数学建模论文-悬崖跳水的水池深度设定问题.doc

数学建模论文论文题目：悬崖跳水的水池深度设定问题摘要：在红牛悬崖跳水世界杯比赛的背景下，本文经讨论研究了跳水水池的深度设定问题，以保证运动员的安全及投资建设水池的成本合理为目的。对此，本文将建立物理模型，运用物理学，微分方程学，理论力学等数学方法，结合MATLAB编程进行求解。对于问题一：我们假设将人体设为一个均匀圆柱体，并建立物理模型，用物理动力学解题思路，对悬崖跳水物理过程细化为运动员空中、入水与水中三个运动过程，并逐个对其建立运动状态方程，结合微分方程学简化求解，最后用MATLAB软件画图展示结果，从图中得到：男子的安全水池深度为12.85米，女子的安全水池深度为8.82米；对于问题二：根据物理质量公式，结合运动员悬崖跳水三个具体物理运动的总方程进行分析，得出高度与质量等之间的联系，从而判断体重不同者与水池深度大小的关系，得到结果：体重越大的人跳水时需要更深的水。该模型建立亮点一体现在其充分运用了物理知识，同时结合微分法简化了求解难度；亮点二体现在用MATLAB软件画图展示结果，直观准确。关键字：物理动力学、能量转换、微分方程学、MATLAB1 问题重述近年来世界上新兴一种跳水比赛叫红牛悬崖跳水世界杯比赛。是一种非常危险、挑战人类极限的比赛，比赛规定男子跳台高度为23至28米，女子为18至23米。我国福建连城的冠豸山就举行过这样的比赛，那里的跳台高度是男子28米，女子20米。需要完成的任务：1. 跳台下面的水池要多深才能安全，请大家给以计算；2. 分析两个体重不同的人跳水时哪个需要更深的水。2 问题分析要探讨跳水者的安全问题，需要分析在跳水者进入水之后继续下落的深度，从而在速度减为0时而不撞击池底而受伤，以保证跳水者的绝对安全经分析，把跳水过程具体化，形象化后，悬崖跳水可分为三个阶段，在下文并逐个对其建立了运动状态方程：第一阶段：从跳板到水面;运动员从跳板跳下，再空中完成一系列的动作之后在重力的作用下会快速自由下落，在此期间，会受到恒定不变的重力及方向向上的空气阻力，且空气阻力越来越大，使运动员做加速度减小的加速运动，并且在到达水面时速度最大。第二阶段：从空气到水中;运动员入水后会受到水的阻力，抵消身体所受的重力作用，从而使身体在水中做减速运动，随着运动员入水体积的变化所受水的浮力也会变化；第三阶段：从水中到水池深处；在运动员身体全部进入水中后，浮力可视为与重力相等，则运动员只受到水的阻力引起的速度改变。直到速度为0，人体入水的最深位置即为所求深度。3 模型假设1、假设将人体设为一个均匀圆柱体，重心位于L/2处。且侧面光滑，不受侧面的摩擦阻力，只受到横截面积引起的流体阻力。2、假设人体所受空气浮力较小可忽略；3、假设忽略跳水运动员的蹬板过程；4、假设风速水速为零，不影响运动员下落；5、假设人水短暂碰撞过程没有能量损失；6、假设当运动员落入水中速度降为最小即速度为零时，此时的深度为安全深度零界。7、假设在水中不产生涡流；4 符号说明符号 意义说明H 跳台到水面的高度；L 人的身高；m 运动员的质量；Vt 第一阶段结束时人体刚接触水面的瞬时速度；V2 第二阶段结束时人体完全浸入水面的瞬时速度g 重力加速度=9.8m/s2；0 水的密度=1.0103kg/m3 空气密度=1.293kg/m32 人的密度约为1.0103kg/m3t 与水面撞击的时间；s 将人体视为圆柱体的横截面积；C1 空气阻力系数；C2 水的阻力系数；Wf 空气阻力对人体所做的功； 水的动力粘度=1104pasCd 水的阻力系数，由表查得为0.4f水 在水中水对人的阻力5 模型建立与求解对于问题一：5.11第一阶段：从跳板到水面;建立运动状态方程：根据动能定理， 又空气阻力对人体所做的功 又由可得并由MATLAB编程，分别得出了男子、女子在这一阶段的图，如下图所示：且由以上两图，可得出结论：男子经历第一阶段的最终速度为：23.4172 m/s；女子经历第一阶段的最终速度为：17.6737 m/s；5.12第二阶段：从空气到水中;运动员所受的浮力，由以上两图可得出结论：男子经历第二阶段的最终速度为17.6732m/s；女子经历第二阶段的最终速度为13.1048m/s；5.1.3第三阶段：从水中到水池深处；由以上两图可得出结论：男子最终安全着底的最适深度为：12.85m女子最终安全着底的最适深度为：8.82 m对于问题二：5.2经分析并建模可得：，由第一问分析的过程，同理用MATLAB编程可得出如下两图：由以上公式和图片，可得出结论：体重越大的人跳水时需要更深的水；6 模型评价该模型的建立充分运用了物理学、理论力学、微分方程学，分析了悬崖跳水的整个过程，且该模型基于科学，详细的分析了跳水者在跳水过程中的速度及其受力情况，并根据实时数据计算了出了悬崖跳水的水池深度的合理设计。以保证运动员的安全，又能使建设水池的成本得到更优化。6.1模型的优点（1）、本文在正确、清楚地分析了题意地基础上，建立了合理、科学的可变成实际的模型，为求水池最适合深度准备了条件；（2）、本文的建模充分运用了物理知识，同时结合微分法简化了求解难度；亮点二体现在用MATLAB软件画图展示结果，直观准确。（3）、本文选用了数学专用软件MATLAB编程，且实现了图像与文字相结合的模型解说，直观准确，提升了文章可信度；6.2模型的缺点（1）、在假设中，本文假设将人体设为一个均匀圆柱体，重心位于L/2处，且侧面光滑，不受侧面的摩擦阻力，只受到横截面积引起的流体阻力，而且又假设了人体所受空气浮力较小可忽略，这些因素将会在模型过程中造成必然的误差（2）、在确定空气密度、人的密度、水的动力粘度的时候，我们只是从网上查阅的资料中得到其粗略值，这与实际情况有出入。7参考文献1空气阻力对跳水成绩的影响郭桔光等； 2弹丸在水介质中的运动特性仿真研究姚养元，曹红松，刘晓雷，张会锁等；3张圣勤等，MATLAB实用教程，北京：机械工业出版社，2006。4姜启源，谢金星，数学建模案例选集，北京：高等教育出版社，2006。8附录男一：（1）跳台至水面过程clear allclc H=0:1:28;C=0.865;P=1.29;S=1/24;m=70;g=9.8;v=(-C*P*S*H+sqrt(C2)*(P2)*(S2)*(H.2)+8*(m2)*g*H)/(2*m);plot(H,v)男二：（2）空气到水中clear all clcv1=23.4172;Cd=0.4;g=9.8;L=0:0.05:1.7;v2=sqrt(-g*L+(1-Cd)*v12);plot(L,v2) 男子（3）从水中到水池深处；clear all clcv2=17.6732:-0.1:0;L=1.7;Cd=0.4;h1=v2*L/(6*Cd);plot(v2,h1)女子（1）：跳台至水面过程clear allclc H=0:1:20;C=0.865;P=1.29;m=50;g=9.8;L=1.6;s=m/(P*L);v=(-C*P*s*H+sqrt(C2)*(P2)*(s2)*(H.2)+8*(m2)*g*H)/(2*m);plot(H,v)女子：（2）空气到水中clear all clcv1=17.6737;Cd=0.4;g=9.8;L=0:0.05:1.6;v2=sqrt(-g*L+(1-Cd)*v12);plot(L,v2)女子：（3）从水中到水池深处；clear all clcv2=13.1048:-0.1:0;L=1.6;Cd=0.4;m=50;h1=v2*L/(6*Cd);plot(v2,h1)电度表原始值记录表#1主变主表#1主变副表正向有功正向无功反向有功反向无功正向有功正向无功反向有功反向无功0.0490.0710.0360.0370.0360.0370.0640.032正向有功峰值正向有功平值正向有功谷值有功功率正向有功峰值正向有功平值正向有功谷值有功功率0.04900000.03600反向有功峰值反向有功平值反向有功谷值无功功率反向有功峰值反向有功平值反向有功谷值无功功率0.03600000.06400#2主变主表#2主变副表正向有功正向无功反向有功反向无功正向有功正向无功反向有功反向无功0.0490.0710.0360.0370.0490.0710.0360.037正向有功峰值正向有功平值正向有功谷值有功功率正向有功峰值正向有功平值正向有功谷值有功功率0.0490000.049000反向有功峰值反向有功平值反向有功谷值无功功率反向有功峰值反向有功平值反向有功谷值无功功率0.0360000.036000#1主变主、副表数值不同，#2主变主、副表数值相同，有功单位：Kwh，无功单位：Kvarh。9JWKffwvG#tYM*Jg&6a*CZ7H$dq8KqqfHVZFedswSyXTy#&QA9wkxFyeQ!djs#XuyUP2kNXpRWXmA&UE9aQGn8xp$R#͑GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmUE9aQGn8xp$R#͑GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z8vG#tYM*Jg&6a*CZ7H$dq8KqqfHVZFedswSyXTy#&QA9wkxFyeQ!djs#XuyUP2kNXpRWXmA&UE9aQGn8xp$R#͑GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxG89AmUE9aQGn8xp$R#͑GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z8vG#tYM*Jg&6a*CZ7H$dq8KqqfHVZFedswSyXTy#&QA9wkxFyeQ!djs#XuyUP2kNXpRWXmA&UE9aQGn8xp$R#͑GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmUE9aQGn8xp$R#͑GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNuGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89Amv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz84!z89Amv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNuGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP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