<\/p>\n
\u521d\u671f\u6761\u4ef6\u3068\u89e3<\/h3>\n
\u521d\u671f\u6761\u4ef6\u3092 \\(t = 0\\) \u3067<\/p>\n
$$x = 0, \\quad y = h, \\quad v_x\u00a0 = v_0 \\cos\\theta, \\quad v_y = v_0 \\sin \\theta$$<\/p>\n
\u3068\u3057\u305f\u3068\u304d\u306e\u89e3\u306f<\/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 $\\tau$<\/h3>\n
$t = 0$ \u3067\u9ad8\u3055 $h$ \u306e\u5834\u6240\u304b\u3089\u6295\u5c04\u3057\u3066\u5730\u9762 $y=0$ \u306b\u843d\u3061\u308b\u307e\u3067\u306e\u6ede\u7a7a\u6642\u9593\u3092 $\\tau$ \u3068\u3059\u308b\u3068<\/p>\n
$$y(\\tau, \\theta) =\u00a0 0$$<\/p>\n
\u6c34\u5e73\u5230\u9054\u8ddd\u96e2<\/h3>\n
\u6ede\u7a7a\u6642\u9593 $\\tau$ \u306e\u9593\u306e\u6c34\u5e73\u5230\u9054\u8ddd\u96e2 $\\ell$ \u306f<\/p>\n
$$\\ell(\\tau, \\theta) = x(\\tau, \\theta) $$<\/p>\n
\u6c34\u5e73\u5230\u9054\u8ddd\u96e2\u304c\u6700\u5927\u3068\u306a\u308b\u89d2\u5ea6\u3092\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u306e\u672a\u5b9a\u4e57\u6570\u6cd5\u3067<\/h3>\n
\u6c34\u5e73\u5230\u9054\u8ddd\u96e2\u304c\u6700\u5927\u3068\u306a\u308b\u6253\u3061\u51fa\u3057\u89d2\u5ea6 $\\theta$ \u306f\uff0c<\/p>\n
$$F(\\tau, \\theta, \\lambda) \\equiv\u00a0 x(\\tau, \\theta) + \\lambda\\, y(\\tau, \\theta)$$<\/p>\n
\u3068\u5b9a\u7fa9\u3057\u305f $F(\\tau, \\theta, \\lambda)$ \u306b\u3064\u3044\u3066<\/p>\n
\\begin{eqnarray}
\n\\frac{\\partial F}{\\partial \\tau} &=& \\frac{\\partial x}{\\partial \\tau} + \\lambda \\frac{\\partial y}{\\partial \\tau}
\n= v_0 \\cos\\theta + \\lambda\\, \\left( v_0 \\sin\\theta -g\\, \\tau\\right) = 0\\tag{1} \\\\
\n\\frac{\\partial F}{\\partial \\theta} &=& \\frac{\\partial x}{\\partial \\theta} + \\lambda \\frac{\\partial y}{\\partial \\theta}
\n=-v_0 \\sin\\theta\\cdot \\tau + \\lambda \\left( v_0 \\cos\\theta\\cdot \\tau\\right) = 0 \\tag{2} \\\\
\n\\frac{\\partial F}{\\partial \\lambda} &=& h + v_0 \\sin\\theta\\cdot\\tau -\\frac{1}{2} g\\,\\tau^2 = 0 \\tag{3}
\n\\end{eqnarray}<\/p>\n
\u3092\u89e3\u3051\u3070\u3088\u3044\u3002<\/p>\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} \\left( v_0 \\sin\\theta -g\\, \\tau\\right)
\n&=& \\frac{1}{\\cos\\theta} \\left\\{v_0 -\\sin\\theta\\cdot g\\,\\tau \\right\\} \\\\
\n&=& 0 \\\\
\n\\therefore\\ \\ \\tau &=& \\frac{v_0}{g\\,\\sin\\theta}
\n\\end{eqnarray}<\/p>\n
\u3053\u308c\u3092 $(3)$ \u5f0f\u306b\u4ee3\u5165\u3059\u308b\u3068<\/p>\n
\\begin{eqnarray}
\nh + v_0 \\sin\\theta\\cdot\\frac{v_0}{g\\,\\sin\\theta} -\\frac{1}{2} g\\left( \\frac{v_0}{g\\,\\sin\\theta}\\right)^2 &=&
\nh + \\frac{v_0^2}{g} -\\frac{1}{2 g} \\frac{v_0^2}{\\sin^2\\theta} \\\\
\n&=& 0 \\\\
\n\\therefore \\ \\ \\sin^2\\theta &=& \\frac{\\dfrac{v_0^2}{2 g}}{h + \\dfrac{v_0^2}{g}} = \\frac{v_0^2}{2 v_0^2 + 2 g h} \\\\
\n\\cos^2\\theta &=& 1 -\\sin^2\\theta = \\frac{v_0^2+ 2 g h}{2 v_0^2 + 2 g h} \\\\
\n\\therefore\\ \\ \\tan\\theta &=& \\sqrt{\\dfrac{v_0^2}{v_0^2+ 2 g h}}
\n\\end{eqnarray}<\/p>\n
\u6c34\u5e73\u5230\u9054\u8ddd\u96e2\u3092\u6700\u5927\u306b\u3059\u308b\u6253\u3061\u51fa\u3057\u89d2\u5ea6 $\\theta$ \u304c\u6c42\u307e\u308c\u3070\uff0c\u3042\u3068\u306f\u300c\u9ad8\u3055 h \u304b\u3089\u306e\u659c\u65b9\u6295\u5c04\u306e\u554f\u984c\u3092\u9670\u95a2\u6570\u5b9a\u7406\u3092\u4f7f\u3063\u3066\u89e3\u3044\u3066\u307f\u308b<\/a>\u300d\u3068\u540c\u3058\u3088\u3046\u306b\uff0c\u6700\u5927\u6c34\u5e73\u5230\u9054\u8ddd\u96e2\u3084\u63a5\u5730\u6642\u306e\u4fef\u89d2\u306a\u3069\u3082\u6c42\u307e\u308b\u3002<\/p>\n \u3059\u3067\u306b\u4ee5\u4e0b\u306e\u30da\u30fc\u30b8\u3067\u9670\u95a2\u6570\u5b9a\u7406\u3092\u4f7f\u3063\u3066\u89e3\u3044\u3066\u3044\u308b\u554f\u984c\u3060\u304c\uff0c\u4eca\u56de\u306f\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u306e\u672a\u5b9a\u4e57\u6570\u6cd5\u3092\u4f7f\u3063\u3066\u89e3\u3044\u3066\u307f\u308b\u3002<\/p>![\"\"](\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/hshahou03.svg\")
<\/p>\n","protected":false},"excerpt":{"rendered":"