\u659c\u65b9\u6295\u5c04\u306e\u6700\u5927\u6c34\u5e73\u5230\u9054\u8ddd\u96e2\u3092\u9670\u95a2\u6570\u5b9a\u7406\u3092\u4f7f\u3063\u3066\u6c42\u3081\u308b\u7df4\u7fd2\u554f\u984c\u3002<\/p>\n
<\/p>\n
\u659c\u65b9\u6295\u5c04\u306e\u8ecc\u9053<\/p>\n
\\begin{eqnarray}
\nx(t, \\theta) &=& v_0\\,\\cos\\theta\\cdot t \\\\
\ny(t, \\theta) &=& v_0\\, \\sin\\theta\\cdot t -\\frac{1}{2} g\\,t^2
\n\\end{eqnarray}<\/p>\n
\u521d\u901f\u5ea6\u306e\u5927\u304d\u3055 $v_0$ \u3092\u4e00\u5b9a\u3068\u3057\u305f\u3068\u304d\u306b\uff0c\u6c34\u5e73\u5230\u9054\u8ddd\u96e2\u304c\u6700\u5927\u3068\u306a\u308b\u6295\u5c04\u89d2\u5ea6 $\\theta$ \u306f\uff1f \u3068\u3044\u3046\u554f\u984c\u3092\u9670\u95a2\u6570\u5b9a\u7406\uff08\u3084\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u306e\u672a\u5b9a\u4e57\u6570\u6cd5\uff09\u3092\u4f7f\u308f\u305a\u306b\u3002<\/p>\n
\u6ede\u7a7a\u6642\u9593\u3068\u306f $t=0$ \u3067 $y=0$ \u304b\u3089\u6295\u5c04\u3057\u305f\u7269\u4f53\u304c\u518d\u3073 $y=0$ \u306b\u306a\u308b\u307e\u3067\u306e\u6642\u9593\u3067\u3042\u308a\uff0c$y(\\tau(\\theta), \\theta) = 0$ \u3092\u6e80\u305f\u3059\u967d\u95a2\u6570 $\\tau(\\theta) \\ (>0) $ \u3068\u3057\u3066\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n
\\begin{eqnarray}
\ny(\\tau(\\theta), \\theta) &=& v_0\\, \\sin\\theta\\cdot \\tau(\\theta) -\\frac{1}{2} g\\,\\tau^2(\\theta) = 0 \\ \\ \\mbox{\u3088\u308a} \\\\
\n\\tau(\\theta) &=& \\frac{2 v_0\\, \\sin\\theta}{g}
\n\\end{eqnarray}<\/p>\n
\u6ede\u7a7a\u6642\u9593 $\\tau(\\theta)$ \u3092 $\\theta$ \u306e\u95a2\u6570\u3068\u3057\u3066\u3042\u304b\u3089\u3055\u307e\u306b\u8868\u3057\u305f\u3002<\/p>\n
\u6c34\u5e73\u5230\u9054\u8ddd\u96e2 $\\ell(\\theta)$ \u3068\u306f\uff0c\u6ede\u7a7a\u6642\u9593 $\\tau(\\theta)$ \u306e\u9593\u306b $x$ \u65b9\u5411\u3078\u9032\u3080\u8ddd\u96e2\u3067\u3042\u308a\uff0c<\/p>\n
\\begin{eqnarray}
\n\\ell(\\theta) &\\equiv& x(\\tau(\\theta), \\theta) \\\\
\n&=& v_0\\, \\cos\\theta\\cdot \\tau(\\theta) \\\\
\n&=& v_0\\, \\cos\\theta\\cdot \\frac{2 v_0\\, \\sin\\theta}{g}\\\\
\n&=& \\frac{v_0^2}{g} \\sin 2 \\theta
\n\\end{eqnarray}<\/p>\n
$v_0$ \u304c\u4e00\u5b9a\u3067\u3042\u308c\u3070\uff0c$\\ell(\\theta)$ \u304c\u6700\u5927\u3068\u306a\u308b\u306e\u306f $\\sin 2 \\theta = 1$ \u3059\u306a\u308f\u3061 $\\displaystyle \\theta = \\frac{\\pi}{4}$ \u306e\u3068\u304d\u3002<\/p>\n
\u3088\u308a\u4e00\u822c\u7684\u72b6\u6cc1\u3078\u306e\u5fdc\u7528\u3092\u898b\u3059\u3048\u3066\uff0c\u3042\u3048\u3066\u9670\u95a2\u6570\u5b9a\u7406\u3092\u4f7f\u3063\u3066\u89e3\u3044\u3066\u307f\u308b\u3002<\/p>\n
$\\displaystyle y(\\tau, \\theta) = v_0\\,\\sin\\theta\\cdot\\tau – \\frac{1}{2} g\\,\\tau^2 =0$ \u3092\u6e80\u305f\u3059\u9670\u95a2\u6570\u3092 $\\tau = \\tau(\\theta)$ \u3068\u3059\u308b\u3002\u9670\u95a2\u6570\u5b9a\u7406\u3088\u308a\uff0c$\\tau$ \u306e\u5fae\u5206\u306f<\/p>\n
\\begin{eqnarray}
\n\\frac{d\\tau}{d\\theta} &=& – \\frac{\\ \\ \\ \\frac{\\partial y}{\\partial \\theta}\\ \\ \\ }{\\frac{\\partial y}{\\partial \\tau}} \\\\
\n&=& – \\frac{v_0\\,\\cos\\theta\\cdot\\tau}{v_0\\,\\sin\\theta -g\\,\\tau}
\n\\end{eqnarray}<\/p>\n
\u6ede\u7a7a\u6642\u9593 $\\tau$ \u306e\u9593\u306b $x$ \u65b9\u5411\u3078\u9032\u3080\u8ddd\u96e2 $\\ell$ \u3068\u305d\u306e\u5fae\u5206\u306f<\/p>\n
\\begin{eqnarray}
\n\\ell &=& v_0\\, \\cos\\theta\\cdot \\tau \\\\
\n\\frac{d\\ell}{d\\theta} &=& -v_0\\, \\sin\\theta\\cdot \\tau + v_0\\,\\cos\\theta\\cdot\\frac{d\\tau}{d\\theta}\\\\
\n&=& -v_0\\, \\sin\\theta\\cdot \\tau -v_0\\,\\cos\\theta\\cdot\\frac{v_0\\,\\cos\\theta\\cdot\\tau}{v_0\\,\\sin\\theta -g\\,\\tau}\\\\
\n&=& \\frac{v_0\\,g\\,\\tau^2 \\,\\sin\\theta-v_0^2\\,\\tau}{v_0\\,\\sin\\theta -g\\,\\tau} \\\\ \\ \\\\
\n\\frac{d\\ell}{d\\theta} &=& 0 \\ \\ \\mbox{\u3088\u308a} \\\\
\n\\tau &=& \\frac{v_0}{g\\,\\sin\\theta}
\n\\end{eqnarray}<\/p>\n
\u3053\u308c\u3092\u3042\u3089\u305f\u3081\u3066 $\\displaystyle y(\\tau, \\theta) = v_0\\,\\sin\\theta\\cdot\\tau – \\frac{1}{2} g\\,\\tau^2 =0$ \u306b\u4ee3\u5165\u3059\u308b\u3068\uff0c<\/p>\n
\\begin{eqnarray}
\nv_0\\,\\sin\\theta\\cdot \\tau -\\frac{1}{2} g\\,\\tau^2 &=& 0 \\\\
\nv_0\\,\\sin\\theta -\\frac{1}{2} g\\cdot\\frac{v_0}{g\\,\\sin\\theta} &=& 0 \\\\
\n\\therefore\\ \\ \\sin^2\\theta &=& \\frac{1}{2} \\\\\\
\n\\therefore\\ \\ \\sin\\theta &=& \\frac{1}{\\sqrt{2}} \\\\
\n\\therefore \\ \\ \\theta &=& \\frac{\\pi}{4}
\n\\end{eqnarray}<\/p>\n
\u659c\u65b9\u6295\u5c04\u306e\u8ecc\u9053<\/p>\n
\\begin{eqnarray}
\nx(t, \\theta) &=& v_0\\,\\cos\\theta\\cdot t \\\\
\ny(t, \\theta) &=& h + v_0\\, \\sin\\theta\\cdot t -\\frac{1}{2} g\\,t^2
\n\\end{eqnarray}<\/p>\n
\u6ede\u7a7a\u6642\u9593 $\\tau(\\theta)\\ (>0)$ \u3092 $y(\\tau(\\theta), \\theta) = 0$ \u3092\u6e80\u305f\u3059\u967d\u95a2\u6570\u3068\u3057\u3066\u8868\u3059\u3068\uff0c<\/p>\n
\\begin{eqnarray}
\ny(\\tau(\\theta), \\theta) &=& h + v_0\\, \\sin\\theta\\cdot \\tau(\\theta) -\\frac{1}{2} g\\,\\tau^2(\\theta) = 0 \\ \\ \\mbox{\u3088\u308a} \\\\
\n\\tau(\\theta) &=& \\frac{v_0 \\,\\sin\\theta + \\sqrt{v_0^2\\,\\sin^2\\theta + 2 g\\, h}}{g}
\n\\end{eqnarray}<\/p>\n
\\begin{eqnarray}
\n\\ell(\\theta) &=& v_0\\,\\cos\\theta\\cdot\\tau(\\theta) \\\\
\n&=& v_0\\,\\cos\\theta\\cdot\\frac{v_0 \\,\\sin\\theta + \\sqrt{v_0^2\\,\\sin^2\\theta + 2 g\\, h}}{g}
\n\\end{eqnarray}<\/p>\n
$\\ell(\\theta)$ \u3092\u6700\u5927\u306b\u3059\u308b $\\theta$ \u306f $\\displaystyle \\frac{d\\ell}{d\\theta} = 0$ \u304b\u3089\u6c42\u3081\u3089\u308c\u308b\u30cf\u30ba\u3060\u304c\uff0c\u306a\u3093\u304b\u9762\u5012\u305d\u3046…<\/p>\n
\u9670\u95a2\u6570\u5b9a\u7406\uff08\u3084\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u306e\u672a\u5b9a\u4e57\u6570\u6cd5\uff09\u3092\u4f7f\u3063\u3066\u89e3\u304f\u3068\uff0c\u9ad8\u3055 $h$ \u304b\u3089\u306e\u659c\u65b9\u6295\u5c04\u306e\u554f\u984c\u3082\u308f\u308a\u3068\u3059\u3063\u304d\u308a\u89e3\u304f\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u4ee5\u4e0b\u306e\u30da\u30fc\u30b8\u3092\u53c2\u7167\uff1a<\/p>\n
\u8ffd\u52a0\uff1a<\/p>\n