2011MCM Preliminary Study of Snowboard Course(完整版).doc

Preliminary Study of Snowboard CourseAbstract. We present two mathematics models based on the law of conservation of energy in order to obtain the shape of a snowboard course which can maximize the production of vertical air by a skilled snowboarder. The knowledge about mathematics, physics and computer is involved to analyze the problem above. These models mainly take into account the factors that determine how a snowboarder reach his full height, since the launched heigh plays an important role in evaluating a players action in games. These factors include the average weight of athlete, final velocity, friction produced on slope, sliding vertical distance and so on. By the law of conservation of energy, all the factors considered here are interosculated together , therefore the two models are constructed. Throughout the paper, we assume that the subjective factors of the players and the organizers are negligible and the notations of the variables used in these models are as shown in Table 1. According to the basic formulas about calculus and the numerical simulation schemes, we can get the optimal heigh based on our models, therefore the player can obtain his expected production of vertical air. The results of a numerical experiment demonstrate our models are feasible and reasonable in some way.Compared with other models before, the advantage of our models is obviouslythat they are easy to implement. Nevertheless, their shortcomings inevitable, such as some enviromental conditions and so on. Finally, in view of our models, we list possible future work. Key words: the law of conservation of energy; Snowboard; Vertical air; optimal heighTeam #10789 Page 9 of 9ContentsPreliminary Study of Snowboard Course1Contents2I. Introduction3II. The Description of the Problem3III. Models33.1 Basic Model33.1.1 Terms, Definitions and Symbols33.1.2 Assumptions43.1.3 The Foundation of Model43.1.4 Solution and Result53.1.5 Analysis of the Result53.1.6 Strength and Weakness53.2 Simple discussion about twist in the air5IV. Conclusions84.1 Conclusions of the problem84.2 Applications of our models8VI. References9I. IntroductionSnowboard half-pipe was a very young snow sports. In 1988, it becomes an official event of the Winter Olympic Games in 18th Nagano. In 2003, half-pipe was formally introduced to China. Snowboarding is very popular winter sports with glide, leap, skills. Half-pipe snowboarding is an aerial acrobatic routine performed by snowboard athletes on a half-pipe constructed of snow. The snowboarders use body and feet to control the direction and complete a variety of difficulty routine such as twist and flip in the air. 3 For the sake of finishing the series of breathtakingly difficult moves, the athletes must have enough time, so the snowboarders try their best to increase the commencing height. The higher commencing height, the longer flight distance, so they could get enough hang time to ensure the movement completed. 4 In theory, the snowboarders should have constant inrun velocity corresponding to different movements. But in practical terms, it is impossible that the athletes keep the velocity invariant in each skip.II. The Description of the Problem The objective in recreational and competitive half-pipe snowboarding is to perform well executed and stylish routines consisting of complex aerial acrobatic that are executed as high above the half-pipe lip as possible. The chief purpose is to pick up a higher commencing height, so as to get a longer hang time. 5 We understand the importance of a commencing height, the next we probably think about is how to fly higher within the context of technology. In the objective aspect, it was in connection with the Slope angle of the half-pipe, Height (floor to crown), quality of snow, especially the take-off angle of the half-pipe. Subjective factors were controlled by the athletes, such as the skills, the psychologic status and the physical state. III. Models 3.1 Basic Model 3.1.1 Terms, Definitions and Symbols Table1 shows the variables used in the simulation of the heigh model. Table 1.Variable used in the model.SymbolDefinitionUnitsFriction produced on slopeNAverage weight of athletekgAcceleration of gravityDistance(top of slop to floor)mSlope angledegreeFinal velocitym/sFriction coefficientLength of the slopemSliding vertical distancedegreeFriction happened in half-pipeNLength of sliding pathmSliding vertical distancemHeight of vertical distancem3.1.2 Assumptions 1 Snowboarders has better basic skill, so they can control the snowboard freely . 2 We assume that snowboarder should maximize the level of technology in each slide. 4 Accurate selecting.5 Can accurately glide along the best route and choose the exact point of departure.6 There are no accidents during the glide.3.1.3 The Foundation of Model Snowboard competition is held in a particular space. To simplify the competition space, we overlook the detail in competition space. Altitude difference is the vertical distance between starting point (top of the slope) and the edge of the half-pipe. The sliding friction is produced when the sportsman is moving on the slope.By coasting in the assistant incline, sportsman gets a final velocity before entering the half-pipe. We obtain the final velocity.Assuming that the final velocity which a snowboarder completes accelerating is equal to the primary speed which he get into the half-pipe.is defined as the friction when athlete is gliding on the surface of snowboard course.A skilled snowboarder can perform in a smooth, symmetry half-pipe when finished accelerating in straight snow-slope. Assuming that is a constant value in any snowboard course. In this equation, we regard the action of the sportsman as a curvilinear motion. Depend on energy conservation, we obtain an equation: we conclude the relationship between the angle , which is the tangent through entering spot between level and vertical air is:3.1.4 Solution and Result We collect some available digitals about the international standard track. (Table 2)39.89m170.2Obviously, there is an inverse relationship between and. Friction keeps obstructing motion in the easy model. To maximum the vertical air, the skilled athlete should properly cut down the distance gliding distance, at the same time, reach his ideal height (an athlete not gets injured easily after reach the surface of snow-ground) over the edge of half-pipe. To satisfy the most difficult action (Shaun Whit, U.S. player, can reach 5 meters above the edge of the half-pipe), we can get. When we make, its an extreme situation, players cant finish basic action. We can gain.3.1.5 Analysis of the Result Follow above discussion, angle arranges from to .Its a reasonable and acceptable results for the half-pipe building. Skilled players can have freedom for the production of “vertical air”.3.1.6 Strength and Weakness Strength: Our model is based on easily understood principles and simply expressed assumption. The equation is simple and compact. Weakness: This model just applies to ideal situation to analyze complicated problems. We do not account for the resistance of air.3.2 Simple discussion about twist in the airAll kinds of action in the air depend on the twist. The power of twist comes from the force, which is produced between snowboard and the surface of half-pipe. When athlete is going to rush out of the half-pipe, he takes advantage of one blade of the snowboard in order to find the best angle. Best angle is an important factor that affects performance heigh. From the snowboard video，we find out that the key points of scoring consist of commencing height, the degree of twist and the flip. Towards to second problem, we analysis the twist and body rotation based on the method of biomechanical theory. In practice, a players spin in air can be considered as turning around a axis. Therefore, we can put a plays spinning action into a three-dimensional coordinate system.The trajectory of a board player does flips or twist in air is a parabola. The distance which a player can get is proportional to his vertical displacement and horizontal displacement (Figure2). In a board game, the launched height and the diversity play an important role in evaluating a players action because no height means no jumping distance and so there is no excellent actions. Athletes in the air are mainly caused by and the F, of which the F is analyzed the horizontal velocity X and the ascensional velocity Y.( Figure1) They shows that the larger taking-off velocity, the better the athletes overcoming the gravity and the longer horizontal shift, the higher taking-off altitude.(Figure3) Figure 1 Analysis of excellent snowboarders flip and turning angle Figure 2 Relation between movement cours and vertical displacement X, horizontal displacement in the air Figure 3 Analysis of force Table 2.Name FlipTwistThe angle of flip/The execution timePalstance/ rads-1The angle of Twist/The execution time Palstance/ rads-1Chen XU 54051087204180Zhifeng Sun 54051087204180Jiayu Liu 54041359003300LauraPest54041359003300Kjersti36031209003300HannahTate54041359003300Haozheng Jin 900422510804270国母900422510804270Wanwei Shi72041809004225Xiaoye Zheng72041809004225MArkku Kosk900330010803360Daniel Kas720324010803360Shaun whit900330010803360Through Table 2 above, we analysis the flip and twist of the world excellent snowboarder, we conclude that the velocity of twist is higher than the angle of flip and there is a big distance between the domestic players and the foreign athletes. The Table 2 shows that the velocity of twist is as high as 360rad/s, however, the velocity of flip is only 300rad/s. The domestic snowboarders can only slip 180rad/s and twist 225rad/s. so they need continuous trains to develop their level.IV. Conclusions 4.1 Conclusions of the problem Currently, snowboarding is a developed competition event. So we do not raise a revolutionary program. We based on conservation of energy, gain a range of. It can keep snowboarder perform excellent in ideal venues. The drop heigh of half-piper snowboarding is bound to have some impacts on sliding. So athletes should take full advantage of the drop heigh to increase the taking-of velocity.The athletes slide into the half-pipe for the first time should receive the controllable velocity and accumulate the energy positively. The largest controllable velocity is 15 m/s.While sliding into the under-part of the U-Shape Groove, the body should be tilted tank.4.2 Applications of our models With our model, manager of a ski resort can construct a better skifield for the people entertainment and the athlete training. Even though, our Researches has some insufficiency and ignores some factors. But we think that our mode will be developed through the Later efforts and make a contribution to the athlete. VI. References 1 Baoheng Wang. Thoughts of the Basic Slipping and Flying Height in