{"id":26376,"date":"2021-08-19T00:00:45","date_gmt":"2021-08-19T00:00:45","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=26376"},"modified":"2024-09-16T12:42:05","modified_gmt":"2024-09-16T12:42:05","slug":"%e8%bf%90%e5%8a%a8%e5%ad%a6","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/zh\/%e8%bf%90%e5%8a%a8%e5%ad%a6\/","title":{"rendered":"\u8fd0\u52a8\u5b66\u57fa\u7840\uff1a\u4f4d\u7f6e\u3001\u901f\u5ea6\u4e0e\u52a0\u901f\u5ea6"},"content":{"rendered":"<p><center><\/p>\n<h1>\u8fd0\u52a8\u5b66\u57fa\u7840\uff1a\u4f4d\u7f6e\u3001\u901f\u5ea6\u548c\u52a0\u901f\u5ea6<\/h1>\n<p><em><strong>\u6458\u8981\uff1a<\/strong><br \/>\n\u5728\u8fd9\u8282\u8bfe\u4e2d\uff0c\u6211\u4eec\u5c06\u590d\u4e60\u8fd0\u52a8\u5b66\u7684\u57fa\u672c\u6982\u5ff5\uff1a\u4f4d\u7f6e\u3001\u901f\u5ea6\u548c\u52a0\u901f\u5ea6\u3002\u6211\u4eec\u5c06\u63a2\u8ba8\u5982\u4f55\u8868\u793a\u4f4d\u7f6e\u968f\u65f6\u95f4\u7684\u53d8\u5316\uff0c\u533a\u5206\u77ac\u65f6\u901f\u5ea6\u548c\u5e73\u5747\u901f\u5ea6\uff0c\u4ee5\u53ca\u77ac\u65f6\u52a0\u901f\u5ea6\u548c\u5e73\u5747\u52a0\u901f\u5ea6\u3002\u6b64\u5916\uff0c\u6211\u4eec\u5c06\u63a8\u5bfc\u51fa\u5177\u6709\u6052\u5b9a\u52a0\u901f\u5ea6\u7684\u8fd0\u52a8\u65b9\u7a0b\uff0c\u8fd9\u5bf9\u4e8e\u9884\u6d4b\u7269\u4f53\u7684\u4f4d\u7f6e\u548c\u901f\u5ea6\u81f3\u5173\u91cd\u8981\u3002<\/em><\/p>\n<p><\/center><\/p>\n<p style=\"text-align:center;\"><strong>\u5b66\u4e60\u76ee\u6807\uff1a<\/strong><br \/>\n\u5728\u8fd9\u8282\u8bfe\u7ed3\u675f\u65f6\uff0c\u5b66\u751f\u5c06\u80fd\u591f\uff1a<\/p>\n<ol>\n<li><strong>\u56de\u5fc6<\/strong>\u8fd0\u52a8\u5b66\u4e2d\u4f4d\u7f6e\u3001\u901f\u5ea6\u548c\u52a0\u901f\u5ea6\u7684\u57fa\u672c\u5b9a\u4e49\u3002<\/li>\n<li><strong>\u5206\u6790<\/strong>\u52a0\u901f\u5ea6\u3001\u901f\u5ea6\u548c\u4f4d\u7f6e\u4e4b\u95f4\u7684\u5173\u7cfb\u3002<\/li>\n<li><strong>\u5e94\u7528<\/strong>\u5bfc\u6570\u548c\u79ef\u5206\u4ece\u52a0\u901f\u5ea6\u8ba1\u7b97\u901f\u5ea6\u548c\u4f4d\u7f6e\uff0c\u53cd\u4e4b\u4ea6\u7136\u3002<\/li>\n<li><strong>\u7406\u89e3<\/strong>\u77ac\u65f6\u901f\u5ea6\u548c\u5e73\u5747\u901f\u5ea6\uff0c\u4ee5\u53ca\u77ac\u65f6\u52a0\u901f\u5ea6\u548c\u5e73\u5747\u52a0\u901f\u5ea6\u4e4b\u95f4\u7684\u533a\u522b\u3002<\/li>\n<\/ol>\n<p><center><\/p>\n<p><strong>\u5185\u5bb9\u7d22\u5f15<\/strong><br \/>\n<a href=\"#1\">\u4ecb\u7ecd<\/a><br \/>\n<a href=\"#2\">\u4f4d\u7f6e\u3001\u7a7a\u95f4\u548c\u89c2\u5bdf\u8005<\/a><br \/>\n<a href=\"#3\">\u901f\u5ea6\u548c\u901f\u7387<\/a><br \/>\n<a href=\"#4\">\u52a0\u901f\u5ea6<\/a><br \/>\n<a href=\"#5\">\u8fd0\u52a8\u65b9\u7a0b<\/a><br \/>\n<a href=\"#6\">\u7ed3\u8bba<\/a>\n<\/p>\n<p><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/fjv-qEuPhwU\" title=\"YouTube \u89c6\u9891\u64ad\u653e\u5668\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>\u4ecb\u7ecd<\/h2>\n<p style=\"text-align: justify;\">\u901a\u8fc7\u79ef\u5206\u52a0\u901f\u5ea6\u53ef\u4ee5\u8ba1\u7b97\u901f\u5ea6\u548c\u4f4d\u7f6e\uff0c\u800c\u901a\u8fc7\u6c42\u5bfc\u4f4d\u7f6e\u53ef\u4ee5\u5f97\u5230\u901f\u5ea6\u548c\u52a0\u901f\u5ea6\u3002\u8fd9\u4e9b\u8bdd\u6982\u62ec\u4e86\u6211\u4eec\u5c06\u63a2\u8ba8\u7684\u8fd0\u52a8\u5b66\u5185\u5bb9\uff0c\u6211\u4eec\u7684\u4e3b\u8981\u76ee\u6807\u4e4b\u4e00\u662f\u7406\u89e3\u8fd9\u4e9b\u672f\u8bed\u7684\u542b\u4e49\u3002\u8fd0\u52a8\u4ee3\u8868\u4e86\u4e00\u79cd\u53d8\u5316\uff0c\u81ea\u7136\u754c\u7684\u4e00\u5207\u90fd\u5904\u4e8e\u53d8\u5316\u4e4b\u4e2d\u3002\u56e0\u6b64\uff0c\u7814\u7a76\u53d8\u5316\u53ca\u5176\u53d8\u5316\u5f62\u5f0f\u662f\u7269\u7406\u5b66\u7684\u57fa\u672c\u652f\u67f1\u4e4b\u4e00\u3002<\/p>\n<p style=\"text-align: justify;\">\u6709\u5f88\u591a\u53d8\u91cf\u662f\u5bb9\u6613\u53d8\u5316\u7684\uff1a\u65b0\u4e8b\u7269\u53d8\u5f97\u9648\u65e7\uff0c\u4eba\u4eec\u53ef\u4ee5\u4ece\u4e00\u79cd\u804c\u4e1a\u8f6c\u6362\u5230\u53e6\u4e00\u79cd\u804c\u4e1a\uff0c\u4ece\u5065\u5eb7\u5230\u751f\u75c5\uff0c\u53cd\u4e4b\u4ea6\u7136\uff0c\u4ee5\u53ca\u4ece\u767d\u5929\u5230\u9ed1\u591c\uff0c\u7b49\u7b49\u3002\u8fd9\u4e9b\u90fd\u662f\u53d8\u5316\u7684\u4f8b\u5b50\uff0c\u4f46\u5728\u7814\u7a76\u8fd0\u52a8\u5b66\u65f6\uff0c\u6211\u4eec\u5c06\u4e13\u6ce8\u4e8e\u5176\u4e2d\u4e00\u79cd\uff1a\u4f4d\u7f6e\u6216\u8fd0\u52a8\u7684\u53d8\u5316\u3002\u5728\u5bf9\u8fd0\u52a8\u7684\u7814\u7a76\u4e2d\uff0c\u6211\u4eec\u53ef\u4ee5\u770b\u5230\u4e24\u79cd\u4e92\u8865\u7684\u65b9\u6cd5\uff1a\u4e00\u79cd\u662f\u57fa\u4e8e\u4ea7\u751f\u8fd0\u52a8\u7684\u539f\u56e0\uff0c\u53e6\u4e00\u79cd\u662f\u57fa\u4e8e\u8fd0\u52a8\u7684\u65b9\u5f0f\uff0c\u5206\u522b\u5f62\u6210\u4e86\u52a8\u529b\u5b66\u548c\u8fd0\u52a8\u5b66\uff0c\u8fd9\u4e24\u8005\u5171\u540c\u6784\u6210\u4e86\u529b\u5b66\u3002<\/p>\n<p style=\"text-align: justify;\">\u5728\u7269\u7406\u5b66\u4e2d\uff0c\u6211\u4eec\u5c06\u7269\u7406\u7a7a\u95f4\u5efa\u6a21\u4e3a\u4e00\u4e2a\u5411\u91cf\u7a7a\u95f4\uff0c\u4ee5\u65b9\u4fbf\u8868\u793a\u8bf8\u5982\u4f4d\u7f6e\u3001\u901f\u5ea6\u548c\u52a0\u901f\u5ea6\u7b49\u6982\u5ff5\u3002\u901a\u5e38\uff0c\u6211\u4eec\u4f7f\u7528\u4e09\u7ef4\u7a7a\u95f4 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3<\/span><\/span> \u6765\u5b9e\u73b0\u8fd9\u4e2a\u76ee\u7684\uff0c\u5c3d\u7ba1\u7406\u8bba\u4e0a\u4efb\u4f55\u7ef4\u5ea6\u7684\u7a7a\u95f4\u90fd\u53ef\u4ee5\u6839\u636e\u4e0a\u4e0b\u6587\u6765\u9002\u7528\u3002<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>\u4f4d\u7f6e\u3001\u7a7a\u95f4\u548c\u89c2\u5bdf\u8005<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=fjv-qEuPhwU&amp;t=257s\" target=\"_blank\" rel=\"noopener\"><strong>\u8ba9\u6211\u4eec\u8003\u8651\u4e00\u4e2a\u51fd\u6570<\/strong><\/a><\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rll}\n\n\\vec{r}:\\mathbb{R}[T]&amp;\\longrightarrow&amp;\\mathbb{R}^n[L] \\\\\n\nt &amp;\\longmapsto&amp;\\vec{r}(t)\n\n\\end{array}\n\n<\/span>\n<p style=\"text-align: justify;\">\u8fd9\u662f\u4e00\u4e2a\u5c06\u4f4d\u7f6e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{r}(t)<\/span><\/span> \u5206\u914d\u7ed9\u6bcf\u4e2a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">t\\in\\mathbb{R}<\/span><\/span> \u7684\u51fd\u6570\uff0c\u56e0\u6b64\u6211\u4eec\u79f0\u4e4b\u4e3a\u4f4d\u7f6e\u51fd\u6570\uff08\u6216\u7b80\u5355\u5730\u79f0\u4e3a\u4f4d\u7f6e\uff09\u3002\u81ea\u53d8\u91cf <span class=\"katex-eq\" data-katex-display=\"false\">t<\/span> \u88ab\u79f0\u4e3a\u201c\u65f6\u95f4\u201d\uff0c\u53c2\u6570 <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> \u5bf9\u5e94\u4e8e\u7a7a\u95f4\u7684\u201c\u7ef4\u5ea6\u201d\u3002\u7b26\u53f7 <span class=\"katex-eq\" data-katex-display=\"false\">[T]<\/span> \u548c <span class=\"katex-eq\" data-katex-display=\"false\">[L]<\/span> \u5206\u522b\u6307\u65f6\u95f4\u548c\u957f\u5ea6\u7684\u7269\u7406\u91cf\uff0c\u901a\u5e38\u4ee5\u201c\u79d2\u201d\u548c\u201c\u7c73\u201d\u4e3a\u5355\u4f4d\u3002<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-gCSXLVh_JEI\/YQhm3e26l2I\/AAAAAAAAFXs\/fBu5WvV1PiIhGDkTUZfSAR4EMQqUFneoACLcBGAsYHQ\/s0\/posici%25C3%25B3n%2Ben%2Bel%2Bespacio%2B3D.PNG\" width=\"400\" height=\"300\" alt=\"\u57fa\u672c\u8fd0\u52a8\u5b66\uff1a\u4e09\u7ef4\u7a7a\u95f4\u4e2d\u76f8\u5bf9\u4e8e\u539f\u70b9\u7684\u4f4d\u7f6e\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-gCSXLVh_JEI\/YQhm3e26l2I\/AAAAAAAAFXs\/fBu5WvV1PiIhGDkTUZfSAR4EMQqUFneoACLcBGAsYHQ\/s0\/posici%25C3%25B3n%2Ben%2Bel%2Bespacio%2B3D.PNG\" width=\"400\" height=\"300\" alt=\"\u57fa\u672c\u8fd0\u52a8\u5b66\uff1a\u4e09\u7ef4\u7a7a\u95f4\u4e2d\u76f8\u5bf9\u4e8e\u539f\u70b9\u7684\u4f4d\u7f6e\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/p>\n<p style=\"text-align: justify;\">\u4f4d\u7f6e\uff0c\u50cf\u4efb\u4f55\u7269\u7406\u91cf\u4e00\u6837\uff0c\u662f\u7531\u89c2\u5bdf\u8005\u6d4b\u91cf\u7684\u3002\u5728\u6211\u7ed9\u51fa\u7684\u5173\u4e8e\u4f4d\u7f6e\u7684\u63cf\u8ff0\u4e2d\uff0c\u6211\u9690\u542b\u5730\u5047\u8bbe\u89c2\u5bdf\u8005\u5177\u6709\u96f6\u5411\u91cf\u7684\u5750\u6807\uff0c <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{0}<\/span>\u3002\u5982\u679c\u89c2\u5bdf\u8005 <span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{O}<\/span> \u62e5\u6709\u5411\u91cf <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{r}<\/span> \u7684\u5750\u6807\uff0c\u800c\u70b9\u7269\u4f53\u62e5\u6709\u5750\u6807 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{r}^\\prime,<\/span><\/span> \u90a3\u4e48 <strong>\u76f8\u5bf9\u4e8e\u89c2\u5bdf\u8005\u7684\u4f4d\u7f6e<\/strong> \u5c06\u662f\uff1a<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{r}_\\mathcal{O} = \\vec{r} - \\vec{r}^\\prime<\/span><\/span><\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-NV6VA_yStVo\/YQhpIOk5yCI\/AAAAAAAAFX0\/YW9xYYXY9h0DTnLfkNA0H7deQPkFqsbogCLcBGAsYHQ\/s0\/posicion%2Bdesde%2Bel%2Bobservador.PNG\" width=\"500\" height=\"300\" alt=\"\u57fa\u672c\u8fd0\u52a8\u5b66\uff1a\u76f8\u5bf9\u4e8e\u4efb\u610f\u89c2\u5bdf\u8005\u7684\u4f4d\u7f6e\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-NV6VA_yStVo\/YQhpIOk5yCI\/AAAAAAAAFX0\/YW9xYYXY9h0DTnLfkNA0H7deQPkFqsbogCLcBGAsYHQ\/s0\/posicion%2Bdesde%2Bel%2Bobservador.PNG\" width=\"500\" height=\"300\" alt=\"\u57fa\u672c\u8fd0\u52a8\u5b66\uff1a\u76f8\u5bf9\u4e8e\u4efb\u610f\u89c2\u5bdf\u8005\u7684\u4f4d\u7f6e\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/p>\n<p style=\"text-align: justify;\">\u7a7a\u95f4\u662f\u6240\u6709\u53ef\u80fd\u4f4d\u7f6e\u7684\u96c6\u5408\uff0c\u4e5f\u662f\u76f8\u5bf9\u4e8e\u4efb\u4f55\u89c2\u5bdf\u8005\u7684\u6240\u6709\u53ef\u80fd\u4f4d\u7f6e\u7684\u96c6\u5408\u3002\u4f4d\u7f6e\u662f\u65f6\u95f4\u7684\u51fd\u6570\uff0c\u7528\u4e8e\u6570\u5b66\u4e0a\u8868\u793a\u4e00\u4e2a\u7406\u60f3\u5316\u7269\u4f53\u79f0\u4e3a\u201c\u70b9\u7269\u4f53\u201d\u5728\u6bcf\u4e00\u65f6\u523b <span class=\"katex-eq\" data-katex-display=\"false\">t<\/span> \u6240\u5728\u7684\u5730\u65b9\u3002\u70b9\u7269\u4f53\u662f\u4e00\u79cd\u7406\u60f3\u5316\u7684\u6982\u5ff5\uff0c\u5b83\u662f\u5f53\u6211\u4eec\u5265\u593a\u4e00\u4e2a\u771f\u5b9e\u7269\u4f53\u7684\u6240\u6709\u7279\u6027\uff0c\u5305\u62ec\u5927\u5c0f\u548c\u5f62\u72b6\uff0c\u53ea\u4fdd\u7559\u5b83\u201c\u5360\u636e\u7a7a\u95f4\u7684\u4f4d\u7f6e\u201d\u65f6\u5269\u4e0b\u7684\u4e1c\u897f\u3002<\/p>\n<p style=\"text-align: justify;\">\u4f4d\u7f6e\u901a\u5e38\u662f\u4e00\u4e2a\u5411\u91cf\u3002\u5411\u91cf\u7531\u4e24\u4e2a\u5143\u7d20\u7ec4\u6210\uff1a\u5927\u5c0f\u548c\u65b9\u5411\u3002\u76f8\u5bf9\u4e8e\u89c2\u5bdf\u8005\u7684\u4f4d\u7f6e\u7684\u5927\u5c0f\u662f\u4e0e\u89c2\u5bdf\u8005\u7684\u8ddd\u79bb\uff0c\u7531\u516c\u5f0f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">dist_\\mathcal{O}(t)=\\|\\vec{r}_\\mathcal{O}(t)\\|.<\/span><\/span> \u8868\u793a\u3002<\/p>\n<p style=\"text-align: justify;\">\u4ece\u8fd9\u91cc\u5f00\u59cb\uff0c\u5f3a\u70c8\u63a8\u8350\u638c\u63e1\u5fae\u79ef\u5206\u8bfe\u7a0b\u4e2d\u7684\u5185\u5bb9\uff0c<strong>\u5fae\u5206<a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&#038;list=PL_C8rbeFjqAVaR_sgLJRBvMm5t6E1GxGI\" rel=\"noopener\" target=\"_blank\">\u5fae\u5206<\/a> \u548c <a href=\"https:\/\/www.youtube.com\/watch?v=4wSTxA7zY9k&#038;list=PL_C8rbeFjqAUU5ClOkZMRJbiefHDeXGI2\" rel=\"noopener\" target=\"_blank\">\u79ef\u5206\u3002<\/a><\/strong><\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>\u901f\u5ea6\u548c\u901f\u7387<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=fjv-qEuPhwU&amp;t=1232s\" target=\"_blank\" rel=\"noopener\"><strong>\u5982\u679c\u4f4d\u7f6e\u5bf9\u65f6\u95f4\u53ef\u5fae\u5206\uff0c<\/strong><\/a> \u90a3\u4e48\u53ef\u4ee5\u5b9a\u4e49\u76f8\u5bf9\u4e8e\u89c2\u5bdf\u8005 <span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{O},<\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{v}_\\mathcal{O}(t),<\/span><\/span> \u7684\u901f\u5ea6\u5982\u4e0b\uff1a<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{v}_\\mathcal{O}(t) =\\displaystyle \\lim_{\\Delta t \\to 0}\\frac{\\vec{r}_\\mathcal{O}(t+\\Delta t) - \\vec{r}_\\mathcal{O}(t)}{\\Delta t} = \\frac{d\\vec{r}_\\mathcal{O}(t)}{dt} <\/span><\/span><\/p>\n<p style=\"text-align: justify;\">\u7b80\u5355\u6765\u8bf4\uff0c\u901f\u5ea6\u662f\u4f4d\u7f6e\u7684\u65f6\u95f4\u5bfc\u6570\uff0c\u544a\u8bc9\u6211\u4eec\u5728\u6bcf\u4e2a\u65f6\u523b <span class=\"katex-eq\" data-katex-display=\"false\">t<\/span> \u4f4d\u7f6e\u5982\u4f55\u53d8\u5316\u3002<\/p>\n<p style=\"text-align: justify;\">\u901f\u5ea6\u6709\u4e24\u79cd\u7c7b\u578b\uff1a\u77ac\u65f6\u901f\u5ea6\u548c\u5e73\u5747\u901f\u5ea6\u3002\u77ac\u65f6\u901f\u5ea6\u662f\u521a\u624d\u5448\u73b0\u7684\uff0c\u5e73\u5747\u901f\u5ea6\u5219\u901a\u8fc7\u7701\u7565\u6781\u9650\u8ba1\u7b97\u5f97\u5230\u3002\u5728\u65f6\u95f4\u95f4\u9694\u957f\u5ea6 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Delta t,<\/span><\/span> \u4e0a\u7684\u5e73\u5747\u901f\u5ea6 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left&lt; \\vec{v}_{\\mathcal{O},\\Delta t}(\\overline{t}) \\right&gt;,<\/span><\/span> \u5b9a\u4e49\u4e3a<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left&lt; \\vec{v}_{\\mathcal{O},\\Delta t}(\\overline{t}) \\right&gt; = \\displaystyle \\frac{\\vec{r}_\\mathcal{O}(t+\\Delta t) - \\vec{r}_\\mathcal{O}(t)}{\\Delta t}<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">\u5176\u4e2d <span class=\"katex-eq\" data-katex-display=\"false\">\\overline{t}<\/span> \u662f\u5305\u542b\u5728\u95f4\u9694 <span class=\"katex-eq\" data-katex-display=\"false\">[t,t+\\Delta t]<\/span> \u4e2d\u7684\u4efb\u4f55\u65f6\u523b\u3002<\/p>\n<p style=\"text-align: justify;\">\u4ece\u901f\u5ea6\uff08\u65e0\u8bba\u662f\u77ac\u65f6\u7684\u8fd8\u662f\u5e73\u5747\u7684\uff09\u4e2d\uff0c\u53ef\u4ee5\u5b9a\u4e49\u901f\u7387\u4f5c\u4e3a\u5176\u5bf9\u5e94\u7684\u5927\u5c0f\u3002\u76f8\u5bf9\u4e8e\u89c2\u5bdf\u8005 <span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{O}<\/span> \u7684\u901f\u7387\u662f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">v_\\mathcal{O}(t)=\\|\\vec{v}_\\mathcal{O}(t)\\|,<\/span><\/span> \u5e73\u5747\u901f\u7387 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left&lt; {v}_{\\mathcal{O},\\Delta t}(\\overline{t}) \\right&gt; = \\|\\left&lt; \\vec{v}_{\\mathcal{O},\\Delta t}(\\overline{t}) \\right&gt;\\|.<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">\u901f\u7387\u548c\u901f\u5ea6\u7684\u5355\u4f4d\u662f\u957f\u5ea6\u5355\u4f4d\u9664\u4ee5\u65f6\u95f4\u5355\u4f4d <span class=\"katex-eq\" data-katex-display=\"false\">[L\/T]<\/span>\uff0c\u901a\u5e38\u4ee5\u201c\u7c73\u6bcf\u79d2\u201d\u6765\u8868\u793a\u3002<\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2>\u52a0\u901f\u5ea6<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=fjv-qEuPhwU&amp;t=1522s\" target=\"_blank\" rel=\"noopener\"><strong>\u7c7b\u4f3c\u4e8e\u901f\u5ea6\uff0c<\/strong><\/a> \u5982\u679c\u5b83\u5bf9\u65f6\u95f4\u53ef\u5fae\u5206\uff0c\u90a3\u4e48\u53ef\u4ee5\u5b9a\u4e49\u76f8\u5bf9\u4e8e\u89c2\u5bdf\u8005 <span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{O}<\/span> \u7684\u52a0\u901f\u5ea6 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{a}_\\mathcal{O}(t),<\/span><\/span> \u5982\u4e0b\uff1a<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{a}_\\mathcal{O}(t)= \\displaystyle \\lim_{\\Delta t \\to 0}\\frac{\\vec{v}_\\mathcal{O}(t+\\Delta t) - \\vec{v}_\\mathcal{O}(t)}{\\Delta t} = \\frac{d\\vec{v}_\\mathcal{O}(t)}{dt}<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">\u52a0\u901f\u5ea6\u662f\u901f\u5ea6\u7684\u65f6\u95f4\u5bfc\u6570\uff0c\u56e0\u6b64\u544a\u8bc9\u6211\u4eec\u901f\u5ea6\u5982\u4f55\u968f\u65f6\u95f4\u53d8\u5316\u3002<\/p>\n<p style=\"text-align: justify;\">\u7c7b\u4f3c\u4e8e\u901f\u7387\uff0c\u6709\u77ac\u65f6\u52a0\u901f\u5ea6\u548c\u5e73\u5747\u52a0\u901f\u5ea6\u3002\u77ac\u65f6\u52a0\u901f\u5ea6\u662f\u6211\u4eec\u521a\u521a\u56de\u987e\u7684\uff0c\u5e73\u5747\u52a0\u901f\u5ea6\u662f\u901a\u8fc7\u7701\u7565\u6781\u9650\u8ba1\u7b97\u5f97\u5230\u7684\u3002\u5728\u65f6\u95f4\u95f4\u9694\u957f\u5ea6 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Delta t,<\/span><\/span> \u4e0a\u7684\u5e73\u5747\u52a0\u901f\u5ea6 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left&lt;\\vec{a}_{\\mathcal{O},\\Delta t}\\right&gt;,<\/span><\/span> \u5b9a\u4e49\u4e3a<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left&lt; \\vec{a}_{\\mathcal{O},\\Delta t}(\\overline{t}) \\right&gt; = \\displaystyle \\frac{\\vec{v}_\\mathcal{O}(t+\\Delta t) - \\vec{v}_\\mathcal{O}(t)}{\\Delta t}<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">\u52a0\u901f\u5ea6\u7684\u5355\u4f4d\u662f\u957f\u5ea6\u5355\u4f4d\u9664\u4ee5\u65f6\u95f4\u7684\u5e73\u65b9 <span class=\"katex-eq\" data-katex-display=\"false\">[L\/T^2]<\/span>\uff0c\u901a\u5e38\u4ee5\u201c\u7c73\u6bcf\u79d2\u5e73\u65b9\u201d\u6765\u8868\u793a\u3002<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>\u8fd0\u52a8\u65b9\u7a0b<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=fjv-qEuPhwU&amp;t=1588s\" target=\"_blank\" rel=\"noopener\"><strong>\u5047\u8bbe\u6211\u4eec\u6709\u4e00\u4e2a\u70b9\u7269\u4f53<\/strong><\/a> \u5b83\u76f8\u5bf9\u4e8e\u89c2\u5bdf\u8005 <span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{O}<\/span> \u4ee5\u6052\u5b9a\u52a0\u901f\u5ea6 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{a}_\\mathcal{O}(t) = \\vec{a}_0<\/span><\/span> \u8fd0\u52a8\u3002\u5982\u679c\u53ef\u4ee5\u4ece\u4f4d\u7f6e\u5bfc\u51fa\u901f\u5ea6\u548c\u52a0\u901f\u5ea6\uff0c\u90a3\u4e48\u901a\u8fc7\u79ef\u5206\u52a0\u901f\u5ea6\u53ef\u4ee5\u8ba1\u7b97\u901f\u5ea6\u548c\u4f4d\u7f6e\u3002\u8fd9\u79cd\u65b9\u5f0f\u5f97\u5230\u7684\u7ed3\u679c\u79f0\u4e3a\u8fd0\u52a8\u65b9\u7a0b\u3002<\/p>\n<p style=\"text-align: justify;\">\u5bf9 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{a}_\\mathcal{O}(t) = \\vec{a}_0<\/span><\/span> \u79ef\u5206\u6211\u4eec\u5f97\u5230\uff1a<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{v}_\\mathcal{O}(t) = \\displaystyle \\int \\vec{a}_\\mathcal{O}(t) dt = \\int \\vec{a}_0 dt = \\vec{a}_0 t + \\vec{v}_0 <\/span><\/span><\/p>\n<p style=\"text-align: justify;\">\u518d\u79ef\u5206\u4e00\u6b21\u6211\u4eec\u5f97\u5230<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{r}_\\mathcal{O}(t) = \\displaystyle \\int \\vec{v}_\\mathcal{O}(t) dt = \\int \\vec{a}_0t + \\vec{v}_0 dt = \\displaystyle \\frac{1}{2}\\vec{a}_0 t^2 + \\vec{v}_0t+\\vec{r}_0 <\/span><\/span><\/p>\n<p style=\"text-align: justify;\">\u8fd9\u91cc\uff0c\u5e38\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{v}_0<\/span><\/span> \u548c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{r}_0<\/span><\/span> \u662f\u79ef\u5206\u5e38\u6570\uff0c\u4ee3\u8868\u76f8\u5bf9\u4e8e\u89c2\u5bdf\u8005 <span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{O}<\/span> \u7684\u70b9\u7269\u4f53\u7684\u521d\u901f\u5ea6\u548c\u521d\u4f4d\u7f6e\u3002\u603b\u7ed3\u4e00\u4e0b\uff0c\u8fd0\u52a8\u65b9\u7a0b\u662f\uff1a<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\n\\vec{a}_\\mathcal{O}(t) =&amp; \\vec{a}_0 \\\\\n\n\\vec{v}_\\mathcal{O}(t) =&amp;  \\vec{a}_0t+\\vec{v}_0 \\\\\n\n\\vec{r}_\\mathcal{O}(t) =&amp; \\displaystyle \\frac{1}{2}\\vec{a}_0t^2 + \\vec{v}_0t + \\vec{r}_0\n\n\\end{array}\n\n<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">\u901a\u8fc7\u8fd9\u4e9b\u65b9\u7a0b\uff0c\u53ef\u4ee5\u5b8c\u6574\u63cf\u8ff0\u4efb\u4f55\u4ee5\u6052\u5b9a\u52a0\u901f\u5ea6\u8fd0\u52a8\u7684\u70b9\u7269\u4f53\u7684\u8fd0\u52a8\u3002\u8fd9\u5c31\u5efa\u7acb\u4e86\u901a\u8fc7\u79ef\u5206\u52a0\u901f\u5ea6\u53ef\u4ee5\u8ba1\u7b97\u901f\u5ea6\u548c\u4f4d\u7f6e\uff0c\u800c\u901a\u8fc7\u5bfc\u6570\u4f4d\u7f6e\u53ef\u4ee5\u8ba1\u7b97\u901f\u5ea6\u548c\u52a0\u901f\u5ea6\u7684\u6982\u5ff5\u3002<\/p>\n<p style=\"text-align: justify;\">\u8bf7\u6ce8\u610f\uff0c\u8fd9\u4e9b\u662f\u77e2\u91cf\u65b9\u7a0b\uff0c\u56e0\u6b64\u53ef\u4ee5\u5206\u89e3\u4e3a\u5b83\u4eec\u7684\u5206\u91cf\u3002\u5982\u679c\u6211\u4eec\u5728\u4e09\u7ef4\u7a7a\u95f4\u4e2d\u5bf9\u8fd0\u52a8\u8fdb\u884c\u5efa\u6a21\uff0c\u90a3\u4e48\u6211\u4eec\u5c06\u5f97\u5230\u4ee5\u4e0b\u65b9\u5f0f\u7684\u6bcf\u4e2a\u5206\u91cf\uff1a<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\n\\vec{a}_\\mathcal{O}(t) &amp;= (a_x(t), a_y(t), a_z(t))\\\\\n\n\\vec{v}_\\mathcal{O}(t) &amp;= (v_x(t), v_y(t), v_z(t))\\\\\n\n\\vec{r}_\\mathcal{O}(t) &amp;= (x(t), y(t), z(t))\\\\\n\n\\vec{a}_0 &amp;= (a_{0x}, a_{0y}, a_{0z})\\\\\n\n\\vec{v}_0 &amp;= (v_{0x}, v_{0y}, v_{0z})\\\\\n\n\\vec{r}_0 &amp;= (x_{0}, y_{0}, z_{0})\\\\\n\n\\end{array}\n\n<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">\u8fd9\u6837\uff0c\u5c31\u521b\u5efa\u4e86\u4e00\u7ec49\u4e2a\u65b9\u7a0b\uff0c\u6bcf\u4e2a\u5750\u6807\u8f74\u4e00\u4e2a\u3002\u4f8b\u5982\uff0c\u5bf9\u4e8e <span class=\"katex-eq\" data-katex-display=\"false\">\\hat{x}<\/span> \u8f74\uff0c\u6211\u4eec\u5c06\u6709<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\na_x(t) &amp; = a_{0x}\\\\\n\nv_x(t) &amp; = a_{0x}t + v_{0x} \\\\\n\nx(t) &amp; = \\displaystyle \\frac{1}{2}a_{0x}t^2 + v_{0x}t + x_0\n\n\\end{array}\n\n<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">\u901a\u5e38\uff0c\u5750\u6807 <span class=\"katex-eq\" data-katex-display=\"false\">\\hat{z}<\/span> \u7528\u4e8e\u8868\u793a\u9ad8\u5ea6\uff0c\u56e0\u6b64\u5047\u8bbe <span class=\"katex-eq\" data-katex-display=\"false\">a_{0z}=-g \\approx -9.81[m\/s^2];<\/span> \u5373\u5728\u6b64\u8f74\u4e0a\u7684\u52a0\u901f\u5ea6\u4e0e\u5730\u7403\u91cd\u529b\u4ea7\u751f\u7684\u52a0\u901f\u5ea6\u6709\u5173\u3002\u8fd9\u88ab\u5305\u62ec\u5728\u65b9\u7a0b\u4e2d\u4ee5\u6a21\u62df\u81ea\u7531\u843d\u4f53\u6216\u6295\u5c04\u7269\u7684\u53d1\u5c04\u7b49\u73b0\u8c61\u3002<\/p>\n<p><a name=\"6\"><\/a><\/p>\n<h2>\u7ed3\u8bba<\/h2>\n<p style=\"text-align: justify;\">\n    \u5728\u5bf9\u8fd0\u52a8\u5b66\u57fa\u7840\u7684\u8fd9\u4e2a\u65c5\u7a0b\u4e2d\uff0c\u6211\u4eec\u63a2\u7d22\u4e86\u5982\u4f55\u4f7f\u7528\u6570\u5b66\u6765\u63cf\u8ff0\u548c\u7406\u89e3\u7269\u7406\u7a7a\u95f4\u4e2d\u7684\u8fd0\u52a8\u3002\u4ece\u5728\u4efb\u610f\u7ef4\u5ea6\u7684\u7a7a\u95f4\u4e2d\u8868\u793a\u4f4d\u7f6e\u4e3a\u4e00\u4e2a\u5411\u91cf\u5230\u5bf9\u77e2\u91cf\u51fd\u6570\u8fdb\u884c\u5bfc\u6570\u548c\u79ef\u5206\u4ee5\u83b7\u5f97\u901f\u5ea6\u3001\u52a0\u901f\u5ea6\u548c\u8fd0\u52a8\u65b9\u7a0b\uff0c\u6211\u4eec\u770b\u5230\u4e86\u8fd9\u4e9b\u6982\u5ff5\u5982\u4f55\u4ea4\u7ec7\u5728\u4e00\u8d77\u5e76\u5e94\u7528\u4e8e\u8fd0\u52a8\u7684\u5206\u6790\u3002\n<\/p>\n<p style=\"text-align: justify;\">\n    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