\u9670\u95a2\u6570\u5b9a\u7406<\/strong><\/span><\/a>\u3088\u308a<\/p>\n$$\\frac{d\\tau}{d\\theta} = – \\frac{\\frac{\\partial}{\\partial\\theta} y(\\tau, \\theta)}{\\frac{\\partial}{\\partial\\tau} y(\\tau, \\theta)} = – \\frac{v_0 \\cos\\theta\\cdot \\tau}{v_0\\sin\\theta -g \\tau}$$<\/p>\n
\u6c34\u5e73\u5230\u9054\u8ddd\u96e2 $\\ell$<\/h3>\n
\u6ede\u7a7a\u6642\u9593 $\\tau$ \u306e\u9593\u306e\u6c34\u5e73\u5230\u9054\u8ddd\u96e2 $\\ell$ \u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b $\\theta$ \u306e\u95a2\u6570\u3068\u306a\u308b\u3002<\/p>\n
$$\\ell(\\theta) = x(\\tau(\\theta), \\theta)$$<\/p>\n
\u6b21\u306b\u4f7f\u3046\u306e\u3067\uff0c$\\ell$ \u306e\u5fae\u5206\u3092\u8a08\u7b97\u3057\u3066\u304a\u304f\u3002<\/p>\n
\\begin{eqnarray}
\n\\frac{d\\ell}{d\\theta} &=& \\frac{\\partial}{\\partial\\theta} x(\\tau, \\theta)
\n+ \\frac{\\partial}{\\partial\\tau} x(\\tau, \\theta) \\frac{d\\tau}{d\\theta} \\\\
\n&=& -v_0 \\sin\\theta\\cdot \\tau + v_0\\cos\\theta\\cdot\\left\\{ – \\frac{v_0 \\cos\\theta\\cdot \\tau}{v_0\\sin\\theta -g \\tau}\\right\\}\\\\
\n&=& \\frac{-v_0^2 \\tau + v_0 \\sin\\theta\\ g \\tau^2}{v_0 \\sin\\theta -g \\tau}
\n\\end{eqnarray}<\/p>\n
\u6c34\u5e73\u5230\u9054\u8ddd\u96e2\u304c\u6700\u5927\u3068\u306a\u308b\u89d2\u5ea6<\/h3>\n
\u6c34\u5e73\u5230\u9054\u8ddd\u96e2\u304c\u6700\u5927\u3068\u306a\u308b\u89d2\u5ea6 $\\theta$ \u306f $\\displaystyle \\frac{d\\ell}{d\\theta} = 0$ \u3092\u6e80\u305f\u3059\u306e\u3067<\/p>\n
\\begin{eqnarray}
\n-v_0^2\\, \\tau + v_0 \\sin\\theta\\, g\\, \\tau^2 &=& 0 \\\\
\n\\therefore\\ \\ \\tau &=& \\frac{v_0}{g \\sin\\theta}
\n\\end{eqnarray}<\/p>\n
\u3053\u308c\u3092 $y(\\tau, \\theta)$ \u306b\u5165\u308c\u3066<\/p>\n
\\begin{eqnarray}
\ny(\\tau, \\theta) &=&
\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 \\\\
\n&=& h + \\frac{v_0^2}{g} -\\frac{v_0^2}{2 g} \\frac{1}{\\sin^2\\theta} \\\\
\n&=& 0 \\\\
\n\\therefore\\ \\ \\sin^2\\theta &=& \\frac{v_0^2}{2 v_0^2 + 2 g h} \\\\
\n\\cos^2\\theta &=& 1 – \\sin^2\\theta \\\\
\n&=& \\frac{v_0^2 + 2 g h}{2 v_0^2 + 2 g h} \\\\
\n\\therefore\\ \\ \\tan\\theta &=& \\sqrt{\\frac{v_0^2}{v_0^2 + 2 g h}} \\quad < 1 \\quad \\mbox{for}\\quad\u00a0 h > 0
\n\\end{eqnarray}<\/p>\n
$h > 0$ \u306b\u5bfe\u3057\u3066\uff0c\u6700\u5927\u6c34\u5e73\u5230\u9054\u8ddd\u96e2\u3092\u4e0e\u3048\u308b\u6253\u3061\u51fa\u3057\u89d2\u5ea6 $\\theta$ \u3092\u3042\u3089\u305f\u3081\u3066 $\\theta_{\\rm m}$ \u3068\u66f8\u304f\u3068 $\\theta_{\\rm m}$ \u306f $45^{\\circ}$ \u3088\u308a\u5c0f\u3055\u304f\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n
\u6700\u5927\u6c34\u5e73\u5230\u9054\u8ddd\u96e2<\/h3>\n
\u6700\u5927\u6c34\u5e73\u5230\u9054\u8ddd\u96e2 $\\ell_{\\rm m} $ \u306f $\\tau_{\\rm m} \\equiv \\tau(\\theta_{\\rm m})$ \u3068\u3057\u3066<\/p>\n
\\begin{eqnarray}
\n\\ell_{\\rm m} &=& x(\\tau_{\\rm m}, \\theta_{\\rm m}) \\\\
\n&=& v_0 \\cos\\theta_{\\rm m} \\cdot \\frac{v_0}{g \\sin\\theta_{\\rm m}} \\\\
\n&=& \\frac{v_0^2}{g} \\frac{1}{\\tan\\theta_{\\rm m}} \\\\
\n&=& \\frac{v_0^2}{g} \\sqrt{1 + \\frac{2gh}{v_0^2}}
\n\\end{eqnarray}<\/p>\n
\u63a5\u5730\u6642\u306e\u89d2\u5ea6\uff08\u4fef\u89d2\uff09<\/h3>\n
\u6295\u5c04\u6642\u306e\u89d2\u5ea6\uff08\u4ef0\u89d2\uff09\u3092\u6700\u5927\u6c34\u5e73\u5230\u9054\u8ddd\u96e2\u3068\u306a\u308b\u89d2\u5ea6 $\\theta_{\\rm m}$ \u306b\u3057\u3066\u6295\u5c04\u3055\u308c\u305f\u7269\u4f53\u306f\uff0c\u6ede\u7a7a\u6642\u9593 $\\tau_{\\rm m}$ \u306e\u5f8c\u306b\u5730\u9762\u306b\u5230\u9054\u3059\u308b\u3002\u3053\u306e\u6642\u306e\u89d2\u5ea6\uff08\u4fef\u89d2\uff09$\\theta_{\\rm f}$ \u306f<\/p>\n
\\begin{eqnarray}
\nv_x(\\tau_{\\rm m}, \\theta_{\\rm m}) &=& v_0 \\cos\\theta_{\\rm m} \\\\
\nv_y(\\tau_{\\rm m}, \\theta_{\\rm m}) &=& v_0 \\sin\\theta_{\\rm m} -g \\tau_{\\rm m} \\\\
\n&=& v_0 \\sin\\theta_{\\rm m} -g \\frac{v_0}{\\sin\\theta_{\\rm m}} \\\\
\n&=& -\\frac{v_0 \\cos^2\\theta_{\\rm m}}{\\sin\\theta_{\\rm m}} \\\\
\n\\therefore\\ \\ \\tan \\theta_{\\rm f} &\\equiv& \\frac{|v_y(\\tau_{\\rm m}, \\theta_{\\rm m})|}{v_x(\\tau_{\\rm m}, \\theta_{\\rm m})}\\\\
\n&=& \\frac{1}{\\tan\\theta_{\\rm m}}
\n\\end{eqnarray}<\/p>\n
\u3064\u307e\u308a\uff0c<\/p>\n
$$\\theta_{\\rm m} + \\theta_{\\rm f} = \\frac{\\pi}{2}$$<\/p>\n
\u3068\u3044\u3046\u95a2\u4fc2\u304c\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002\u306a\u305c\u304b\u3068\u3044\u3046\u3068\uff0c\u4e00\u822c\u306b<\/p>\n
\\begin{eqnarray}
\n\\tan\\left(\\frac{\\pi}{2} -\\theta\\right) &=&
\n\\frac{\\sin\\left(\\frac{\\pi}{2} -\\theta\\right)}{\\cos\\left(\\frac{\\pi}{2} -\\theta\\right)} \\\\
\n&=& \\frac{\\cos\\left(\\theta\\right)}{\\sin\\left( \\theta\\right)} \\\\
\n&=& \\frac{1}{\\tan\\theta}
\n\\end{eqnarray}<\/p>\n
\u3060\u304b\u3089\uff0c$\\displaystyle \\tan\\theta_{\\rm f} = \\frac{1}{\\tan\\theta_{\\rm m}}$\u00a0 \u3068\u3044\u3046\u3053\u3068\u306f<\/p>\n
\\begin{eqnarray}
\n\\theta_{\\rm f} &=& \\frac{\\pi}{2} -\\theta_{\\rm m} \\\\
\n\\therefore\\ \\\u00a0 \\theta_{\\rm m} + \\theta_{\\rm f} &=& \\frac{\\pi}{2}
\n\\end{eqnarray}<\/p>\n
![\"\"](\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/hshahou03.svg\")
<\/p>\n
\u6700\u5927\u5230\u9054\u8ddd\u96e2\u306e\u30b0\u30e9\u30d5\u4f8b<\/h3>\n
\u76ee\u76db\u304b\u3089\u308f\u304b\u308b\u3088\u3046\u306b\uff0c\u6a2a\u8ef8 $x$\uff0c\u7e26\u8ef8 $y$ \u306f $h = 0$ \u306e\u6642\u306e\u6700\u5927\u5230\u9054\u8ddd\u96e2 $\\displaystyle \\frac{v_0^2}{g}$ \u3067\u898f\u683c\u5316\u3055\u308c\u305f\u9577\u3055\u306b\u306a\u3063\u3066\u3044\u308b\u3002<\/p>\n
![\"\"](\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/hshahou01.svg\")
<\/p>\n","protected":false},"excerpt":{"rendered":"
\u3059\u3067\u306b\u4ee5\u4e0b\u306e\u30da\u30fc\u30b8\u3067\u89e3\u3044\u3066\u3044\u308b\u554f\u984c\u3060\u304c\uff0c\u4eca\u56de\u306f\u9670\u95a2\u6570\u5b9a\u7406\u3092\u7a4d\u6975\u7684\u306b\u4f7f\u3044\u306a\u304c\u3089\u3082\uff0c\u624b\u8a08\u7b97\u306e\u307f\u3067\u6c42\u3081\u3066\u307f\u308b\u3002<\/p>\n
\n- \u9ad8\u3055 h \u304b\u3089\u306e\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<\/li>\n
- \u9ad8\u3055 h \u304b\u3089\u306e\u659c\u65b9\u6295\u5c04\u306e\u6700\u5927\u5230\u9054\u8ddd\u96e2\u306f\u89d2\u5ea645\u00b0\u306e\u3068\u304d\u3067\u306f\u306a\u3044<\/li>\n<\/ul>\n
\u4e0a\u8a18\u306e\u30da\u30fc\u30b8\u3067\u306f\uff0c\u30b3\u30f3\u30d4\u30e5\u30fc\u30bf\u6f14\u7fd2\u7528\u306e\u554f\u984c\u3068\u3057\u3066\u8003\u3048\u3066\u3044\u305f\u304c\uff0c\u3042\u3089\u305f\u3081\u3066\u7406\u5de5\u7cfb\u306e\u6570\u5b66\u306e\u9670\u95a2\u6570\u5b9a\u7406\u306b\u95a2\u3059\u308b\u6f14\u7fd2\u554f\u984c\u3068\u3057\u3066\u3069\u3046\u304b\u306a\u3068\u601d\u3063\u3066\u3002<\/p>
\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":33,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[25,21],"tags":[],"class_list":["post-8779","post","type-post","status-publish","format-standard","hentry","category-25","category-21","nodate","item-wrap"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/posts\/8779","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/users\/33"}],"replies":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/comments?post=8779"}],"version-history":[{"count":40,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/posts\/8779\/revisions"}],"predecessor-version":[{"id":10362,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/posts\/8779\/revisions\/10362"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=8779"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/categories?post=8779"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/tags?post=8779"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}