. 7.

Fig .7  relationship between displacement and time step

 

When the time steps are 0.001，0.0005 and 0.0001，It can be seen that the displacement and velocity curves basically coincide. In this paper, the 0.0005 solution is taken as the reference solution to calculate the error value. Fig. 10 shows the relationship between the downward tangential and binormal displacements at different time steps with time.

Fig .8  peak point error of tangential and auxiliary

normal displacements of the first period

         Fig .9  error of key points of the first cycle speed

Fig. 10: tangential and subnormal displacement time histories of the first perio

◆Analysis of 3d models

  Three-dimensional coordinate axes are established for the three-dimensional model of the half-pipe slide (as shown in Fig.11):

 

Fig.11  3d model of half-pipe slide

In which, being the Angle between the bottom plane of the track and the ZOY plane. The height h above the vertical wall in the slide is set as the slope Angle of the model with zero scriber.

So,

                           （24）

When:

1Vmax is the velocity when the athletes are losing control.

2It’s balanced when there’s maximum gravitational acceleration.

3Athletes can change its direction and start perform actions in reaction time.

If the take-off velocity is ，for Newton’s law of motion， is  .           

        

Fig.12  Displacement versus velocity

 

  According to the initial speed into the characteristics of movement, the influence of groove Angle when v0 = 15 m/s, a ϵ (0 40), players complete a periodic motion displacement calculation, the characteristics of the change with t:        

       

Fig.13  First period tangential and subnormal displacements

 

As the direction of the initial velocity is more and more close to the direction of the edge of the parallel slide, the time to complete the first movement becomes longer. As the initial velocity is gradually perpendicular to the edge of the slide, the tangential displacement and velocity increase, while the binormal displacement and velocity decrease, and the total velocity decreases. Study the influence of the size and direction of the initial velocity on the motion characteristics : when v0ϵ[10,18],αϵ[0,20], study the effects of initial velocity on flight height and flight time.(shown in Fig.14)

 

Fig.14  The relationship between initial velocity and flight height and flight time

 

◆Characteristic analysis of the whole process

When orbital tilt Angle θ= 18 °, selecting initial velocity v0 is 15 m/s, a0= 0 , friction coefficient 0.05 calculating the basic model to complete six empty action trajectory and the displacement, speed change over time, as shown in figure 15, sum up speed of the model under the condition of the tank, hang time, distance, suspension lift height, velocity and Angle and pool (see annex 1);In order to further study the movement characteristics and conduct a comparative study, when reposition =0, the movement track and displacement under 6 flight actions and the velocity change with time are calculated in the model, as shown in Fig.16(see annex 2).

   

 

Fig.15  Trajectory and displacement（

 

 

Fig.16  Trajectory and displacement（）

）

As can be seen from the above two groups of figures, when the initial velocity Angle of the first time entering the pool is perpendicular to the u-shaped pool wall and has a certain size, there is a field with a gradient Angle, which makes the movement track, displacement and velocity different from that of the field without inclination Angle. Mainly reflected in:

• Under the u-shaped field without inclination Angle, only tangential motion exists. The overall motion trend is attenuated, and both velocity and displacement attenuate rapidly.

• For the u-shaped field with inclination, the movement is in two directions, and the tangential displacement and velocity attenuation are slower than those of the field without inclination, which is beneficial for the players to complete the movement.

However, at the same time, the speed and displacement along the side normal direction of the length of the field will appear. Under the current analysis condition, the speed and displacement in this direction will increase too fast, which will cause the players to leave the field and is not conducive to the players to complete the competition.

To this, we will track Angle as 18 °, annex 1 data, using the methods of the same contrast data in attachment 2, in the same vertical velocity into the u-shaped track, there are inclination of the track every discrepancy pool speed increasing, the aclinic site is diminishing, attention fly high in the ratings, and height to a certain extent depends on the suspension hang time, obviously, there are various aspects are better than no inclination Angle track circuit.

However, the results in annex 1 and 2 show that, with the increase in the number of lifts, the lift distance under the pitch will increase with the increase in the Angle of entry into the pool, indicating that the direction of speed is increasingly turning to the direction of the length of the field, which is not conducive to the completion of the action and will not be repeated.(detailed complete motion trajectory and tangential direction, time history of binormal displacement and total velocity of complete motion, tangential direction and binormal direction are given in annex 3).

5.1.4.Model solution

• The greater the initial speed Angle, the higher the take-off speed before taking off, and the farther the single deviation from the vertical direction of the pool wall, resulting in the phenomenon of running too fast in the direction of length, which is not conducive to the completion of the race;

• According to the theoretical formula calculation, combined with the analysis of skaters' sliding movements, the optimal configuration of the "half-pipe" track parameters within the scope of this paper is finally obtained：l（slide length）=118.6m, h（height of a vertical part）=1.2m, R（radius of the circle similar to transition）=2.4m, d（flat width）=15m, H（height of slide without leaning）=3.5m, µ（dynamic friction coefficient in slide）=0.05，θ（angle of inclination）=18°, the curvature of the transition is 0.41667.  

5.2 . Establishment and solution of model 2

5.2.1.Model preparation

Model description: The athletes will lose energy when they leave the U-shaped pool, and we use the loss rate as a measure. In the air, the athlete will adjust the direction of speed and perform a series of technical movements, so there will be a certain energy change or energy loss. And according to the literature, the degree of loss of different athletes will be different. At the same time, the athletes to complete rotation and somersault need to have the corresponding axis around the body angular speed and height. The higher the height, the greater the speed required, the greater the change in the velocity of the center of mass, and the greater the amount of energy consumed.

5.2.2.Establishment and solution of the model

◆ The assumption of the modified model

Make further assumptions based on the previous model：

1）The skater passes through a vertical segment as he slides out and into the u-shaped chute. At this point, there is only gravity, mechanical energy conservation.so:

 is the vertical section of a slide， is initial velocity into the vertical segment，  is the velocity away from the vertical segment.

2）Assuming that the athlete's rotation in the air is a cylinder, the rotation around the center of mass is a constant rotation moment of inertia formula as follows:

among：P is the radius of rotation.

3）During the takeoff，keep the total energy of sliding out of the slide constant, In addition to getting the centroid speed, the athletes also need to obtain the angular velocity of the completed rotation and somersault，Velocity（） of centroid sliding out of the slide It can be calculated according to the law of conservation of energy：

（27）

and  are the rotation of athletes and the moment of inertia in somersault, respectively,  and  are respectively the minimum angular velocity required for athletes to complete the rotation and somersault. Assuming uniform rotation, it can be obtained from the following equation:

Where and  are respectively the number of turns and somersaults of athletes.

Time:

By (tangential initial velocity into vertical segment) approximate。

When landing, it is assumed that the kinetic energy of the center of mass remains unchanged, while the rotational energy generated by the somersault and the rotating body is offset by the resistance when entering the u-shaped pool, and the velocity loss rate is the velocity loss rate of the center of mass.

1. ◆ Analysis of flight correction model

  Considering only the impact of the loss of take-off and landing velocity, the initial velocity of 15m/s, the loss rate of take-off and landing velocity of 17% and 13% respectively, the Angle of take-off of . and , and the Angle of entering the u-shaped pool without control are selected to simulate the motion trajectory and tangential and subnormal velocities. The results are shown in figure 15:

 

                        

                        

Fig.17  Trajectory curves at different take-off angles

 

As can be seen from figure 17,When the velocity loss in and out of the U-shaped cell is considered, the overall variation law is similar to that without energy loss，But in addition to taking off angle 70° does not control the landing angle, even if the take-off angle is 70°, the athlete can complete the complete set of actions successfully and will not fly off the track.

 

                             

Fig.18  Tangential velocity curves at different take-off angles

 

Tangential velocity changes with time (figure 18)，After taking into account the velocity loss, the overall velocity trend remains stable, regardless of the circumstances. The change of the gap between each jump is reduced without considering the speed loss. This is more in line with the actual situation, taking into account the speed loss, the gap between each take-off becomes smaller. This is more realistic, and at the same time, taking into account the speed loss, it is in the range of 10-12m/ s each time the cut-off and the floor-to-ground cutting are basically maintained, and the actual observation results are met.

 

                               

Fig.19  Subnormal velocity curves at different take-off angles

 

As can be seen from figure 19, when the velocity loss of the initial velocity in and out of the u-shaped pool is taken into account, the change of the secondary normal velocity is also stable and there will be no rapid growth that is not in line with the reality.

5.2.3.Model conclusion

•In order to complete the high difficulty movement such as overturning in the air, the athletes should enter the slide track at a speed greater than or equal to 15 m / s, the angle between the velocity direction and the edge of the slide track is greater than 70 °, and the speed direction should be taxied at least 10 m in the x-axis direction.

•When designing the shape of the slide track in Problem 1, the condition is 15m/s, the entry angle is arbitrary, the length of the x axis is 118.6m > 10m, so the shape of the slide track designed in Problem 1 satisfies the condition

5.3.Solution of problem 3

As a chute for skilled skiers, the construction of half-pipe slide should meet the following practical conditions:

    ◆Safety：

    •The slope is consistent with the rolling line (the rolling line is the shortest falling track from the peak to the foot of the mountain);Runs back to Angle is greater than 160 °;Gradient between the 16 ° to 30 °.

•There shall be no obstacles within 5 meters around the u-shaped site.

•The u-shaped site cannot form a "steep wall" on both sides.

•U-shaped site termination area should be wide and safe.

◆Environment：

•Construction is carried out on slopes with suitable slopes and perennial snow to ensure adequate use value of ski runs.

•It is better to be built against the hillside to reduce the impact of wind on the sliding effect.

•The scale of ski slope construction is in line with the market environment to maximize the benefit output of ski resort under this scale.

•There are material transport corridors to facilitate the necessary artificial snow or other maintenance measures.

6. Evaluations And Improvement

6.1. Models’ Advantages

•Model 1

1 For sliding sections, the method is reasonable, accurate and convergent.

2 From simple to complex, abstract to practical, fully embodies the modeling "simple means to solve complex problems" idea.

•Model 2

1 The trajectory of the athletes is simulated by computer in many places, which not only reproduces the process of the athletes' flying, but also makes the problem more intuitive, and helps to find out the conditions that the athletes need to meet when they complete the actions such as flipping.

2 Using the conclusion of the first question to pave the way for their own, broaden the perspective of thinking.

6.2. Models’ Disadvantages

•Model 1

In the actual situation, the wind direction has an obvious influence on the athlete's sliding, but in model 1, the influence of air resistance on the movement is ignored, which may lead to some deviation in the results.

•Model 2

When the conclusion of model 1 has not been fully verified, it is easy to make two consecutive mistakes by using the conclusion of model 1 to pave the way for oneself.

6.3. Models’ improvements

◆ The effect of air resistance on the gliding process

In the modeling process of this paper, air resistance is not considered due to the requirement of simplified model. However, in the actual situation, the high-speed straight glide landing speed can reach 62.5m/s, so the influence of air resistance cannot be ignored.

Air resistance,

where is air drag coefficient which is between 0.3 and 0.95【6】; is air density; i stands for human body or snowboard; is the taking-off velocity;  is the velocity relative to air for i;  is the projection onto .

The flight,

In transition,

From (31) and (32), we know,

Establish functional optimization model to work out the maximum of vertical height,

Since the transition part connects the flat part and the vertical part, in order to avoid the skater hitting the slope on the board, the transition surface needs to be connected with these two parts smoothly and tangentially. The constraint conditions are obtained:     

 Combining the air resistance expressed by this model with the snow resistance mentioned above, the accurate resistance model is finally obtained.

7. References

[1] Curvature radius solving method and physical application 312000, Yang guoping.

[2] From partial derivative identity transformation to partial differential equation solution vol.31.no.2, school of science, national university of defense technology, 410073, Liu majestic, Wang xiao.

[3] Consorium for Mathematics and Its Applications, Inc., COMAP (www.comap.com) UMAP geometric PROGRAMMING (GEMETRIC PROGRAMMING)

[4] Mathematical modeling fifth edition binary search method.

[5] Optimal estimation theory (science press).

[6] Model design of air brake system in vacuum pipeline -- calculation of air resistance coefficient and model design of constant air resistance, middle school affiliated to Beijing normal university, Zhang minghao.

8. Appendices

1.Annex 1:

	a0=10°	Off v （ms）	Off（°）	Lift t（s）	Lift d（m）	Lift h(m)	In v(m)	In (°)
1   2   3   4   5   6	θ=18. θ=0. θ=18. θ=0. θ=18. θ=0. θ=18. θ=0. θ=18. θ=0. θ=18. θ=0.	13.00 12.00 17.9 9.12 25.10 6.26 32.70 2.63 40．30 0.0 47.80 0.0	23.50 10.50 56．2 11.40 69.80 13.50 76.20 26.20 79.90 0.0 82.20 0.0	2.55 2.40 2.14 1.83 1.87 1.24 1.67 0.48 1.51 0.0 1.39 0.0	23.00 5.23 38.80 3.28 49.30 1.82 57.20 0.56 63.50 0.0 68.60 0.0	7.57 7.07 5.32 4.08 4.05 1.89 3.24 0.28 2.67 0.0 52.00 0.0	17.50 12.00 47.30 10.50 65.00 11.40 73.40 13.50 78.10 26.20 80.90 0.0	47.30 10.50 65.00 11.40 73.40 13.50 78.10 26.20 80.90 0.0 82.90 0.0

 

2.Annex 2

	α0=0。	Off m/s	Off(°)	Lift (s)	Lift(m)	Lift h	In v(m)	In(°)
1   2   3	θ=18. θ=0. θ=18. θ=0. θ=18. θ=0. θ=18. θ=0. θ=18. θ=0. θ=18. θ=0.	12.40 11.90 16.30 9.07 23.30 6.20 30.80 2.48 38.30 0.0 45.80 0.0	13.80 0.0 16.30 9.07 23.30 6.20 30.80 2.48 38.30 0.0 45.80 0.0	2.58 2.44 2.13 1.85 1.84 1.26 1.63 0.51 1.47 0.0 1.34 0.0	17.60 0.0 34.30 0.0 44.90 0.0 52.60 0.0 58.70 0.0 63.40 0.0	7.74 7.27 5.29 4.20 3.95 1.96 3.10 0.31 2.52 0.0 2.09 0.0	16.10 11.90 21.70 9.07 28.50 6.20 35.60 2.48 42.70 0.0 49.80 0.0	41.80 0.0 62.80 0.0 72.50 0.0 77.70 0.0 80.80 0.0 82.80 0.0

 

3.Problem 1 model results in the required code

a=u*g*cos(angb);

vB1(j)=xp(p,2);

vcl(j)=spart(vb1(j)^2-2*a*1)

t0=(vB1(1)-Vc1(1))/a;

n2=2*t0/dt2

[T,Y2]=ode45(pzxl,[tp(p):dt2:tp(p)+2*t0],[xp(p,1) xp(p,2) xp(p,3)

 xp(p,4)] );

for i=1:n2

if(Y2(i,1)<1/2)

>>f

if(Y2(i+1,1)>1/2)

k2(j)=i;

end

end

end

for i=1:k2(j)-1

xp(p+i,1)=Y2(i+1,1);

xp(p+i,2)=Y2(i+1,2);

xp(p+i,3)=Y3(i+1,3);

xp(p+i,4)=Y4(i+1,4);

tp(p+i)=T(i+1)

end