{"id":28210,"date":"2023-08-16T18:59:05","date_gmt":"2023-08-16T10:59:05","guid":{"rendered":"http:\/\/www.qiusir.com\/?p=28210"},"modified":"2026-03-06T11:06:37","modified_gmt":"2026-03-06T03:06:37","slug":"%e5%be%ae%e7%a7%af%e5%88%86%e5%9c%a8%e7%89%a9%e7%90%86%e4%b8%ad%e7%9a%84%e5%ba%94%e7%94%a8%e5%88%9d%e6%ad%a5-2","status":"publish","type":"post","link":"https:\/\/www.qiusir.com\/?p=28210","title":{"rendered":"\u5fae\u79ef\u5206\u5728\u7269\u7406\u4e2d\u7684\u5e94\u7528\u521d\u6b65"},"content":{"rendered":"

\/\u5b59\u6d5a\u8c6a<\/strong>
\n\u6c42\u5e08\u5f97\u6570\u4f4d\u5b66\u4e60\u65b0\u8bfe\u9898<\/em>
\n\u9ad8\u4e2d\u7269\u7406\u5bf9\u7528\u5fae\u79ef\u5206\u89e3\u51b3\u95ee\u9898\u4e00\u822c\u4e0d\u4f5c\u8981\u6c42\uff0c\u4f46\u5f88\u591a\u7269\u7406\u95ee\u9898\u53c8\u65e0\u6cd5\u56de\u907f\u9ad8\u7b49\u6570\u5b66\u7684\u80cc\u666f\u3002\u672c\u5468\u8bfe\u9898\u5c06\u901a\u8fc7\u82e5\u5e72\u9898\u76ee\u548c\u793a\u4f8b\u6765\u5c55\u793a\u4e00\u4e9b\u5fae\u79ef\u5206\u5728\u9ad8\u4e2d\u7269\u7406\u4e2d\u7684\u5e94\u7528\u3002<\/p>\n

\u4e00\u3001\u4ece\u5300\u53d8\u901f\u76f4\u7ebf\u8fd0\u52a8\u8bf4\u8d77<\/strong><\/p>\n

\u672c\u5468\u8bfe\u7a0b\u5c31\u4ece\u9ad8\u4e2d\u7269\u7406\u7684\u5165\u95e8\u7bc7\u2014\u2014\u5300\u53d8\u901f\u76f4\u7ebf\u8fd0\u52a8\u5c55\u5f00\u3002
\n\u5bf9\u4e8e\u66f4\u7b80\u5355\u7684\u5300\u901f\u76f4\u7ebf\u8fd0\u52a8\uff0c\u6211\u4eec\u5728\u5c0f\u5b66\u5c31\u5b66\u4e60\u8fc7\uff1a\u8def\u7a0b=\u901f\u5ea6*\u65f6\u95f4\u3002\u6beb\u65e0\u7591\u95ee\u5730\uff0c\u5300\u901f\u76f4\u7ebf\u8fd0\u52a8\u7684v\u2014t\u56fe\u50cf\u662f\u4e00\u6761\u6c34\u5e73\u7684\u7ebf\u6bb5(\u5373\u901f\u5ea6\u6052\u5b9a)\uff0c\u90a3\u4e48\uff0c\u4ece0~t\u65f6\u95f4\u5185\uff0c\u7269\u4f53\u8fd0\u52a8\u7684\u8def\u7a0bs=vt\u7684\u51e0\u4f55\u610f\u4e49\u4fbf\u662f\u4e00\u4e2a\u77e9\u5f62\u7684\u9762\u79ef\u3002
\n\"2023-8\"
\n\u800c\u5bf9\u4e8e\u5300\u53d8\u901f\u76f4\u7ebf\u8fd0\u52a8\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u8fd0\u52a8\u5206\u89e3\u6210n\u4e2a\u65f6\u95f4\u95f4\u9694\uff0c\u6bcf\u4e2a\u95f4\u9694\u7684\u65f6\u95f4\u4e3a\uff1a$\\Delta t$\uff0c\u6bcf\u4e2a\u95f4\u9694\u5185\u7684\u6700\u5c0f\u901f\u5ea6\u4e3a$v_i$(i=1\uff0c2\uff0c3\uff0c\u2026\uff0cn)\u5219\u6bcf\u4e2a\u95f4\u9694\u7684\u4f4d\u79fb\u8fd1\u4f3c\u4e3a$x_i=v_i\\Delta t_i$\uff0c\u5168\u7a0b\u7684\u603b\u4f4d\u79fb\u8fd1\u4f3c\u4e3a$x=\\Sigma x_i=x_1+x_2+x_3+…+x_n$\uff0c\u5728\u51e0\u4f55\u4e0a$\\Delta t_i$\u4f53\u73b0\u4e3a\u6240\u6709\u4e0a\u7684\u77e9\u5f62\u9762\u79ef\u548c\uff0c\u6b64\u65f6\u7684x\u5c0f\u4e8e\u771f\u5b9e\u603b\u4f4d\u79fb\u3002\u589e\u5927n\u4ece\u800c\u51cf\u5c0f$\\Delta t_i$\uff0c$v_i$\u66f4\u52a0\u63a5\u8fd1\u5168\u7a0b\u7684\u771f\u5b9e\u901f\u5ea6\uff0c\u5219$x_i$\u66f4\u52a0\u63a5\u8fd1\u5bf9\u5e94\u7684\u771f\u5b9e\u4f4d\u79fb\uff0cx\u4e5f\u66f4\u52a0\u63a5\u8fd1\u771f\u5b9e\u603b\u4f4d\u79fb\uff0c\u77e9\u5f62\u9762\u79ef\u548c\u4e5f\u66f4\u52a0\u63a5\u8fd1\u68af\u5f62\u9762\u79ef\uff1b\u4ee4n\u8d8b\u8fd1\u4e8e\u65e0\u7a77\uff0c\u5219$x_i$\u548cx\u8d8b\u8fd1\u4e8e\u771f\u5b9e\u503c\uff0c\u5bf9\uff4e\u53d6\u6781\u9650\u540e\uff0c\u77e9\u5f62\u9762\u79ef\u548c\u7b49\u4e8e\u68af\u5f62\u9762\u79ef\u3002\u8fd9\u4e5f\u5c31\u662f\u8bf4\uff1a\u56fe\u7ebf\u4e0e\u6a2a\u7eb5\u8f74\u6240\u56f4\u6210\u56fe\u5f62\u7684\u9762\u79ef\u5373\u4e3a\u771f\u5b9e\u7684\u4f4d\u79fb\u3002\u56e0\u6b64\u76f4\u63a5\u6c42\u5f97\u68af\u5f62\u9762\u79ef\uff0c\u5c31\u53ef\u7b97\u51fa\u5bf9\u5e94\u7684\u5300\u53d8\u901f\u8fd0\u52a8\u7684\u4f4d\u79fb\u3002\u5176\u4ed6\u7269\u7406\u8fc7\u7a0b\u540c\u7406\u3002<\/p>\n

\u5f53\u7136\uff0c\u5982\u679c$v_i$\u8868\u793a\u7684\u662f\u6bcf\u4e2a\u95f4\u9694\u7684\u6700\u5927\u901f\u5ea6\uff0c\u53d6\u548c\u540ex\u5927\u4e8e\u771f\u5b9e\u503c\uff0c\u4f46\u53d6\u6781\u9650\u540e\uff0cx\u8f6c\u5316\u4e3a\u771f\u5b9e\u503c\uff0c\u4ecd\u65e7\u4f53\u73b0\u4e3a\u56fe\u7ebf\u4e0e\u6a2a\u7eb5\u8f74\u6240\u56f4\u6210\u56fe\u5f62\u7684\u9762\u79ef\u3002
\n\u5206\u5272\uff0c\u5316\u201c\u53d8\u201d\u4e3a\u201c\u6052\u201d\u83b7\u5f97\u7269\u7406\u610f\u4e49\uff1b\u6c42\u548c\uff0c\u83b7\u5f97\u5b8f\u89c2\u8fd1\u4f3c\u503c\uff1b\u53d6\u6781\u9650\uff0c\u83b7\u5f97\u7cbe\u786e\u503c\u3002<\/p>\n

\u26051 \u6a21\u62df\u56de\u5230\u51fa\u53d1\u70b9\u7684\u5300\u53d8\u901f\u76f4\u7ebf\u8fd0\u52a8\uff1av=-t+5
\n1.1 \u9996\u5148\u5728\u6307\u4ee4\u680f\u8f93\u5165\u201cf(x) = \u5982\u679c(0\uff1c=x\uff1c=10,-x+5)\u201d\u753b\u51fav\u2014t\u56fe\u50cf\u3002
\n1.2 \u53d6\u539f\u70b9A(0,0)\uff0cB(10,0)\uff0c\u4f5c\u7ebf\u6bb5AB\uff0c\u5e76\u5728\u5c04\u7ebfAB\u4e0a\u4efb\u53d6\u4e00\u70b9C\uff0c\u9650\u5b9aC\u7684\u6a2a\u5750\u6807\u5927\u4e8e0\uff0c\u7528C\u7684\u6a2a\u5750\u6807\u6765\u8868\u793a\u65f6\u95f4\u3002
\n1.3 \u8fc7C\u70b9\u4f5cx\u8f74\u7684\u5782\u7ebf\uff0c\u4ea4f(x)\u4e8eD\uff0cD\u70b9\u7684\u7eb5\u5750\u6807\u5c31\u4ee3\u8868t\u65f6\u523b\u7269\u4f53\u7684\u901f\u5ea6\u3002
\n1.4 \u53d6E(0,5)\u4ee3\u8868\u7269\u4f53\u7684\u521d\u901f\u5ea6$v_0=5$\u3002
\n1.5 \u7528\u591a\u8fb9\u5f62\u5de5\u5177\u4f9d\u6b21\u9009\u53d6\u70b9C\u3001D\u3001E\u3001A\u3001C\u5f97\u5230q1\u3002
\n1.6 \u5728\u6307\u4ee4\u680f\u8f93\u5165(0,q1)\uff0c\u7eb5\u5750\u6807\u591a\u8fb9\u5f62\u7684\u9762\u79efq1\u5c31\u662f\u6211\u4eec\u63d0\u5230\u7684\u201cv\u2014t\u56fe\u4e0ex\u8f74\u6240\u6210\u7684\u9762\u79ef\u201d\u3002\u81f3\u6b64\uff0c\u6211\u4eec\u5c31\u5f97\u5230\u4e86\u4e00\u4e2a\u56de\u5230\u8d77\u59cb\u70b9\u7684\u5300\u53d8\u901f\u76f4\u7ebf\u8fd0\u52a8\u3002
\n1.7 \u5728\u6307\u4ee4\u680f\u8f93\u5165(x(C),q1)\u5e76\u8ffd\u8e2a\u5f97\u5230\u7684G\u70b9\u7684\u8f68\u8ff9\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5f97\u51fa\u8fd0\u52a8\u7684x\u2014t\u56fe\u50cf\u3002
\n1.8 \u542f\u52a8\u70b9C\u7684\u52a8\u753b\u5c31\u53ef\u4ee5\u770b\u5230\u8fd0\u52a8\u7684\u52a8\u6001\u7684\u8fc7\u7a0b\u3002
\n\"2023-8\"
\n\u26052 \u63a2\u7a76\u77e9\u5f62\u9762\u79ef\u548c\u4e0e\u68af\u5f62\u9762\u79ef\u7684\u5173\u7cfb<\/p>\n

2.1\u8fd9\u91cc\u6211\u4eec\u4f7f\u7528\u53e6\u4e00\u79cd\u521d\u901f\u5ea6\u4e0d\u4e3a0\u7684\u5300\u52a0\u901f\u76f4\u7ebf\u8fd0\u52a8\u6765\u8fdb\u884c\u9a8c\u8bc1\u8ba1\u7b97\uff1a\u5728\u6307\u4ee4\u680f\u8f93\u5165\u201cf(x) = \u5982\u679c(0\uff1c=x\uff1c=10, x+1)\u201d\u753b\u51fav\u2014t\u56fe\u50cf\u3002
\n2.2 \u521b\u5efa\u6ed1\u52a8\u6761n\uff0c\u6700\u5c0f\u503c\u8bbe\u4e3a10\uff0c\u6700\u5927\u503c\u8bbe\u4e3a100\uff0c\u589e\u91cf\u4e3a1\u3002\u5c06\u8fd0\u52a8\u5e73\u5206\u4e3an\u4e2a\u65f6\u95f4\u95f4\u9694\u3002
\n2.3 \u8f93\u5165\u201c\u4e0b\u548c(f,0,10,n)\u201d\u5f97\u5230\u82e5\u5e72\u77e9\u5f62\u7684\u9762\u79ef\u548c\u5373\u4e3a\uff1a\u7528\u6bcf\u6bb5\u65f6\u95f4\u95f4\u9694\u5185\u901f\u5ea6\u7684\u6700\u5c0f\u503c\u6765\u4f30\u7b97\u603b\u4f4d\u79fb\u3002
\n2.4 \u8f93\u5165\u201c\u4e0a\u548c(f,0,10,n)\u201d\u5f97\u5230\u82e5\u5e72\u77e9\u5f62\u7684\u9762\u79ef\u548c\u5373\u4e3a\uff1a\u7528\u6bcf\u6bb5\u65f6\u95f4\u95f4\u9694\u5185\u901f\u5ea6\u7684\u6700\u5927\u503c\u6765\u4f30\u7b97\u603b\u4f4d\u79fb\u3002
\n2.5 \u8f93\u5165\u201c\u79ef\u5206(f,0,10)\u201d\u5c55\u793a\u51fa\u68af\u5f62\u9762\u79ef\uff0c\u5373\u771f\u5b9e\u4f4d\u79fb\u5927\u5c0f\u3002
\n2.6\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u6587\u672c\u6846\u7684\u5de5\u5177\u5c06\u8fd9\u51e0\u4e2a\u91cf\u663e\u793a\u51fa\u6765\uff0c\u4fbf\u4e8e\u6211\u4eec\u89c2\u5bdf\u3002\u62d6\u52a8\u6ed1\u52a8\u6761n\uff0c\u4e0d\u96be\u53d1\u73b0\uff0c\u5f53n\u8d8a\u6765\u8d8a\u5927\u503c\uff0c\u6211\u4eec\u505a\u51fa\u7684\u201c\u4e0a\u548c\u201d\u548c\u201c\u4e0b\u548c\u201d\u7684\u503c\u4e5f\u8d8a\u63a5\u8fd1\u79ef\u5206\u503c\u3002\u8fd9\u4e5f\u7b26\u5408\u6211\u4eec\u7684\u8ba1\u7b97\u3002
\n\"2023-8\"<\/p>\n

\u4e8c\u3001\u53d8\u52a0\u901f\u76f4\u7ebf\u8fd0\u52a8<\/strong><\/p>\n

\u26053 \u6a21\u62df\u8fd0\u52a8\uff1a$v=-t^2+2t+3$.
\n3.1 \u4e0e\u4e0a\u9762\u4e00\u6837\u5730\uff0c\u5728\u6307\u4ee4\u680f\u8f93\u5165\u201cf(x) = \u5982\u679c(0\uff1c=x\uff1c=10, -x^2+2x+3)\u201d\u753b\u51fav\u2014t\u56fe\u50cf\u3002
\n3.2 \u53d6\u70b9A(0,0)\uff0cB(10,0)\u5728\u7ebf\u6bb5AB\u4e0a\u53d6\u70b9C\uff0c\u70b9C\u7684\u6a2a\u5750\u6807\u8868\u793a\u65f6\u95f4t.
\n3.3 \u8fc7C\u70b9\u4f5cx\u8f74\u7684\u5782\u7ebf\uff0c\u4ea4f(x)\u4e8eD\u70b9\uff0cD\u70b9\u7684\u7eb5\u5750\u6807\u8868\u793at\u65f6\u523b\u7269\u4f53\u7684\u901f\u5ea6\u3002
\n3.4 \u8f93\u5165\u201c\u79ef\u5206(f,0,x(C))\u201d\u5f97\u5230f(x)\u4e0ex\u8f74\u6240\u6210\u56fe\u5f62\u7684\u9762\u79ef\u3002\u5982\u679c\u5728\u4e00\u4e2a\u8fd0\u52a8\u4e2d\u6211\u4eec\u89c4\u5b9a\u4e86\u6b63\u65b9\u5411\uff0c\u90a3\u4e48\u5f53v<0\u65f6\u4ea7\u751f\u7684\u4f4d\u79fb\u4e5f\u4e3a\u8d1f\uff0c\u6240\u4ee5\u5728x\u8f74\u4e0b\u65b9\u7684\u9762\u79ef\u8bb0\u4e3a\u8d1f\u6570\u3002
\n3.5 \u8f93\u5165\u201c(0,\u79ef\u5206(f,0,x(C)))\u201d\u5f97\u5230\u7684E\u70b9\u5c31\u53ef\u4ee5\u6a21\u62df\u8fd9\u4e00\u8fd0\u52a8\uff0c\u8f93\u5165\u201c(x(C),\u79ef\u5206(f,0,x(C)))\u201d\uff0c\u5f97\u5230F\u70b9\u5e76\u8ffd\u8e2a\u5176\u8f68\u8ff9\uff0c\u53ef\u4ee5\u5f97\u5230x\u2014t\u56fe\u50cf\u3002
\n\"2023-8\"
\n3.6 \u542f\u52a8C\u70b9\u52a8\u753b\u53ef\u4ee5\u770b\u5230E\u70b9\u8fd0\u52a8\u60c5\u51b5\u3002
\n3.6\u7531\u4e8e\u901f\u5ea6\u5173\u4e8e\u65f6\u95f4\u7684\u5bfc\u6570\u8868\u793a\u8fd0\u52a8\u7684\u52a0\u901f\u5ea6\uff0c\u5728\u6307\u4ee4\u680f\u8f93\u5165\u201c\u5bfc\u6570(f(x))\u201d\u5f97\u5230\u7684\u66f2\u7ebf\u4fbf\u662fa\u2014t\u56fe\u50cf\u3002<\/p>\n

\u4e09\u3001\u7b80\u8c10\u8fd0\u52a8<\/strong><\/p>\n

\u26054 \u7b80\u8c10\u8fd0\u52a8\uff1av=sin(t)
\n4.1 \u6839\u636e\u4e0a\u9762\u7684\u77e5\u8bc6\uff0c\u6211\u4eec\u753b\u51fa\u4e00\u4e2a\u5468\u671f\u5185\u7684v\u2014t\u56fe\u50cf\u3002\u201cf(x) = \u5982\u679c(0 \u2264 x \u2264 2\u03c0, sin(x))\u201d
\n4.2 \u5728x\u8f74\u4e0a\u53d6\u70b9\uff1aA(0,0)\uff0cB(2\u03c0,0)\uff0c\u8fde\u63a5\u7ebf\u6bb5AB\uff0c\u5e76\u5728AB\u4e0a\u53d6\u70b9C\uff0c\u70b9C\u7684\u6a2a\u5750\u6807\u8868\u793a\u65f6\u95f4\u3002
\n4.3 \u8fc7C\u505ax\u8f74\u7684\u5782\u7ebf\uff0c\u4ea4f(x)\u4e8e\u70b9D\uff0c\u70b9D\u7684\u7eb5\u5750\u6807\u8868\u793a\u7269\u4f53\u7684\u901f\u5ea6\uff0cCD\u626b\u8fc7\u7684\u9762\u79ef\u5373\u4e3a\u7269\u4f53\u7684\u4f4d\u79fb(x\u8f74\u4e0a\u65b9\u4e3a\u6b63)\u3002
\n4.4 \u8f93\u5165\u201c\u79ef\u5206(f, 0, x(C))\u201d\u8868\u793a\u51fa\u7269\u4f53\u7684\u4f4d\u79fb\uff0c\u6211\u4eec\u4e0d\u96be\u53d1\u73b0\uff0c\u5f53x(C)= \u03c0\u65f6\uff0c\u7269\u4f53\u7684\u4f4d\u79fb\u8fbe\u5230\u6700\u5927\u503c\uff0c\u6b64\u540e\uff0cD\u70b9\u7eb5\u5750\u6807\u503c\u4e3a\u8d1f\uff0c\u7269\u4f53\u4f4d\u79fb\u5f00\u59cb\u51cf\u5c0f\u3002
\n4.5 \u8f93\u5165\u201c\u79ef\u5206(f, 0, \u03c0)\u201d\uff0cGeoGebra\u4f1a\u7ed9\u51fa\uff1ab=2\uff0c\u5373\u4e3a\u7269\u4f53\u505a\u7b80\u8c10\u8fd0\u52a8\u7684\u4e24\u4e2a\u6700\u8fdc\u4f4d\u7f6e\u4e4b\u95f4\u7684\u8ddd\u79bb\uff0c\u4e5f\u5373\u4e3a\u632f\u5e45\u7684\u4e8c\u500d\u3002
\n4.6 \u8f93\u5165(a-1,0)\uff0c\u5373\u5f97\u5230\u4ee5\u539f\u70b9\u4e3a\u5e73\u8861\u4f4d\u7f6e\u7684\u4e00\u4e2a\u7b80\u8c10\u8fd0\u52a8\u3002
\n\"2023-8\"<\/p>\n

\u56db\u3001\u96e8\u6ef4\u4e0b\u843d\u6a21\u578b<\/strong><\/p>\n

\u5728\u96e8\u5929\uff0c\u6211\u4eec\u53ef\u4ee5\u770b\u5230\u65e0\u6570\u96e8\u6ef4\u843d\u4e0b\uff0c\u6211\u4eec\u4e0d\u7981\u5c31\u4f1a\u60f3\u5230\uff0c\u96e8\u6ef4\u4ece\u5982\u6b64\u9ad8\u7684\u4e91\u5c42\u843d\u4e0b\uff0c\u4e3a\u4ec0\u4e48\u901f\u5ea6\u5374\u6ca1\u6709\u6211\u4eec\u60f3\u8c61\u4e2d\u7684\u90a3\u4e48\u5feb\u5462\uff1f\u8fd9\u5176\u5b9e\u662f\u56e0\u4e3a\u96e8\u6ef4\u6240\u53d7\u5230\u7684\u7a7a\u6c14\u963b\u529b\u5e76\u4e0d\u662f\u6052\u5b9a\u7684\uff0c\u800c\u662f\u4e0e\u901f\u5ea6\u76f8\u5173\u8054\uff0c\u6240\u4ee5\u96e8\u6ef4\u4e0b\u843d\u5e76\u4e0d\u662f\u5300\u53d8\u901f\u76f4\u7ebf\u8fd0\u52a8\u3002\u5f53\u96e8\u6ef4\u901f\u5ea6\u589e\u52a0\uff0c\u6240\u53d7\u5230\u7684\u7a7a\u6c14\u963b\u529b\u4e5f\u4f1a\u53d8\u5927\uff0c\u4ece\u800c\u96e8\u6ef4\u7684\u52a0\u901f\u5ea6\u51cf\u5c0f\uff0c\u76f4\u81f3\u8fbe\u5230\u6700\u5927\u901f\u5ea6\uff0c(\u5373\u201c\u6536\u5c3e\u901f\u5ea6\u201d)\u3002
\n

\"qiutopia\"<\/center>
\n\u00b7\u963b\u529b\u4e0e\u901f\u5ea6\u6210\u6b63\u6bd4\u7684\u8fd0\u52a8\uff1a
\n\u6211\u4eec\u5047\u8bbe\u96e8\u6ef4\u4e0b\u843d\u8fc7\u7a0b\u4e2d\u963b\u529b\u4e0e\u901f\u5ea6\u6210\u6b63\u6bd4\uff0cf=-kv
\n\u6839\u636e\u52a0\u901f\u5ea6\u7684\u5b9a\u4e49$a=\\frac{dv}{dt}$\uff1a\u2028\u6211\u4eec\u53ef\u4ee5\u5217\u51fa\u5fae\u5206\u65b9\u7a0b\uff1a\u2028$mg-kv=m\\frac{dv}{dt}$
\n\u5728GeoGebra\u4e2d\u6709\u5de5\u5177\uff1a\u201c\u89e3\u5e38\u5fae\u5206\u65b9\u7a0b( \uff1cf ‘(x, y)\uff1e, \uff1cf \u4e0a\u7684\u70b9\uff1e )\u201d
\n\u5176\u4e2d\uff1af\u2019(t,v)= (mg-kv)\/m\uff0c\u518d\u6839\u636e\u521d\u59cb\u6761\u4ef6\uff1a\u5f53t=0\u65f6\u521d\u901f\u5ea6\u4e3av_0\uff1b\u6240\u4ee5\u6211\u4eec\u53ef\u4ee5\u786e\u5b9a(0,v_0)\u5728v\u2014t\u56fe\u50cf\u4e0a\u3002
\n\u540c\u6837\u5730\uff0c\u6839\u636e\uff1a\u2028$v=\\frac{dx}{dt}$
\n\u6211\u4eec\u53ef\u4ee5\u6c42\u51fax\u4e0et\u7684\u5173\u7cfb\u3002
\n\u4e0b\u9762\u6211\u4eec\u6765\u64cd\u4f5c\u4e00\u4e0b\uff1a<\/p>\n

\u26055 \u963b\u529b\u4e0e\u901f\u5ea6\u6210\u6b63\u6bd4\u7684\u8fd0\u52a8\u3002
\n5.1 \u5728\u4ee3\u6570\u533a\u5efa\u7acb\u6ed1\u52a8\u6761g=10\uff0cm=1\uff0ck=2\uff0cv_0=0.\u6211\u4eec\u4ee5\u7ad6\u76f4\u5411\u4e0a\u4e3a\u6b63\u65b9\u5411\u3002
\n5.2 \u5728\u6307\u4ee4\u680f\u8f93\u5165\u201c\u89e3\u5e38\u5fae\u5206\u65b9\u7a0b((-mg+ky)\/m,(0,v_0))\u201d\u53ef\u4ee5\u5f97\u5230v\u2014t\u7684\u5173\u7cfb\uff1b\u518d\u8f93\u5165\u201c\u89e3\u5e38\u5fae\u5206\u65b9\u7a0b(f(x), (0, 0))\u201d \u53ef\u4ee5\u5f97\u5230x\u2014t\u7684\u5173\u7cfbh(x)
\n\"2023-8\"
\n5.3 \u5efa\u7acb\u6ed1\u52a8\u6761t\uff0c\u8868\u793a\u8fd0\u52a8\u65f6\u95f4\uff0c\u5728\u6307\u4ee4\u680f\u8f93\u5165\u201c(0,h(t))\u201d\u63cf\u51fa\u96e8\u6ef4\u7684\u4f4d\u7f6e\u3002
\n\u7531\u4e8e\u8fd0\u52a8\u7684\u901f\u5ea6\u8f83\u5927\uff0c\u4e0d\u4fbf\u4e8e\u5c4f\u5e55\u4e2d\u663e\u793a\uff0c\u56e0\u6b64\u6211\u4eec\u4e5f\u53ef\u4ee5\u8fdb\u884c\u9002\u5f53\u7684\u7f29\u653e\u3002
\n5.4 \u8fd9\u91cc\u9762\uff0cf(t)\u5c31\u4ee3\u8868\u96e8\u6ef4\u7684\u77ac\u65f6\u901f\u7387\uff0c|k f(t)|\u5373\u4e3a\u96e8\u6ef4\u53d7\u5230\u52a0\u901f\u5ea6\u7684\u5927\u5c0f\u3002
\n5.5 \u6211\u4eec\u53ef\u4ee5\u5728\u56fe\u4e2d\u7528\u77e2\u91cf\u6807\u8bb0\u51fa\u96e8\u6ef4\u7684\u53d7\u529b\u60c5\u51b5\uff0c\u8fd9\u91cc\u7559\u7ed9\u5927\u5bb6\u81ea\u884c\u5c1d\u8bd5\u3002
\n\"2023-8\"
\n5.6 \u6539\u53d8v_0\u7684\u503c\uff0c\u53ef\u4ee5\u6a21\u62df\u8003\u8651\u7a7a\u6c14\u963b\u529b\u7684\u4e0a\u629b\u8fd0\u52a8\uff0c\u627e\u5230x\u2014t\u56fe\u50cf\u4e0ex\u8f74\u7684\u4ea4\u70b9\uff0c\u53ef\u4ee5\u770b\u51fa\u7269\u4f53\u843d\u56de\u5730\u9762\u7684\u65f6\u95f4\uff0c\u8fdb\u800c\u8fd8\u53ef\u4ee5\u770b\u51fa\u843d\u56de\u5730\u9762\u7684\u901f\u5ea6\u3002\u53e6\u5916\uff0c\u7269\u4f53\u843d\u56de\u5730\u9762\uff0c\u4e5f\u5373\u4f4d\u79fbx=0\uff0c\u6240\u4ee5\u56fe\u4e2d\u4e24\u5757\u9634\u5f71\u9762\u79ef\u5927\u5c0f\u76f8\u7b49\u3002
\n\"2023-8\"
\n\u5230\u8fd9\u91cc\uff0c\u6211\u4eec\u770b\u4f3c\u5df2\u7ecf\u89e3\u51b3\u4e86\u96e8\u6ef4\u4e0b\u843d\u95ee\u9898\uff0c\u7136\u800c\uff0c\u7a7a\u6c14\u963b\u529b\u4e0e\u901f\u5ea6\u6210\u6b63\u6bd4\u53ea\u662f\u6211\u4eec\u7684\u4e00\u79cd\u8fd1\u4f3c\u3002\u5176\u5b9e\uff0c\u5728\u5b9e\u9645\u95ee\u9898\u4e2d\uff0c\u6211\u4eec\u8fd8\u4f1a\u9047\u5230\u67d0\u4e9b\u6a21\u578b\u7684\u963b\u529b\u8fd1\u4f3c\u4e0e\u901f\u5ea6\u7684\u5e73\u65b9\u6210\u6b63\u6bd4\u3002
\n\u26065.a\u5927\u5bb6\u53ef\u4ee5\u5c1d\u8bd5\u6a21\u62dff=-kv^2\u7684\u8fd0\u52a8\u6a21\u578b\u3002
\n\u26065.b\u5982\u679c\u521d\u901f\u5ea6\u4e0d\u662f\u7ad6\u76f4\u65b9\u5411\uff0c\u90a3\u4e48\u5bf9x\u65b9\u5411\u4e5f\u662f\u76f8\u540c\u7684\u65b9\u6cd5\uff0c\u5927\u5bb6\u53ef\u4ee5\u5c1d\u8bd5\u521d\u901f\u5ea6\u4e3av_0\uff0c\u4e0e\u6c34\u5e73\u65b9\u5411\u5939\u89d2\u4e3a\u03b8\u7684\u659c\u4e0a\u629b\uff0cf=-kv
\n\u26065.c \u66f4\u5e7f\u6cdb\u5730\uff0c\u963b\u529b\u4e0e\u901f\u5ea6\u76f8\u5173\u8054\u7684\u8fd0\u52a8\u4e0d\u6b62\u96e8\u6ef4\u4e0b\u843d\u6a21\u578b\u8fd9\u4e00\u79cd\uff0c\u5728\u7535\u78c1\u611f\u5e94\u6ed1\u8f68\u6a21\u578b\u4e2d\u5b89\u57f9\u529b\u4e0e\u901f\u5ea6\u4e5f\u662f\u5173\u8054\u7684\u3002\u73b0\u5728\u5217\u51fa\u4ee5\u4e0b\u51e0\u79cd\u7535\u8def\u7684\u7b54\u6848\uff0c\u629b\u7816\u5f15\u7389\uff0c\u611f\u5174\u8da3\u7684\u540c\u5b66\u53ef\u4ee5\u5c1d\u8bd5\uff1a
\n\"2023-8\"
\n$v=v_0e^{-\\frac{B^2L^2}{mR}t}$
\n$x=-\\frac{mRv_0}{B^2L^2}e^{-\\frac{B^2L^2}{mR}t}+\\frac{mRv_0}{B^2L^2}$<\/p>\n

$v=\\frac{B^2L^2v_0}{B^2L^2C+m}e^{-\\frac{B^2L^2C+m}{RmC}t}+\\frac{mv_0}{B^2L^2C+m}$
\n$x=-\\frac{B^2L^2v_0mRC^2}{{(B^2L^2C+m)}^2}e^{-\\frac{B^2L^2C+m}{RmC}t}+\\frac{mv_0}{B^2L^2C+m}t+\\frac{B^2L^2v_0mRC^2}{{(B^2L^2C+m)}^2}$<\/p>\n

$v=\\frac{E}{BL}(1-e^{-\\frac{BL}{mR}t})$
\n$x=\\frac{E}{BL}t+\\frac{mRE}{B^2L^2}e^{-\\frac{BL}{mR}t}-\\frac{mRE}{B^2L^2}$
\n\u5f20\u73c9\u7855:\u52a8\u6746\u95ee\u9898\u7684\u4e00\u822c\u65b9\u7a0b\u7ec4[?<\/a>]<\/p>\n

\u4e94\u3001\u536b\u661f\u8f68\u9053<\/strong><\/p>\n

\u5982\u56fe\u6240\u793a\uff0c\u5728\u79bb\u5730\u9762\u4e00\u5b9a\u9ad8\u5ea6\u6c34\u5e73\u629b\u51fa\u4e00\u7269\u4f53\uff0c\u5f53\u521d\u901f\u5ea6\u5c0f\u4e8e\u7b2c\u4e00\u5b87\u5b99\u901f\u5ea6\u65f6\uff0c\u7269\u4f53\u6cbf\u692d\u5706\u66f2\u7ebfa1\u3001a2\u843d\u5730\uff1b\u5f53\u901f\u5ea6\u4e3a\u7b2c\u4e00\u5b87\u5b99\u901f\u5ea6\u65f6\uff0c\u7269\u4f53\u6cbf\u5706\u8f68\u9053b\u8fd0\u884c\uff1b\u5f53\u521d\u901f\u5ea6\u4ecb\u4e8e\u7b2c\u4e00\u5b87\u5b99\u901f\u5ea6\u548c\u7b2c\u4e8c\u5b87\u5b99\u901f\u5ea6\u4e4b\u95f4\u65f6\uff0c\u7269\u4f53\u6cbf\u692d\u5706\u8f68\u9053\uff43\u8fd0\u884c\uff1b\u521d\u901f\u5ea6\u7b49\u4e8e\u7b2c\u4e8c\u5b87\u5b99\u901f\u5ea6\u65f6\uff0c\u7269\u4f53\u6cbf\u629b\u7269\u7ebf\u8f68\u9053\uff44\u79bb\u5f00\u5730\u7403\uff0c\u5927\u4e8e\u7b2c\u4e8c\u5b87\u5b99\u901f\u5ea6\u65f6\u7269\u4f53\u6cbf\u53cc\u66f2\u7ebf\uff45\u79bb\u5f00\u5730\u7403\u3002
\n\"2023-8\"
\n\u4e3a\u4e86\u6a21\u62df\u8fd9\u4e00\u73b0\u8c61\uff0c\u53ef\u5148\u7531\u725b\u987f\u7b2c\u4e8c\u5b9a\u5f8b\u548c\u51e0\u4f55\u5173\u7cfb\uff0c\u5efa\u7acb\u5fae\u5206\u65b9\u7a0b\u7ec4\uff1a
\n$\\left\\{\\begin{array} { c } { v _ { x } = \\frac { d x } { d t } } \\\\{ v _ { y } = \\frac { d y } { d t } } \\\\{ r = \\sqrt { x ^ { 2 } + y ^ { 2 } } }\\end{array} \\quad \\left\\{\\begin{array}{c}a=\\frac{d v}{d t}=-\\frac{G M}{r^{2}} \\\\a_{x}=\\frac{d v_{x}}{d t}=-\\frac{G M}{r^{2}} \\frac{x}{r} \\\\a_{y}=\\frac{d v_{y}}{d t}=-\\frac{G M}{r^{2}} \\frac{y}{r}\\end{array}\\right.\\right.$
\n\u4e0b\u9762\u6211\u4eec\u5c31\u53ef\u4ee5\u5229\u7528GeoGebra\u8fdb\u884c\u52a8\u753b\u6a21\u62df\uff1a
\n\u6211\u4eec\u57fa\u4e8e\u201c\u89e3\u5e38\u5fae\u5206\u65b9\u7a0b\u7ec4\u201d\u7684\u6307\u4ee4\uff0c\u6a21\u62df\u5929\u4f53\u8fd0\u52a8\u3002<\/p>\n

6.1 \u5728\u4ee3\u6570\u533a\u4e2d\u5b9a\u4e49\u53c2\u6570\u6ed1\u52a8\u6761\uff0c\u4e3a\u65b9\u4fbf\u8d77\u89c1\uff0c\u53ef\u8bbeG\uff1d1\uff0cM\uff1d10\uff0cm=1\uff0c\u73af\u7ed5\u5929\u4f53\u521d\u59cb\u4f4d\u7f6e\u7684\u6a2a\u5750\u6807x_{01}=10\uff0c\u7eb5\u5750\u6807y_{01}=0\uff0c\u521d\u901f\u5ea6\u6c34\u5e73\u5206\u91cfv_{x01}=0\uff0c\u7ad6\u76f4\u5206\u91cfv_{y01}=1.<\/p>\n

6.2 \u6839\u636e\u5fae\u5206\u65b9\u7a0b\u5206\u522b\u8f93\u5165:
\n\u201cv_{x1}'(t, x_1, y_1, v_{x1}, v_{y1}) = -(G M) x_1 \/ ((x_1)\u00b2 + (y_1)\u00b2)^(3\/2)\u201d
\n\u201cv_{y1}'(t, x_1, y_1, v_{x1}, v_{y1}) = -(G M) y_1 \/ ((x_1)\u00b2 + (y_1)\u00b2)^(3\/2)\u201d
\n\u201cx_{1}'(t, x_1, y_1, v_{x1}, v_{y1}) = v_{x1}\u201d
\n\u201cy_{1}'(t, x_1, y_1, v_{x1}, v_{y1}) = v_{y1}\u201d<\/p>\n

6.3 \u518d\u4f7f\u7528GeoGebra\u7684\u201c\u89e3\u5e38\u5fae\u5206\u65b9\u7a0b\u7ec4\u201d\u6307\u4ee4\u8f93\u5165\uff1a
\n\u201c\u89e3\u5e38\u5fae\u5206\u65b9\u7a0b\u7ec4({x_{1}’,y_{1}’,v_{x1}’,v_{y1}’},0,{x_{01},y_{01},v_{x01},v_{y01}},1000)\u201d<\/p>\n

\u76f4\u63a5\u8f93\u5165\u540e\uff0c\u8f6f\u4ef6\u4f1a\u81ea\u52a8\u5206\u522b\u7ed9\u51fa\u201cx_{1}–t\u201d\u3001\u201cy_{1}–t\u201d\u3001\u201cv_{x1}–t\u201d\u3001\u201cv_{y1}–t\u201d\u7684\u56fe\u50cf\uff0c\u81ea\u52a8\u547d\u540d\u4e3a\uff1a\u201cnumericalIntegral1\u201d\u3001\u201cnumericalIntegral2\u201d\u3001\u201cnumericalIntegral3\u201d\u3001\u201cnumericalIntegral4\u201d\u3002\u6307\u4ee4\u4e2d\u7b2c\u4e00\u90e8\u5206\u662f\u5bfc\u6570\u5217\u8868\uff0c\u7b2c\u4e8c\u90e8\u5206\u7684\u201c0\u201d\u4e3at\u7684\u521d\u59cb\u503c\uff0c\u7b2c\u4e09\u90e8\u5206\u5206\u522b\u662ft=0\u65f6\u7684\u201cx_{1}\u201d\u3001\u201cy_{1}\u201d\u3001\u201cv_{x1}\u201d\u3001\u201cv_{y1}\u201d\u7684\u503c\uff0c\u7b2c\u56db\u90e8\u5206\u76841000\u4e3at\u7684\u7ec8\u6b62\u503c\u3002
\n\"2023-8\"
\n6.4 \u8f93\u5165\u201cl = \u957f\u5ea6(numericalIntegral1)\u201d\u7ed9\u51fa\u6784\u6210\u56fe\u50cf\u7684\u70b9\u7684\u6570\u76ee\u3002
\n\u8f93\u5165\u201ct’=\u6ed1\u52a8\u6761(0, 1, 1\/l)\u201d\uff0c\u5176\u4e2d1\/l\u4e3a\u589e\u91cf\uff0c\u8fd9\u6837t’\u6bcf\u6b21\u589e\u52a0\u90fd\u4f1a\u5bf9\u5e94\u4e8e\u201cnumericalIntegral1\u201d\u4e2d\u7684\u4e0b\u4e00\u4e2a\u70b9\u3002<\/p>\n

6.5 \u63a5\u7740\u8bbe\u4e2d\u5fc3\u5929\u4f53\u6240\u5728\u4f4d\u7f6e\u4e3a\u201cE=(0,0)\u201d\u3002
\n\u4e3a\u4e86\u5f97\u5230\u73af\u7ed5\u5929\u4f53\u7684\u4f4d\u7f6e\uff0c\u5148\u8f93\u5165\u201cB =\u63cf\u70b9(numericalIntegral1, t’)\u201d\uff0c\u5f97\u5230(t , x_{1})\u70b9\uff0c\u8be5\u63cf\u70b9\u6307\u4ee4\u4e2d\u7b2c\u4e8c\u4e2at’\u662f\u8def\u5f84\u503c\uff0c\u53d6\u503c\u57280~1\u4e4b\u95f4\uff0c\u51b3\u5b9a\u4e86\u53d6\u503c\u5728numericalIntegral1\u4e2d\u7684\u76f8\u5e94\u4f4d\u7f6e\uff0c\u4f8b\u5982\uff0c\u82e5\u53d6\u201c0\u201d\u5219\u4e3a\u7b2c\u4e00\u4e2a\u70b9\uff0c\u201c1\u201d\u5219\u4e3a\u6700\u540e\u4e00\u4e2a\u70b9\uff0c\u201c1\/l\u201d\u5219\u4e3a\u7b2c\u4e8c\u4e2a\u70b9\u3002
\n\u518d\u8f93\u5165\u201cC =\u63cf\u70b9(numericalIntegral2, t’)\u201d\uff0c\u5f97\u5230(t , y_{1})\u70b9
\n\u8f93\u5165\u201cD=(y(B),y(C))\u201d\u5373\u53ef\u5f97\u5230\u73af\u7ed5\u5929\u4f53\u7684\u4f4d\u7f6e\u5750\u6807\u3002
\n\u6700\u540e\uff0c\u542f\u52a8t’\u52a8\u753b\uff0c\u5373\u53ef\u89c2\u5bdf\u5230\u73af\u7ed5\u5929\u4f53\u7684\u8fd0\u52a8\u3002t’\u4e58\u4e0a1000\u5c31\u662f\u6a21\u62df\u65f6\u95f4t\u3002<\/p>\n

6.6 \u8fd9\u6837\u6211\u4eec\u5c31\u5f97\u5230\u4e86\u5730\u7403\u536b\u661f\u7684\u8fd0\u52a8\u6a21\u62df\u52a8\u753b\uff0c\u8c03\u6574\u6ed1\u52a8\u6761t’\u7684\u901f\u5ea6\u4ee5\u4fbf\u4e8e\u89c2\u6d4b\uff0c\u4e5f\u53ef\u4ee5\u9009\u62e9\u8ddf\u8e2a\u536b\u661fS\u7684\u8f68\u8ff9\uff0c\u6dfb\u52a0\u6309\u94ae\u66f4\u65b9\u4fbf\u64cd\u4f5c\u2026\u2026
\n6.7 \u89e3\u5fae\u5206\u65b9\u7a0b\u540e\u5f97\u5230\u56db\u6761\u66f2\u7ebf\uff0c\u5176\u4e2d1\u548c2\u8868\u793ax\uff0cy\u7684\u4f4d\u7f6e\u5750\u6807\uff0c3\u548c4\u5206\u522b\u8868\u793a\u901f\u5ea6\u5728x\uff0cy\u65b9\u5411\u4e0a\u7684\u5206\u91cf\uff0c\u6240\u4ee5\u536b\u661f\u7684\u901f\u5ea6\u5e94\u8be5\u5c31\u662f:
\n\"2023-8\"
\n\"2023-8\"<\/p>\n

6.8 \u901a\u8fc7\u8fd9\u4e2a\u6a21\u578b\uff0c\u6211\u4eec\u53ef\u4ee5\u63a2\u7a76\u5f00\u666e\u52d2\u7b2c\u4e8c\u5b9a\u5f8b\u3002\u5982\u679c\u60f3\u8981\u63cf\u51fa\u7b49\u65f6\u95f4\u95f4\u9694\u7684\u536b\u661f\u4f4d\u7f6e\u5c31\u9700\u8981\u7528\u5230\u6307\u4ee4\u201c\u5e8f\u5217( \uff1c\u4ee3\u6570\u5f0f\uff1e, \uff1c\u53d8\u91cf\uff1e, \uff1c\u8d77\u59cb\u503c\uff1e,\uff1c\u7ec8\u6b62\u503c\uff1e, \uff1c\u589e\u91cf\uff1e )\u201d
\n\u5efa\u7acb\u6ed1\u52a8\u6761\u95f4\u9694\u6570\u91cfn\u3002\u5728\u6307\u4ee4\u680f\u8f93\u5165\uff1a\u201cl1=\u5e8f\u5217(\u63cf\u70b9(numericalIntegral1, a), a, 0, 0.5, 0.01)\u201d\u63cf\u51fa\u6bcf\u4e2a\u65f6\u523b\u5bf9\u5e94\u7684B\u70b9\uff1b\u8f93\u5165\u201cl2=\u5e8f\u5217(\u63cf\u70b9(numericalIntegral2, a), a, 0, 0.5, 0.01)\u201d\u63cf\u51fa\u6bcf\u4e2a\u65f6\u523b\u5bf9\u5e94\u7684C\u70b9\uff1b\u8f93\u5165\u201cl3=\u5e8f\u5217((y(\u5143\u7d20(l1, b)), y(\u5143\u7d20(l2, b))), b, 1, n, 1)\u201d\u63cf\u51fa\u6bcf\u4e2a\u65f6\u523b\u7684\u536b\u661f\u4f4d\u7f6e\u3002
\n\u8fd9\u91cc\u6709\u53e6\u5916\u4e00\u4e2a\u6307\u4ee4\u201c\u5143\u7d20( \uff1c\u5217\u8868\uff1e, \uff1c\u5143\u7d20\u4f4d\u7f6e\uff1e )\u201d\u4f8b\u5982\uff0c\u5217\u8868l1\u7684\u7b2c\u4e00\u9879\u53ef\u4ee5\u7528\u201c\u5143\u7d20(l1,1)\u201d\u6765\u8868\u793a\u3002<\/p>\n

6.9 \u8f93\u5165\u201c\u5706\u9525\u66f2\u7ebf(\u5143\u7d20(l3, 1), \u5143\u7d20(l3, 2), \u5143\u7d20(l3, 3), \u5143\u7d20(l3, 4), \u5143\u7d20(l3, 5))\u201d\u6765\u63cf\u51fa\u536b\u661f\u8fd0\u52a8\u8f68\u9053\u3002\u7531\u4e8e\u6ca1\u6709\u5de5\u5177\u53ef\u4ee5\u76f4\u63a5\u753b\u51fa\u692d\u5706\u7126\u70b9\u4e0e\u692d\u5706\u4e0a\u4e24\u70b9\u8fde\u6210\u7684\u5c01\u95ed\u56fe\u5f62\uff0c\u6240\u4ee5\u6211\u4eec\u9700\u8981\u505a\u51fa\u4e00\u5b9a\u7684\u53d8\u5316\u3002\u53d6\u5706\u9525\u66f2\u7ebfc\u4e0ex\u8f74\u7684\u4e24\u4e2a\u4ea4\u70b9\uff0c\u5e76\u53d6\u8fd9\u4e24\u4e2a\u70b9\u7684\u4e2d\u70b9M\u3002\u8f93\u5165\u201cl4=\u5e8f\u5217(\u6247\u5f62(c, \u5143\u7d20(l3, t), \u5143\u7d20(l3, t + 1)), t, 1, n – 1, 1)\u201d\u53ef\u4ee5\u63cf\u51faM\u70b9\u4e0ec\u4e0a\u76f8\u90bb\u4e24\u70b9\u7ec4\u6210\u7684\u5c01\u95ed\u56fe\u884c\uff1b\u8f93\u5165\u201cl5=\u5e8f\u5217(\u591a\u8fb9\u5f62(M, \u5143\u7d20(l3, t), \u5143\u7d20(l3, t + 1)), t, 1, n, 1)\u201d\u53ef\u4ee5\u63cf\u51faM\u70b9\u4e0ec\u4e0a\u76f8\u90bb\u4e24\u70b9\u8fde\u6210\u4e09\u89d2\u5f62\u7684\u9762\u79ef\u3002\u4e8c\u8005\u76f8\u89c1\u5c31\u5f97\u5230c\u4e0a\u76f8\u90bb\u4e24\u70b9\u5f62\u6210\u7684\u7c7b\u4f3c\u201c\u5f13\u5f62\u201d\u7684\u9762\u79ef\uff0c\u5176\u518d\u52a0\u4e0a\u8be5\u4e24\u70b9\u4e0e\u7126\u70b9\u8fde\u6210\u4e09\u89d2\u5f62\u7684\u9762\u79ef\uff0c\u5373\u4e3a\u5730\u7403\u4e0e\u536b\u661f\u8fde\u7ebf\u626b\u8fc7\u7684\u9762\u79ef\u3002\u6240\u4ee5\uff0c\u8f93\u5165\u201cl6=\u5e8f\u5217(\u591a\u8fb9\u5f62(E,\u5143\u7d20(l3, t),\u5143\u7d20(l3, t + 1)), t, 1, n, 1)\u201d\u3001\u201cl7=\u5e8f\u5217(\u5143\u7d20(l4, t) – \u5143\u7d20(l5, t) + \u5143\u7d20(l6, t), t, 1, n, 1)\u201d\u5176\u4e2d\uff0cl7\u4e2d\u5404\u9879\u5373\u4e3a\u4e00\u4e2a\u65f6\u95f4\u95f4\u9694\u5185\u626b\u8fc7\u9762\u79ef\u3002
\n\"2023-8\"
\n\"2023-8\"<\/p>\n

6.10 \u6211\u4eec\u6253\u5f00\u4ee3\u6570\u533a\uff0c\u67e5\u627el7\u6570\u636e\uff0c\u53d1\u73b0l7\u4e2d\u6bcf\u4e00\u9879\u90fd\u76f8\u7b49\uff0c\u8fd9\u4fbf\u9a8c\u8bc1\u4e86\u5f00\u666e\u52d2\u7b2c\u4e8c\u5b9a\u5f8b\u3002
\n\"2023-8\"
\n\"2023-8\"
\n\u516d\u3001\u4f20\u9001\u5e26\u2014\u6ed1\u5757\u6a21\u578b<\/strong><\/p>\n

Tip\uff1a\u8be5\u9898\u76ee\u6d89\u53ca\u8f83\u591a\u7684\u53c2\u91cf\uff0c\u6a21\u62df\u8f83\u4e3a\u590d\u6742\uff0c\u9650\u4e8e\u7bc7\u5e45\uff0c\u672c\u6587\u65e0\u6cd5\u4e00\u4e00\u9610\u8ff0\u5176\u4e2d\u539f\u7406\uff0c\u7ec6\u8282\u90e8\u5206\u7559\u7ed9\u5404\u4f4d\u8bfb\u8005\u601d\u8003\u5176\u80cc\u540e\u539f\u7406\u3002<\/p>\n

7.1 \u5efa\u7acb\u6ed1\u52a8\u6761(\u6216\u6570\u5b57)\uff1a\u659c\u9762\u957f\u5ea6L\uff1b\u4f20\u9001\u5e26\u901f\u5ea6v\uff1b\u91cd\u529b\u52a0\u901f\u5ea6g=10\uff1b\u6469\u64e6\u7cfb\u6570\u03bc\uff1b\u6ed1\u5757\u521d\u901f\u5ea6v_0\uff1b\u76f8\u5bf9\u901f\u5ea6\uff1adv_0=v_0-v\uff1b\u8ddd\u659c\u9762\u5e95\u7aef\u8ddd\u79bbb\uff1b\u659c\u9762\u503e\u89d2\u03b8\uff1b\u65f6\u523bt.<\/p>\n

7.2 \u505a\u51fa \u51fd\u6570q_1(x)=\u5982\u679c(x\u22650, L – b)\uff1b
\n\u6570\u5b57t_2=(-dv_0) \/ (\u03bc g cos(\u03b8) – g sin(\u03b8))\uff1b
\n\u6570\u5b57a_1=g sin(\u03b8) + \u03bc g cos(\u03b8)\uff1b
\n\u51fd\u6570h_1=\u5982\u679c(0 \u2264 x \u2264 dv_0 \/ a_1, v_0 x – 0.5a_1 x\u00b2, \u5982\u679c(x > dv_0 \/ a_1, v_0 dv_0 \/ a_1 – 0.5a_1 (dv_0 \/ a_1)\u00b2 + v (x – dv_0 \/ a_1)))\uff1b
\n\u6570\u5b57a_3=\u03bc g cos(\u03b8) – g sin(\u03b8)\uff1b
\n\u51fd\u6570s_1=\u5982\u679c(0 \u2264 x \u2264 t_2, v_0 x + 0.5a_3 x\u00b2, \u5982\u679c(x > t_2, v_0 t_2 + 0.5a_3 t_2\u00b2 + v (x – t_2)))\uff1b
\n\u6570\u5b57a_2=a_4=g sin(\u03b8) – \u03bc g cos(\u03b8)\uff1b
\n\u51fd\u6570p_1=\u5982\u679c(0 \u2264 x \u2264 dv_0 \/ a_1, v_0 x – 0.5a_1 x\u00b2, \u5982\u679c(x > dv_0 \/ a_1, v_0 dv_0 \/ a_1 – 0.5a_1 (dv_0 \/ a_1)\u00b2 + v (x – dv_0 \/ a_1) – 0.5a_2 (x – dv_0 \/ a_1)\u00b2))\uff1b
\n\u51fd\u6570f_2=v_0 x – 0.5a_4 x\u00b2\uff1b
\n\u51fd\u6570g_2=\u5982\u679c(\u03bc \u2265 tan\u03b8, h_1, p_1)\uff1b
\n\u51fd\u6570h_2=\u5982\u679c(\u03bc \u2265 tan\u03b8, s_1, f_2)\uff1b
\n\u51fd\u6570p_2=\u5982\u679c(dv_0 \u2265 0, g_2, h_2)\uff1b
\n\u51fd\u6570r_1=\u5982\u679c(x \u2265 0, -b)\u3002<\/p>\n

7.3 \u505a\u51fa \u70b9Z=\u4ea4\u70b9(p_2, r_1, 0, 50)\uff1b\u5efa\u7acb\u6ed1\u52a8\u6761\u6ed1\u5757\u5c3a\u5bf8s\uff1b\u6eda\u8f6e\u534a\u5f84r\uff1b\u4efb\u610f\u53d6\u70b9A\u4f5c\u4e3a\u4f20\u9001\u5e26\u5e95\u90e8\u8f74\u5fc3\uff1b\u4ee5A\u4e3a\u5706\u5fc3\uff0cr\u4e3a\u534a\u5f84\u4f5c\u5706d\uff1b\u4f5c\u70b9B=A + (L cos(\u03b8), L sin(\u03b8))\u4f5c\u4e3a\u4f20\u9001\u5e26\u9876\u90e8\u8f74\u5fc3\uff1b\u4ee5B\u4e3a\u5706\u5fc3\uff0cr\u4e3a\u534a\u5f84\u4f5c\u5706c\uff1b\u8fde\u63a5\u7ebf\u6bb5AB(f)\uff1b\u8fc7\u70b9A\u4f5cAB\u7684\u5782\u7ebfh\uff1bc\u4e0eh\u7684\u4ea4\u70b9E\u3001F\uff1b\u65b9\u5411\u5411\u91cfs=(cos(\u03b8), sin(\u03b8))\uff1b\u70b9J=F + (b cos(\u03b8), b sin(\u03b8))\uff1b\u8fc7\u70b9A\u4f5cAB\u7684\u5782\u7ebfg_1\uff1bd\u4e0eg_1\u4ea4\u70b9C\u3001D\uff1b\u8fde\u63a5\u7ebf\u6bb5DF(i)\uff1b\u4f5c\u76f4\u7ebfeq1\uff1ax = x(A)\uff1beq1\u4e0e\u5706c\u4ea4\u70b9G\u3001H\uff1b\u70b9I= H + (L, 0)\uff1b\u8fde\u63a5\u7ebf\u6bb5HI(k)\uff1b\u6570\u5b57b_1=1\/2 s\uff1b\u70b9K= J – b_1 (cos(\u03b8), sin(\u03b8))\uff1b\u70b9M= J + b_1 (cos(\u03b8), sin(\u03b8))\uff1b\u505a\u51fapoly1=\u591a\u8fb9\u5f62(K, M, 4)\uff1b<\/p>\n

7.4 \u4f5c\u70b9X=\u5982\u679c(0 \u2264 t \u2264 dv_0 \/ a_1, J + s (v_0 t – 0.5a_1 t\u00b2), \u5982\u679c(t > dv_0 \/ a_1, J + s (v_0 dv_0 \/ a_1 – 0.5a_1 (dv_0 \/ a_1)\u00b2 + v (t – dv_0 \/ a_1))))\uff1b
\n\u70b9P=\u5982\u679c(0 \u2264 t \u2264 dv_0 \/ a_1, J + s (v_0 t – 0.5a_1 t\u00b2), \u5982\u679c(t > dv_0 \/ a_1, J + s (v_0 dv_0 \/ a_1 – 0.5a_1 (dv_0 \/ a_1)\u00b2 + v (t – dv_0 \/ a_1) – 0.5a_2 (t – dv_0 \/ a_1)\u00b2)))\uff1b
\n\u70b9Q=\u5982\u679c(\u03bc \u2265 tan\u03b8, X, P)\uff1b\u8fde\u63a5\u7ebf\u6bb5CE(j)\uff1b
\n\u4f5ck\u4e0ej\u7684\u5939\u89d2\u03b1\uff1b
\n\u70b9R=\u5982\u679c(0 \u2264 t \u2264 t_2, J + s (v_0 t + 0.5a_3 t\u00b2), \u5982\u679c(t > t_2, J + s (v_0 t_2 + 0.5a_3 t_2\u00b2 + v (t – t_2))))\uff1b
\n\u70b9S= J + s (v_0 t – 0.5a_4 t\u00b2)\uff1b
\n\u70b9T=\u5982\u679c(\u03bc \u2265 tan\u03b8, R, S)\uff1b
\n\u70b9U=\u5982\u679c(dv_0\u22650, Q ,T)\uff1b
\n\u591a\u8fb9\u5f62poly1’=\u5e73\u79fb(poly1, \u5411\u91cf(J, U))\uff1b<\/p>\n

7.5 \u8bbe\u6570\u5b57t_1=dv_0 \/ a_1\uff1b\u70b9W=\u4ea4\u70b9(p_2, q_1, 0, 50)\uff1b
\n\u6570\u5b57o=\u5982\u679c(x(W) \/ x(W) \u225f 1, x(W), 120)\uff1b
\n\u6570\u5b57u=\u5982\u679c(x(Z) \/ x(Z) \u225f 1, x(Z), 120)\uff1b
\n\u6570\u5b57w=\u6700\u5c0f\u503c(o, u)\uff1b
\n\u51fd\u6570q_2=\u5bfc\u6570(p_2, x)\uff1b
\n\u51fd\u6570r_2=\u5982\u679c(0 \u2264 x \u2264 w, q_2(x))\uff1b
\n\u51fd\u6570s_2=\u5982\u679c(0 \u2264 x \u2264 w, p_2(x))\uff1b
\n\u70b9V= (t, s_2(t))\uff1b\u70b9A_1= (t, r_2(t))\uff1b\u70b9B_1= (t, v)\uff1b\u70b9C_1= (t, v t)\uff1b\u70b9D_1= (w, v)\uff1b\u70b9E_1= (0, v)\uff1b\u70b9F_1= (t, v t)\uff1b\u70b9G_1(0,0)\uff1b\u8fde\u63a5\u7ebf\u6bb5A_1B_1(a)\uff1b\u8fde\u63a5\u7ebf\u6bb5C_1V(e)\uff1b\u8fde\u63a5\u7ebf\u6bb5B_1E_1(f_1)\uff1b\u8fde\u63a5\u7ebf\u6bb5F_1G_1(i_1)\uff1b<\/p>\n

7.6 \u8fc7\u70b9J\u4f5ci\u7684\u5782\u7ebf\uff1b\u4ee5J\u4e3a\u5706\u5fc3\uff0cb_1\u4e3a\u534a\u5f84\u4f5c\u5706c_1\uff1bc_1\u4e0ej_1\u7684\u4ea4\u70b9I_1\u3001H_1\uff1b\u4f5c\u5411\u91cfu_1=\u5e73\u79fb(\u5411\u91cf(J, U), I_1)\uff1b\u70b9K_1=\u5f62\u5fc3(poly1′)\uff1b\u5411\u91cfJ_1=\u5e73\u79fb(\u5411\u91cf(r_2(t) s), K_1)\uff1b\u51fd\u6570r_2\u2019=\u5bfc\u6570(r_2)\uff1b\u4f5c\u70b9L_1=(t, r_2′(t))\uff1b\u70b9M_1=(t, 0)\uff1b\u4f5c\u7ebf\u6bb5L_1M_1(k_1)\uff1bf_3=v\uff1b\u70b9N_1=\u4ea4\u70b9(r_2, f_3, 0, 100)\uff1b\u70b9O_1=(x(N_1), s_2(x(N_1)))\uff1b\u70b9P_1=(x(N_1), r_2′(N_1))\uff1b\u70b9Q_1=(x(N_1), 0)\uff1b\u8fde\u63a5\u7ebf\u6bb5O_1Q_1(l_1)\uff1b\u7ebf\u6bb5N_1Q_1(m_1)\uff1b\u7ebf\u6bb5P_1Q_1(n_1)\uff1b\u6570\u5b57d_1=\u7ebf\u6bb5(O_1, Q_1)\uff1be_1=\u5982\u679c(d_1 \/ d_1 \u225f 1, d_1)\uff1b\u4f5c\u70b9G\u2019=\u65cb\u8f6c(G, -v t \/ r, A)\uff1bH\u2019=\u65cb\u8f6c(H, -v t \/ r, A)\uff1bc_2=\u5706\u6247\u5f62(A, G’, H’)\uff1bd_2=\u5706\u6247\u5f62(A, H’, G’)\uff1be_2=\u5e73\u79fb(d_2, \u5411\u91cf(A, B))\uff1bk_2=\u5e73\u79fb(c_2, \u5411\u91cf(A, B))\uff1b\u51fd\u6570g_3=\u5982\u679c(0 \u2264 x \u2264 t, v x)\uff1b<\/p>\n

7.7\u5176\u5b9e\uff0c\u6ed1\u5757\u76f8\u5bf9\u4e8e\u4f20\u9001\u5e26\u7684\u4f4d\u79fb\u4e5f\u662f\u53ef\u4ee5\u7528\u4e00\u4e2a\u91cf\u6765\u8868\u793a\u7684\uff0c\u5e94\u7528GeoGebra\u7684\u201c\u79ef\u5206\u4ecb\u4e8e( \uff1c\u51fd\u6570\uff1e, \uff1c \u51fd\u6570\uff1e, \uff1cx -\u8d77\u59cb\u503c\uff1e, \uff1c\/x\uff1e\uff1cx -\u7ec8\u6b62\u503c\uff1e )\u201d\u6307\u4ee4\uff0c\u53ef\u4ee5\u8868\u793a\u5939\u5728\u4e24\u51fd\u6570\u4e4b\u95f4\u7684\u56fe\u5f62\u9762\u79ef\uff0c\u5927\u5bb6\u53ef\u4ee5\u81ea\u884c\u5c1d\u8bd5\u3002
\n\"2023-8\"
\n\"2023-8\"
\n\"2023-8\"<\/p>\n

\u00b7\u4f5c\u4e1a\u5185\u5bb9\uff1a<\/strong>
\n\u2605\u5fc5\u505a\u9898\u76ee\uff1a
\n1\uff0c\u56de\u5230\u51fa\u53d1\u70b9\u7684\u5300\u53d8\u901f\u76f4\u7ebf\u8fd0\u52a8\uff1av=-t+5
\n2\uff0c\u63a2\u7a76\u77e9\u5f62\u9762\u79ef\u548c\u4e0e\u68af\u5f62\u9762\u79ef\u7684\u5173\u7cfb
\n3\uff0c\u53d8\u52a0\u901f\u76f4\u7ebf\u8fd0\u52a8\uff1a
\n4\uff0c\u7b80\u8c10\u8fd0\u52a8
\n5\uff0c\u96e8\u6ef4\u4e0b\u843d\u6a21\u578b<\/p>\n

\u2605\u9009\u505a\u9898\u76ee\uff1a
\n1\uff0c\u96e8\u6ef4\u4e0b\u843d\u62d3\u5c55\uff1af=-kv^2\u7684\u8fd0\u52a8
\n2\uff0c\u521d\u901f\u5ea6\u4e3av_0\uff0c\u4e0e\u6c34\u5e73\u65b9\u5411\u5939\u89d2\u4e3a\u03b8\u7684\u659c\u4e0a\u629b\uff0cf=-kv
\n3\uff0c\u63a2\u7a76\u51e0\u79cd\u7535\u78c1\u611f\u5e94\u91d1\u5c5e\u68d2\u7684\u8fd0\u52a8
\n4\uff0c\u5c1d\u8bd5\u6a21\u62df\u6c7d\u8f66\u6052\u529f\u7387\u542f\u52a8\u8fc7\u7a0b
\n5\uff0c\u6784\u9020\u5730\u7403\u536b\u661f\u8fd0\u52a8\u6a21\u578b<\/p>\n

\u2606\u6311\u6218\u9898\u76ee\uff1a
\n\u5c1d\u8bd5\u6a21\u62df\u4f20\u9001\u5e26\u2014\u6ed1\u5757\u52a8\u753b<\/p>\n

\u81f4\u8c22\uff1a
\n\u5047\u671f\u4f0a\u59cb\uff0c\u65b0\u8bfe\u9898\u7684\u7f16\u5199\u88ab\u63d0\u4e0a\u65e5\u7a0b\uff0c\u5230\u8fd9\u7bc7\u8ba9\u6211\u6389\u4e86\u4e0d\u5c11\u5934\u53d1\u7684\u6587\u7ae0\u6b63\u5f0f\u5b8c\u7a3f\uff0c\u611f\u8c22\u6240\u6709\u9f0e\u529b\u652f\u6301\u65b0\u8bfe\u9898\u63a8\u8fdb\u7684\u670b\u53cb\u3002\u5728\u6700\u540e\u4e00\u5468\u8bfe\u9898\u201c\u4ed8\u6893\u201d\u4e4b\u9645\uff0c\u9996\u5148\u6211\u8981\u4e07\u5206\u611f\u8c22\u6211\u7684\u6307\u5bfc\u8001\u5e08\u2014\u2014qiusir\u3002\u4ece\u4efb\u52a1\u7684\u9009\u53d6\u5230\u6846\u67b6\u7684\u5236\u5b9a\uff0c\u4ece\u5b66\u672f\u652f\u6301\u5230\u6280\u672f\u7f16\u8f91\uff0cqiusir\u90fd\u7ed9\u4e88\u4e86\u6211\u524d\u6240\u672a\u6709\u7684\u652f\u6301\u3002\u540c\u65f6\u611f\u8c22\u6c5f\u82cf\u4eea\u5f81\u674e\u590f\u5ca9\u540c\u5b66\u4e3a\u672c\u5468\u8bfe\u9898\u8fdb\u884c\u9996\u6b21\u8bd5\u9a8c\uff0c\u5e76\u63d0\u51fa\u5b9d\u8d35\u5efa\u8bae\uff1b\u611f\u8c22\u4faf\u660e\u7384\u540c\u5b66\u548c\u9ec4\u946b\u9633\u540c\u5b66\u4e3a\u672c\u5468\u8bfe\u9898\u5f55\u5236\u9996\u6279\u52a9\u6559\u89c6\u9891\u3002<\/em><\/p>\n

\u53c2\u8003\u6587\u732e\uff1a
\n[1]\u9648\u6d77\u6d9b,\u91d1\u60e0\u5409,\u6768\u79c0\u53d1\u7b49.\u57fa\u4e8eGeoGebra\u548cVPython\u7684\u5929\u4f53\u8fd0\u884c\u6570\u503c\u6a21\u62df[J].\u7269\u7406\u901a\u62a5,2022(09):133-137+140.
\n[2]\u79e6\u667a,\u66f2\u5cf0\u6797.\u57fa\u4e8eGeoGebra\u7684\u5fae\u79ef\u5206\u53ef\u89c6\u5316\u6559\u5b66[J].\u9ad8\u7b49\u6570\u5b66\u7814\u7a76,2022,25(06):83-86.
\n[3]\u9648\u5a67,\u5f20\u4e1c\u6d77,\u738b\u6653\u950b.GeoGebra\u5728\u9ad8\u804c\u5fae\u79ef\u5206\u6982\u5ff5\u6559\u5b66\u4e2d\u7684\u5e94\u7528\u4e3e\u4f8b[J].\u4e2d\u56fd\u6559\u80b2\u6280\u672f\u88c5\u5907,2017(10):42-45.
\n[4]\u6731\u4e9a\u519b.\u501f\u52a9GeoGebra\u6df1\u5ea6\u5256\u6790\u4e00\u9053\u8c03\u7814\u6d4b\u8bd5\u9898[J].\u7269\u7406\u6559\u5e08,2022,43(12):90-91+94.
\n[5]\u4f55\u79cb\u9759,\u6768\u6587\u97ec,\u738b\u6653\u96c5\u7b49.\u65b0\u51a0\u75ab\u60c5\u4e0b\u201c\u505c\u8bfe\u4e0d\u505c\u5b66\u201d\u7684\u5927\u5b66\u7269\u7406GeoGebra\u4eff\u771f\u5355\u6446\u5b9e\u9a8c[J].\u5927\u5b66\u7269\u7406\u5b9e\u9a8c,2021,34(01):87-91.DOI:10.14139\/j.cnki.cn22-1228.2021.01.023.
\n[6]\u6d82\u4e16\u8d85. MBI\u6a21\u5f0f\u4e0b\u5229\u7528GeoGebra\u8f6f\u4ef6\u5728\u9ad8\u4e2d\u7269\u7406\u4e2d\u7684\u53ef\u89c6\u5316\u6559\u5b66\u7814\u7a76[D].\u8d35\u5dde\u5e08\u8303\u5927\u5b66,2023.DOI:10.27048\/d.cnki.ggzsu.2023.000425.
\n[7<\/a>]\u963b\u529b\u4e0e\u901f\u5ea6\u6210\u6b63\u6bd4\u7684\u8fd0\u52a8<\/p>\n

\u4e3a\u4fdd\u969c\u65b0\u8bfe\u9898\u7684\u6309\u65f6\u4ea4\u4ed8\uff0c\u672c\u4eba\u4ed4\u7ec6\u7814\u8bfb\u4e86\u6709\u5173\u95ee\u9898\u7684\u5404\u7c7b\u6587\u7ae0\uff0c\u5199\u4f5c\u671f\u95f4\u672a\u6562\u4e0d\u5168\u529b\u4ee5\u8d74\u3002\u5c3d\u7ba1\u5982\u6b64\uff0c\u53ea\u6015\u672c\u4eba\u80fd\u529b\u6240\u9650\uff0c\u5230\u5934\u6765\u6216\u96be\u514d\u771f\u77e5\u707c\u89c1\u9c9c\u6709\uff0c\u9519\u8bef\u7eb0\u9732\u4e0d\u65ad\uff0c\u6240\u4ee5\u8bf7\u5404\u4f4d\u8bfb\u8005\u53ef\u7528\u6279\u5224\u4e4b\u773c\u5149\u770b\u5f85\u6b64\u6587\u3002\u6b22\u8fce\u5404\u4f4d\u8bfb\u8005\u63d0\u4f9b\u5b9d\u8d35\u610f\u89c1\u3002<\/p>\n

\u65b0\u8bfe\u9898\u4ecd\u5728\u5b8c\u5584\u4e2d\u2026\u2026
\n\u201cMe importa m\u00e1s c\u00f3mo ser una persona amable que ser una persona excelente.\u201d
\n(\u6bd4\u8d77\u505a\u4e00\u4e2a\u4f18\u79c0\u7684\u4eba\uff0c\u6211\u66f4\u5728\u4e4e\u505a\u4e00\u4e2a\u5584\u826f\u7684\u4eba\u3002\u2014\u6885\u897f)<\/p>\n

2023\u5e748\u670815\u65e5\u6df1\u591c \u4e8e\u5bb6\u4e2d
\n\u8c6a<\/p>\n","protected":false},"excerpt":{"rendered":"

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