MCM A 题答案.docx

1、MCM A 题答案Design the half-pipe for Better PerformanceAbstractIn 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 negligib

2、le, 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 m

3、odels to gradually meet the requirement of the practical course.In the basic model, to simplify the calculation, we havent 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 fric

4、tion 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 traj

5、ectory 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 Intern

6、ationale 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 in

7、to 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 C

8、hina, and we work it out.The final result is that the radium of the courses 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,

9、energy conservation,vertical airContents1 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.52 Problem Description and Analysis.63 Models.63.1 Basic Model.63.1.1 Model Overview.63.1.2 Terms Definition

10、s and Symbols.73.1.3 The Model .73.1.4 Strength and Weakness.113.2 Improved Model.113.2.1 Model Overview.113.2.2 Extra Symbols.113.2.3 The Model .123.2.4 Strength and Weakness.15 3.3 Final Model.153.3.1 Model Overview.153.3.2 Extra Symbols.153.3.3 The Model . 153.3.4 Strength and Weakness.204 Analys

11、is of the Result.205 Future Work. . .216 References . . 211 IntroductionIn order to indicate the origin of snowboarding , the following background is worth mentioning.1.1 History of SnowboardingSnowboarding is a very recent sport and is similar to Surfing, skateboarding, and Skiing. Although it is h

12、ard 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, thes