MCM A 题答案.docx

MCM A 题答案.docx

《MCM A 题答案.docx》由会员分享，可在线阅读，更多相关《MCM A 题答案.docx（19页珍藏版）》请在冰豆网上搜索。

MCMA题答案

Designthehalf-pipeforBetterPerformance

Abstract

Inordertodesignanideamodelofthehalf-pipecoursetomeetdifferentrequirements,theconversionrelationshipbetweenworkandenergyinthesystemconsistingofsnowboardersandthecoursecomesfirst.Sincetheairdragisnegligible,theworkthatfrictiondoesbecomesthekeypointofsolvingtheproblem.Wemainlyusethedifferentialequationsandphysicaltheoriessuchasenergyconversionandenergyconservationtosolveforthemaximumvalueoftheverticalairandthespecificationsofthecourse.Andwebuildthreemodelstograduallymeettherequirementofthepracticalcourse.

Inthebasicmodel,tosimplifythecalculation,wehaven’tconsidertheslantangleofthecourse,andwedesignonewithasemicirculartransversalsurface.Weremovetheflatbottompartofthecoursetominimizetheworkdonebyfrictionaimingatmaximizetheverticalair,meanwhile,thetrajectoryofsnowboarderscanbeconsideredasasemi-circle.

Intheimprovedmodel,wetakeslantangleofthecourseintoconsideration,foritcanprovidetheathletewithanextraaccelerationintherace.Underthiscircumstance,thetrajectoryofathletesisalmostaparabola.Andwedividetheathletes’movementintotwoparts,theinclinedoneandtheverticalone.Butwedrawtheconclusionthatwithoutupperlimit,wecannotdeterminetheverticalair.SowereferencetherecommendspecificationssetbytheFIS（FederationInternationaleDeSki）andmakethetrad-offbylimitthesnowboarders’maximumspeed,thus,wecansolveforthewidthandlengthofthecourse.

Inthefinalmodel,weconsiderthereactiontimetheathletesneedtoadjusttheirbodiesandchoosetheidealpath,consequently,weaddtheflatbottompartintothecourse.Inthismodel,weuseanewintegrationmethod,meantime,tosolvefortheworkthatfrictiondoes,wedividethetrajectoryofathletesintothreeindividualparts.Tosolveforthenumericalvalueofspecificationsofthecourse,weusethedataofsomeoutstandingsnowboardersinChina,andweworkitout.

Thefinalresultisthattheradiumofthecourse’stransversalsurfaceshouldbe3.5meters.Thelengthofthecourseshouldbe133.7meters,theslantangleofitshouldbe16degreesandthewidthofitshouldbe15.2meters.

Keywords:

differentialmodel,energyconversion,energyconservation,verticalair

Contents

1Introduction...............................................................................................................................................3

1.1HistoryofSnowboarding…………………………………………….……………………......31.2half-pipeSnowboarding………………………………………………………………………...3

1.3Whatisverticalair?

……………………………………………………………………….……...5

1.4Howaskilledsnowboarderwillperform………………………………………………......5

2ProblemDescriptionandAnalysis…………………………………….………………………..6

3Models……………........................................................................................................................................6

3.1BasicModel..........................................................................................................................................6

3.1.1ModelOverview………………………………..…...……………………….......................6

3.1.2TermsDefinitionsandSymbols……………………………………………………….7

3.1.3TheModel……………………………………………………….………….………….……7

3.1.4StrengthandWeakness………………………………………………….………….......11

3.2ImprovedModel…………………………………….……………………………………..............11

3.2.1ModelOverview……………………………………..………………………...………………….............11

3.2.2ExtraSymbols……………………………………………….……………….…................11

3.2.3TheModel……………………………………………………….………….………….…..12

3.2.4StrengthandWeakness………………………………………………….………….......15

3.3FinalModel…………………………………….…………………………………….........................15

3.3.1ModelOverview……………………………………..………………………...…………………............15

3.3.2ExtraSymbols……………………………………………….……………….…................15

3.3.3TheModel……………………………………………………….………….………….…15

3.3.4StrengthandWeakness………………………………………………….………….......20

4AnalysisoftheResult.........................................................................................................................20

5FutureWork…………………………………….…………………………………….……………....21

6References…………....……………………….………………………………….......................21

1Introduction

Inordertoindicatetheoriginofsnowboarding,thefollowingbackgroundisworthmentioning.

1.1HistoryofSnowboarding

SnowboardingisaveryrecentsportandissimilartoSurfing,skateboarding,andSkiing.AlthoughitishardtopinpointthepioneerofSnowboarding,theHistoryofSnowboardingtellsusthatitwasinitiatedaroundthe1950sbyafewsurfandskateenthusiastswhousedself-madeboardstoconveytheirskillstoanewterrain:

thesnow.

SnowboardingandSkiingaresimilarinseveralways.Ontheotheraspect,thes