{"id":9005,"date":"2024-06-27T14:13:11","date_gmt":"2024-06-27T05:13:11","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=9005"},"modified":"2025-05-31T16:19:50","modified_gmt":"2025-05-31T07:19:50","slug":"%e6%96%9c%e6%96%b9%e6%8a%95%e5%b0%84%e3%82%92%e3%83%a9%e3%82%b0%e3%83%a9%e3%83%b3%e3%82%b8%e3%83%a5%e3%81%ae%e6%9c%aa%e5%ae%9a%e4%b9%97%e6%95%b0%e6%b3%95%e3%81%a7","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e7%90%86%e5%b7%a5%e7%b3%bb%e3%81%ae%e6%95%b0%e5%ad%a6c\/%e5%81%8f%e5%be%ae%e5%88%86%ef%bc%9a%e5%a4%9a%e5%a4%89%e6%95%b0%e9%96%a2%e6%95%b0%e3%81%ae%e5%be%ae%e5%88%86\/%e3%83%a9%e3%82%b0%e3%83%a9%e3%83%b3%e3%82%b8%e3%83%a5%e3%81%ae%e6%9c%aa%e5%ae%9a%e4%b9%97%e6%95%b0%e6%b3%95%ef%bc%882%e6%ac%a1%e5%85%83%ef%bc%89\/%e6%96%9c%e6%96%b9%e6%8a%95%e5%b0%84%e3%82%92%e3%83%a9%e3%82%b0%e3%83%a9%e3%83%b3%e3%82%b8%e3%83%a5%e3%81%ae%e6%9c%aa%e5%ae%9a%e4%b9%97%e6%95%b0%e6%b3%95%e3%81%a7\/","title":{"rendered":"\u659c\u65b9\u6295\u5c04\u3092\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u306e\u672a\u5b9a\u4e57\u6570\u6cd5\u3067"},"content":{"rendered":"

\u659c\u65b9\u6295\u5c04\u306e\u6700\u5927\u6c34\u5e73\u5230\u9054\u8ddd\u96e2\u3092\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u306e\u672a\u5b9a\u4e57\u6570\u6cd5\u3092\u4f7f\u3063\u3066\u6c42\u3081\u308b\u7df4\u7fd2\u554f\u984c\u3002<\/p>\n

\u5730\u4e0a\u9ad8\u30bc\u30ed\u304b\u3089\u306e\u659c\u65b9\u6295\u5c04\u3092\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u306e\u672a\u5b9a\u4e57\u6570\u6cd5\u3092\u4f7f\u3063\u3066<\/h3>\n

\u5fc5\u8981\u306a\u5f0f\u306f\u3059\u3067\u306b\u300c\u659c\u65b9\u6295\u5c04\u3092\u9670\u95a2\u6570\u5b9a\u7406\u3067<\/a>\u300d\u306e\u30da\u30fc\u30b8\u306b\u307e\u3068\u3081\u3066\u3042\u308b\u3002\u8981\u306f\uff0c<\/p>\n

$$f(\\tau, \\theta) \\equiv y = v_0\\, \\sin\\theta\\cdot \\tau -\\frac{1}{2} g\\,\\tau^2 = 0$$<\/p>\n

\u306e\u62d8\u675f\u6761\u4ef6\u306e\u3082\u3068\u3067\uff0c\u6c34\u5e73\u5230\u9054\u8ddd\u96e2<\/p>\n

$$g(\\tau, \\theta) \\equiv \\ell = v_0\\, \\cos\\theta \\cdot \\tau$$<\/p>\n

\u3092\u6700\u5927\u306b\u3059\u308b $\\theta$\u00a0 \u306f\uff1f\u3068\u3044\u3046\u554f\u984c\u3002<\/p>\n

\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u306e\u672a\u5b9a\u4e57\u6570\u6cd5<\/h4>\n

\\begin{eqnarray}
\nF(\\tau, \\theta) &\\equiv& g(\\tau, \\theta) + \\lambda\\, f(\\tau, \\theta) \\\\
\n&=& v_0\\, \\cos\\theta \\cdot \\tau + \\lambda\\left( v_0\\, \\sin\\theta\\cdot \\tau -\\frac{1}{2} g\\,\\tau^2\\right) \\\\ \\ \\\\
\n\\frac{\\partial F}{\\partial \\tau} &=& v_0\\,\\cos\\theta + \\lambda\\,\\left(v_0\\,\\sin\\theta -g\\,\\tau \\right) = 0 \\tag{1} \\\\
\n\\frac{\\partial F}{\\partial \\theta} &=&\u00a0 -v_0\\,\\sin\\theta\\cdot\\tau + \\lambda\\, v_0\\,\\cos\\theta\\cdot\\tau = 0 \\tag{2} \\\\
\n\\frac{\\partial F}{\\partial \\lambda} &=&v_0\\, \\sin\\theta\\cdot \\tau -\\frac{1}{2} g\\,\\tau^2 = 0 \\tag{3}
\n\\end{eqnarray}<\/p>\n

\u9023\u7acb\u65b9\u7a0b\u5f0f\u306e\u89e3<\/h4>\n

\u307e\u305a (2) \u5f0f\u3088\u308a<\/p>\n

$$\\lambda = \\frac{\\sin\\theta}{\\cos\\theta}$$<\/p>\n

\u3053\u308c\u3092 (1) \u5f0f\u306b\u4ee3\u5165\u3057\u3066<\/p>\n

\\begin{eqnarray}
\nv_0\\,\\cos\\theta + \\frac{\\sin\\theta}{\\cos\\theta}\\cdot\\left(v_0\\,\\sin\\theta -g\\,\\tau \\right) &=& 0 \\\\
\nv_0\\,\\cos^2\\theta + v_0\\,\\sin^2\\theta -g\\,\\tau\\,\\sin\\theta &=& 0 \\\\
\n\\therefore\\ \\ \\sin\\theta &=& \\frac{v_0}{g\\,\\tau}
\n\\end{eqnarray}<\/p>\n

\u3053\u308c\u3092 (3) \u5f0f\u306b\u4ee3\u5165\u3059\u308b\u3068\uff0c<\/p>\n

\\begin{eqnarray}
\nv_0\\, \\frac{v_0}{g\\,\\tau}\\cdot \\tau -\\frac{1}{2} g\\,\\tau^2 &=& 0 \\\\
\n\\therefore\\ \\ \\tau &=& \\frac{\\sqrt{2}\u00a0 v_0}{g} \\\\ \\ \\\\
\n\\therefore\\ \\ \\sin\\theta &=& \\frac{v_0}{g\\,\\tau} = \\frac{1}{\\sqrt{2}} \\\\
\n\\therefore\\ \\ \\theta &=& \\frac{\\pi}{4}
\n\\end{eqnarray}<\/p>\n

\u3068\u306a\u308a\uff0c\u6c34\u5e73\u5230\u9054\u8ddd\u96e2\u304c\u6700\u5927\u3068\u306a\u308b\u89d2\u5ea6\u306f $\\displaystyle \\frac{\\pi}{4}$ \u30e9\u30b8\u30a2\u30f3\u3059\u306a\u308f\u3061 $45^{\\circ}$ \u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u3063\u305f\u3002<\/p>\n

\u9ad8\u3055 $h$ \u304b\u3089\u306e\u659c\u65b9\u6295\u5c04\u3082\u540c\u69d8\u306b…<\/h3>\n

$$f(\\tau, \\theta) \\equiv y =h+ v_0\\, \\sin\\theta\\cdot \\tau -\\frac{1}{2} g\\,\\tau^2$$<\/p>\n

\u3068\u3059\u308c\u3070\uff0c\u9ad8\u3055 $h$ \u304b\u3089\u306e\u659c\u65b9\u6295\u5c04\u306e\u554f\u984c\u3082\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u306e\u672a\u5b9a\u4e57\u6570\u6cd5\u3092\u4f7f\u3063\u3066\u89e3\u3044\u3066\uff0c\u6c34\u5e73\u5230\u9054\u8ddd\u96e2\u304c\u6700\u5927\u3068\u306a\u308b\u89d2\u5ea6\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3068\u601d\u3044\u307e\u3059\u306e\u3067\uff0c\u3084\u3063\u3066\u307f\u3066\u304f\u3060\u3055\u3044\u3002<\/p>\n

\u8ffd\u8a18\uff1a\u4ee5\u4e0b\u306e\u30da\u30fc\u30b8\u306b\u307e\u3068\u3081\u3066\u307f\u307e\u3057\u305f\u3002<\/p>\n