Mathematic结课论文.doc

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问题1 根据侦察,发现离我军大炮阵地水平距离10km的前方有一敌军的坦克群正以每小时50km向我军阵地驶来,现欲发射炮弹摧毁敌军坦克群. 为在最短时间内有效摧毁敌军坦克,要求每门大炮都能进行精射击,这样问题就可简化为单门大炮对移动坦克的精确射击问题. 假设炮弹发射速度可控制在0.2km/s至0.6km/s之间,问应选择怎样的炮弹发射速度和怎样的发射角度可以最有效摧毁敌军坦克. 说明 假设不考虑空气阻力,则炮弹的运动轨迹由参数方程 ,给出,其中是炮弹发射的初速度,是炮弹的发射角,是重力加速度（9.8m/）. 上面第一个方程描述炮弹在时刻的水平位置,而第二个方程描述炮弹在时刻的垂直位置. 在上述假设下,进一步研究下列问题: (1) 选择一个初始速度和发射角,利用Mathematica画出炮弹运行的轨迹.(2) 假定坦克在大炮前方10km处静止不动,炮弹发射的初速度为0.32km/s,应选择什么样的发射角才能击中坦克？画出炮弹运行的几个轨迹图,通过实验数据和图形来说明你的结论的合理性.(3) 假定坦克在大炮前方10km处静止不动,探索降低或调高炮弹发射的初速度的情况下,应如何选择炮弹的发射角？从上述讨论中总结出最合理有效的发射速度和发射角. (4) 在上题结论的基础上,继续探索,假定坦克在大炮前方10km处以每小时50km向大炮方向前进,此时应如何制定迅速摧毁敌军坦克的方案？(1)问题一先设时间为t，发射角是a=30度即a=Pi/6,炮弹发射的初速度是v=0.3km/s,以下的单位都以km计。炮弹的水平位移为x = 0.3*t*SinPi/6竖直位移y = 0时求得的非零的时间t就是炮弹从发出到击中目标所用的时间即：0.3*t*CosPi/6 - (1/2)*9.8*t2*0.001 = 0得t = (2*0.3*CosPi/6)/(9.8*0.001)= 53.02196349700644通过mathematics做出函数的图像ParametricPlot0.3*t*SinPi/6, 0.3*t* CosPi/6 - (1/2)*9.8*t2*0.001, t, 0, 53.022, PlotStyle - RGBColor0, 1, 0 , Thickness0.01以下是函数的图像问题二 若坦克在大炮前方10km处而且静止不动，那么要击中目标即满足水平位移x = 0.32*Sina*t = 10km1式而竖直方向上若要击中目标同样要满足y = 0可得所用为t = (2*0.32*Cosa)/(9.8*0.001).2式 1、2两式联立得： 20.89795918367347 Cosa Sina = 10km 即 Sin2a = 0.9570312499999999 (0a RGB Color0, 1, 0 , Thickness0.01;bb = ParametricPlot0.32*t*Sin36.57147687017 5026*Pi/180, 0.32*t* Cos36.571476870175026*Pi/180 - (1/2) *9.8*t2*0.001, t, 0, 52.448273624397544, PlotStyle - RGB Color1, 0, 1 , Thickness0.01;cc = ParametricPlot0.32*t*Sin30*Pi/180, 0.32*t*Cos30*Pi/180 - (1/2)*9.8*t2*0.001, t, 0, 38.91102958213058, PlotStyle - RGBColor0, 1, 0, Thickness0.01dd = ParametricPlot0.32*t*Sin70*Pi/180, 0.32*t*Cos70*Pi/180 - (1/2)*9.8*t2*0.001, t, 0, 38.91102958213058, PlotStyle - RGBColor0, 1, 0, Thickness0.01Showaa, bb, cc, dd, DisplayFunction - $DisplayFunction可知当角度为53.4度时击中目标所用的时间要更少一些为38.9秒。按照实际应用的角度考虑应该取这个角度比较合适。问题三可知当坦克在大炮前方10km处不动时则炮弹的水平位移是确定的，x = v*Sina*t = 10.3式时间t = (2*v*Cosa)/g4式3，4两式结合有 v*Sina*(2*v*Cosa)/g = 10则 Sin2a = (10*g)/v2 = 1 可得出炮弹发射的最小速度Vmin = (10*9.8*0.001)(1/2) = 0.3130495168499705 km/s从而再次限定了炮弹发射的速度范围为：0.31304951684997057 = v = 0.6km/s但对于实际问题还需要考虑炮弹从发射到击中目标所需要的时间，Sin2a = (10*g)/v2带入v的值求得角a的范围为：7.898272455484096 = a RGBColor0, 1, 1 , Thickness0.01;bb = ParametricPlot0.31304951684997057 *t*Sin45*Pi/180, 0.31304951684997057*t* Cos45*Pi/180 - (1/2) *9.8*t2*0.001, t, 0, 45.175395145262556, PlotStyle - RGBColor0, 0, 1 , Thickness0.01;cc = ParametricPlot0.6*t*Sin7.89827 2455484096*Pi/180, 0.6*t* Cos7.898272455484096*Pi/180 - (1/2) *9.8*t2*0.001, t, 0, 121.28738043983, PlotStyle - RGBColor1, 0, 0 , Thickness0.01;Showaa, bb, cc, DisplayFunction - $DisplayFunction问题四要迅速摧毁目标则取最大速度0.6km/sSolve0.6*SinAlpha*t + 50/3600*t = 10, 1/2*9.8* 0.001*t2 - 0.6*CosAlpha*t = 0, Alpha, tt - -120.85036637838965, Alpha - -2.979828334000835, t - -17.23732619303437, Alpha - -1.7120369554135506, t - 16.43493586871528, Alpha - 1.436171398065404, t - 121.65275670270874, Alpha - 0.11410123775918844则时间应该取正的且最小的t - 16.43493586871528, Alpha - 1.436171398065404做出炮弹的轨迹图ParametricPlot0.6*t*Sin1.436171398065404, 0.6*t*Cos1.436171398065404 - (1/2)*9.8*t2*0.001, t, 0, 16.43493586871528, PlotStyle - RGBColor1, 0, 0, Thickness0.01参考书目全美经典Mathematic使用指南Mathematic数学实验.电子工业出版社 徐安农Mathematic 4 教程.电子工业出版社 丁大正用Mathematic 做微积分实验.清华大学出版社 A.D.Andrew G.L.Morley 俞正光、章纪民翻译 羁膃蒇螆螄聿蒆蒆罿羅薅薈螂芄薅蚀羈膀薄袃螀膆薃薂肆肂薂蚅衿莁薁螇肄芇薀衿袇膃薀蕿肃聿芆蚁袅羅芅螄肁芃芄蒃袄艿芃蚆腿膅芃螈羂肁节袀螅莀芁薀羀芆芀蚂螃膂荿螄羈肈莈蒄螁羄莇薆羇莂莇蝿袀芈莆袁肅膄莅薁袈肀莄蚃肃羆莃螅袆芅蒂蒅肂膁蒁薇袄肇蒁蚀肀肃蒀袂羃莁葿薂螆芇蒈蚄羁膃蒇螆螄聿蒆蒆罿羅薅薈螂芄薅蚀羈膀薄袃螀膆薃薂肆肂薂蚅衿莁薁螇肄芇薀衿袇膃薀蕿肃聿芆蚁袅羅芅螄肁芃芄蒃袄艿芃蚆腿膅芃螈羂肁节袀螅莀芁薀羀芆芀蚂螃膂荿螄羈肈莈蒄螁羄莇薆羇莂莇蝿袀芈莆袁肅膄莅薁袈肀莄蚃肃羆莃螅袆芅蒂蒅肂膁蒁薇袄肇蒁蚀肀肃蒀袂羃莁葿薂螆芇蒈蚄羁膃蒇螆螄聿蒆蒆罿羅薅薈螂芄薅蚀羈膀薄袃螀膆薃薂肆肂薂蚅衿莁薁螇肄芇薀衿袇膃薀蕿肃聿芆蚁袅羅芅螄肁芃芄蒃袄艿芃蚆腿膅芃螈羂肁节袀螅莀芁薀羀芆芀蚂螃膂荿螄羈肈莈蒄螁羄莇薆羇莂莇蝿袀芈莆袁肅膄莅薁袈肀莄蚃肃羆莃螅袆芅蒂蒅肂膁蒁薇袄肇蒁蚀肀肃蒀袂羃莁葿薂螆芇蒈蚄羁膃蒇螆螄聿蒆蒆罿羅薅薈螂芄薅蚀羈膀薄袃螀膆薃薂肆肂薂蚅衿莁薁螇肄芇薀衿袇膃薀蕿肃聿芆蚁袅羅芅螄肁芃芄蒃袄艿芃蚆腿膅芃螈羂肁节袀螅莀芁薀羀芆芀蚂螃膂荿螄羈肈莈蒄螁羄莇薆羇莂莇蝿袀芈莆袁肅膄莅薁袈肀莄蚃肃羆莃螅袆芅蒂蒅肂膁蒁薇袄肇蒁蚀肀肃蒀袂羃莁葿薂螆芇蒈蚄羁膃蒇螆螄聿蒆蒆罿羅薅薈螂芄薅蚀羈膀薄袃螀膆薃薂肆肂薂蚅衿莁薁螇肄芇薀衿袇膃薀蕿肃聿芆蚁袅羅芅螄肁芃芄蒃袄艿芃蚆腿膅芃螈羂肁节袀螅莀芁薀羀芆芀蚂螃膂荿螄羈肈莈蒄螁羄莇薆羇莂莇蝿袀芈莆袁肅膄莅薁袈肀莄蚃肃羆莃螅袆芅蒂蒅肂膁蒁薇袄肇蒁蚀肀肃蒀袂羃莁葿薂螆芇蒈蚄羁膃蒇螆螄聿蒆蒆罿羅薅薈螂芄薅蚀羈膀薄袃螀膆薃薂肆肂薂蚅衿莁薁螇肄芇薀衿袇膃薀蕿肃聿芆蚁袅羅芅螄肁芃芄蒃袄艿芃蚆腿膅芃螈羂肁节袀螅莀芁薀羀芆芀蚂螃膂荿螄羈肈莈蒄螁羄莇薆羇莂莇蝿袀芈莆袁肅膄莅薁袈肀莄蚃肃羆莃螅袆芅蒂蒅肂膁蒁薇袄肇蒁蚀肀肃蒀袂羃莁葿薂螆芇蒈蚄羁膃蒇螆螄聿蒆蒆罿羅薅薈螂芄薅蚀羈膀薄袃螀膆薃薂肆肂薂蚅衿莁薁螇肄芇薀衿袇膃薀蕿肃聿芆蚁袅羅芅螄肁芃芄蒃袄艿芃蚆腿膅芃螈羂肁节袀螅莀芁薀羀芆芀蚂螃膂荿螄羈肈莈蒄螁羄莇薆羇莂莇蝿袀芈莆袁肅膄莅薁袈肀莄蚃肃羆莃螅袆芅蒂蒅肂膁蒁薇袄肇蒁蚀肀肃蒀袂羃莁葿薂螆芇蒈蚄羁膃蒇螆螄聿蒆蒆罿羅薅薈螂芄薅蚀羈膀薄袃螀膆薃薂肆肂薂蚅衿莁薁螇肄芇薀衿袇膃薀蕿肃聿芆蚁袅羅芅螄肁芃芄蒃袄艿芃蚆腿膅芃螈羂肁节袀螅莀芁薀羀芆芀蚂螃膂荿螄羈肈莈蒄螁羄莇薆羇莂莇蝿袀芈莆袁肅膄莅薁袈肀莄蚃肃羆莃螅袆芅蒂蒅肂膁蒁薇袄肇蒁蚀肀肃蒀袂羃莁葿薂螆芇蒈蚄羁膃蒇螆螄聿蒆蒆罿羅薅薈螂芄薅蚀羈膀薄袃螀膆薃薂肆肂薂蚅衿莁薁螇肄芇薀衿袇膃薀蕿肃聿芆蚁袅羅芅螄肁芃芄蒃袄艿芃蚆腿膅芃螈羂肁节袀螅莀芁薀羀芆芀蚂螃膂荿螄羈肈莈蒄螁羄莇薆羇莂莇蝿袀芈莆袁肅膄莅薁袈肀莄蚃肃羆莃螅袆芅蒂蒅肂膁蒁薇袄肇蒁蚀肀肃蒀袂羃莁葿薂螆芇蒈蚄羁膃蒇螆螄聿蒆蒆罿羅薅薈螂芄薅蚀羈膀薄袃螀膆薃薂肆肂薂蚅衿莁薁螇肄芇薀衿袇膃薀蕿肃聿芆蚁袅羅芅螄肁芃芄蒃袄艿芃蚆腿膅芃螈羂肁节袀螅莀芁薀羀芆芀蚂螃膂荿螄羈肈莈蒄螁羄莇薆羇莂莇蝿袀芈莆袁肅膄莅薁袈肀莄蚃肃羆莃螅袆芅蒂蒅肂膁蒁薇袄肇蒁蚀肀肃蒀袂羃莁葿薂螆芇蒈蚄羁膃蒇螆螄聿蒆蒆罿羅薅薈螂芄薅蚀羈膀薄袃螀膆薃薂肆肂薂蚅衿莁薁螇肄芇薀衿袇膃薀蕿肃聿芆蚁袅羅芅螄肁芃芄蒃袄艿芃蚆腿膅芃螈羂肁节袀螅莀芁薀羀芆芀蚂螃膂荿螄羈肈莈蒄螁羄莇薆羇莂莇蝿袀芈莆袁肅膄莅薁袈肀莄蚃肃羆莃螅袆芅蒂蒅肂膁蒁薇袄肇蒁蚀肀肃蒀袂羃莁葿薂螆芇蒈蚄羁膃蒇螆螄聿蒆蒆罿羅薅薈螂芄薅蚀羈膀薄袃螀膆薃薂肆肂薂蚅衿莁薁螇肄芇薀衿袇膃薀蕿肃聿芆蚁袅羅芅螄肁芃芄蒃袄艿芃蚆腿膅芃螈羂肁节袀螅莀芁薀羀芆芀蚂螃膂荿螄羈肈莈蒄螁羄莇薆羇莂莇蝿袀芈莆袁肅膄莅薁袈肀莄蚃肃羆莃螅袆芅蒂蒅肂膁蒁薇袄肇蒁蚀肀肃蒀袂羃莁葿薂螆芇蒈蚄羁膃蒇螆螄聿蒆蒆罿羅薅薈螂芄薅蚀羈膀薄袃螀膆薃薂肆肂薂蚅衿莁薁螇肄芇薀衿袇膃薀蕿肃聿芆蚁袅羅芅螄肁芃芄蒃袄艿芃蚆腿膅芃螈羂肁节袀螅莀芁薀羀芆芀蚂螃膂荿螄羈肈莈蒄螁羄莇薆羇莂莇蝿袀芈莆袁肅膄莅薁袈肀莄蚃肃羆莃螅袆芅蒂蒅肂膁蒁薇袄肇蒁蚀肀肃蒀袂羃莁葿薂螆芇蒈蚄羁膃蒇螆螄聿蒆蒆罿羅薅薈螂芄薅蚀羈膀薄袃螀膆薃薂肆肂薂蚅衿莁薁螇肄芇薀衿袇膃薀蕿肃聿芆蚁袅羅芅螄肁芃芄蒃袄艿芃蚆腿膅芃螈羂肁节袀螅莀芁薀羀芆芀蚂螃膂荿螄羈肈莈蒄螁羄莇薆羇