{"id":5138,"date":"2023-01-25T17:20:19","date_gmt":"2023-01-25T08:20:19","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=5138"},"modified":"2025-01-23T10:22:51","modified_gmt":"2025-01-23T01:22:51","slug":"%e7%a9%ba%e6%b0%97%e6%8a%b5%e6%8a%97%e3%81%8c%e3%81%82%e3%82%8b%e5%a0%b4%e5%90%88%e3%81%ae%e6%96%9c%e6%96%b9%e6%8a%95%e5%b0%84%e3%82%92%e8%aa%bf%e3%81%b9%e3%82%8b%e6%ba%96%e5%82%99","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e3%82%b3%e3%83%b3%e3%83%94%e3%83%a5%e3%83%bc%e3%82%bf%e6%bc%94%e7%bf%92\/%e7%a9%ba%e6%b0%97%e6%8a%b5%e6%8a%97%e3%81%8c%e3%81%82%e3%82%8b%e5%a0%b4%e5%90%88%e3%81%ae%e6%96%9c%e6%96%b9%e6%8a%95%e5%b0%84%e3%82%92%e8%aa%bf%e3%81%b9%e3%82%8b%e6%ba%96%e5%82%99\/","title":{"rendered":"\u7a7a\u6c17\u62b5\u6297\u304c\u3042\u308b\u5834\u5408\u306e\u659c\u65b9\u6295\u5c04\u3092\u8abf\u3079\u308b\u6e96\u5099"},"content":{"rendered":"

\u925b\u76f4\u4e0b\u5411\u304d\u306e\u4e00\u69d8\u91cd\u529b\u5834\u4e2d\u3067\uff0c\u3042\u308b\u521d\u901f\u5ea6\u3092\u3082\u3063\u3066\u7269\u4f53\u3092\u7a7a\u4e2d\u306b\u6295\u3052\u51fa\u3059\u3002\u3053\u308c\u3092\u696d\u754c\u7528\u8a9e\u3067\u300c\u659c\u65b9\u6295\u5c04<\/a>\u300d\uff08Wikipedia \u82f1\u8a9e\u7248\u30bf\u30a4\u30c8\u30eb\u3067\u306f Projectile motion<\/a>\uff09\u3068\u3044\u3046\u3002<\/p>\n

\u7a7a\u6c17\u62b5\u6297\u304c\u7121\u8996\u3067\u304d\u308b\u5834\u5408\u306b\u306f\uff0c\u659c\u65b9\u6295\u5c04\u3055\u308c\u305f\u7269\u4f53\u306f\u653e\u7269\u7dda\u3092\u63cf\u3044\u3066\u904b\u52d5\u3057\uff0c\u521d\u901f\u5ea6\u4e00\u5b9a\u306e\u6761\u4ef6\u3067\u6c34\u5e73\u306a\u5730\u4e0a\u9762\u304b\u3089\u6295\u5c04\u89d2 $\\theta$ \u3067\u6295\u5c04\u3055\u308c\u305f\u7269\u4f53\u306e\u6c34\u5e73\u5230\u9054\u8ddd\u96e2\u304c\u6700\u5927\u3068\u306a\u308b\u306e\u306f\uff0c$\\theta = 45$\u00b0 \u3067\u3042\u308b\u3053\u3068\u304c\u77e5\u3089\u308c\u3066\u3044\u308b\u3002<\/p>\n

\u3067\u306f\uff0c\u7a7a\u6c17\u62b5\u6297\u304c\u3042\u308b\u5834\u5408\u306f\u3069\u3046\u306a\u308b\u304b\uff1f<\/p>\n

\u7a7a\u6c17\u62b5\u6297\u304c\u3042\u308b\u5834\u5408\uff0c\u521d\u901f\u5ea6\u4e00\u5b9a\u306e\u6761\u4ef6\u3067\u6295\u5c04\u3055\u308c\u305f\u7269\u4f53\u306e\u6c34\u5e73\u5230\u9054\u8ddd\u96e2\u304c\u6700\u5927\u3068\u306a\u308b\u6295\u5c04\u89d2 $\\theta$ \u3092\u8abf\u3079\u308b\u305f\u3081\u306b\uff0c\u307e\u305a\u306f\u6e96\u5099\u3002<\/p>\n

<\/p>\n

\u554f\u984c\uff083\u629e\uff09<\/h3>\n

\u307e\u305a\u306f\u4e88\u60f3\u3092\u305f\u3066\u3066\u307f\u3088\u3046\u3002<\/p>\n

\u7a7a\u6c17\u62b5\u6297\u304c\u3042\u308b\u5834\u5408\u306e\u659c\u65b9\u6295\u5c04\u306e\u554f\u984c\u3002\u521d\u901f\u5ea6\u306e\u5927\u304d\u3055\u304c\u4e00\u5b9a\u306e\u6761\u4ef6\u3067\uff0c\u6c34\u5e73\u306a\u5730\u4e0a\u9762\u304b\u3089\u6295\u5c04\u3055\u308c\u305f\u7269\u4f53\u306e\u6c34\u5e73\u5230\u9054\u8ddd\u96e2\u304c\u6700\u5927\u3068\u306a\u308b\u6295\u5c04\u89d2 $\\theta$ \u306f…<\/p>\n

    \n
  1. 45\u00b0 \u3088\u308a\u5c0f\u3055\u304f\u306a\u308b\u3002<\/li>\n
  2. \u3084\u3063\u3071\u308a 45\u00b0 \u3067\u3042\u308b\u3002<\/li>\n
  3. 45\u00b0 \u3088\u308a\u5927\u304d\u304f\u306a\u308b\u3002<\/li>\n<\/ol>\n

     <\/p>\n

    \u7a7a\u6c17\u62b5\u6297\u304c\u3042\u308b\u5834\u5408\u306e\u904b\u52d5\u65b9\u7a0b\u5f0f<\/h3>\n

    \u7a7a\u6c17\u62b5\u6297\u304c\u3042\u308b\u5834\u5408\u306f\uff0c\u8cea\u70b9\u306e\u901f\u3055\u304c\u305d\u308c\u307b\u3069\u5927\u304d\u304f\u306a\u3044\u5834\u5408\u306b\u306f\uff0c\u901f\u5ea6\u306b\u6bd4\u4f8b\u3059\u308b\u3068\u3057\u3066\u3088\u3044\u3060\u308d\u3046\u304b\u3089\uff0c\u6bd4\u4f8b\u5b9a\u6570\u3092 $-m \\beta$ \u3068\u3059\u308b\u3068\uff0c\u904b\u52d5\u65b9\u7a0b\u5f0f\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b\u3002\uff08\u30de\u30a4\u30ca\u30b9\u304c\u3064\u304f\u306e\u306f\uff0c\u62b5\u6297\u3068\u3044\u3046\u306e\u306f\u904b\u52d5\u3092\u59a8\u3052\u308b\u5411\u304d\u306b\u50cd\u304f\u304b\u3089\u3002\uff09<\/p>\n

    $$
    \nm \\frac{d^2\\boldsymbol{r}}{dt^2} = -m\\boldsymbol{g} -m \\beta \\frac{d\\boldsymbol{r}}{dt}
    \n$$<\/p>\n

    \u904b\u52d5\u65b9\u7a0b\u5f0f\u306e\u89e3<\/h3>\n

    \u3053\u306e\u904b\u52d5\u65b9\u7a0b\u5f0f\u306e\u89e3\u306f<\/p>\n

    $$
    \n\\boldsymbol{r}(t) = \\boldsymbol{r}_0 + \\frac{1}{\\beta} \\boldsymbol{v}_0\\,(1 -e^{-\\beta t})
    \n-\\frac{1}{\\beta^2} \\boldsymbol{g}\\,\\left\\{\\beta t -\\left(1 -e^{-\\beta t}\\right)\\right\\}
    \n$$<\/p>\n

    \u3068\u306a\u308b\u3002\u3053\u3053\u3067\uff0c$\\boldsymbol{r}_0 = \\boldsymbol{r}(0), \\\u00a0 \\boldsymbol{v}_0 = \\boldsymbol{v}(0)$\u3002\u4ee5\u4e0b\u3067\u8a73\u7d30\u3092\u8aac\u660e\u3059\u308b\u3002<\/p>\n

    \u89e3\u306e\u5c0e\u51fa\u306e\u8a73\u7d30<\/h4>\n

    \u4e0a\u306e\u904b\u52d5\u65b9\u7a0b\u5f0f\u306e\u4e21\u8fba\u3092 $m$ \u3067\u308f\u308a\uff0c$\\boldsymbol{v} = \\frac{d\\boldsymbol{r}}{dt}$ \u3092\u4f7f\u3063\u3066\u66f8\u304f\u3068…<\/p>\n

    \\begin{eqnarray}
    \n\\frac{d\\boldsymbol{v}}{dt} + \\beta \\boldsymbol{v} &=& -\\boldsymbol{g} \\\\
    \ne^{-\\beta t} \\frac{d}{dt} \\left(\\boldsymbol{v}\\, e^{\\beta t}\\right) &=& -\\boldsymbol{g} \\\\
    \n\\frac{d}{dt} \\left(\\boldsymbol{v}\\, e^{\\beta t}\\right) &=& -\\boldsymbol{g}\\,e^{\\beta t}
    \n\\end{eqnarray}<\/p>\n

    \u3068\u306a\u308b\u3002\u4e21\u8fba\u3092 $t$ \u3067\u7a4d\u5206\u3059\u308b\u3068\uff0c<\/p>\n

    \\begin{eqnarray}
    \ne^{\\beta t} \\,\\boldsymbol{v} &=& -\\frac{1}{\\beta} \\boldsymbol{g}\\, e^{\\beta t} + \\boldsymbol{C}\\\\
    \n\\therefore\\ \\boldsymbol{v} &=& -\\frac{1}{\\beta} \\boldsymbol{g} + \\boldsymbol{C}\\, e^{-\\beta t}
    \n\\end{eqnarray}<\/p>\n

    \u7a4d\u5206\u5b9a\u30d9\u30af\u30c8\u30eb $\\boldsymbol{C}$ \u306f\u521d\u671f\u6761\u4ef6\u3092\u4f7f\u3063\u3066\u66f8\u3051\u308b\u3002$t = 0$ \u3067<\/p>\n

    \\begin{eqnarray}
    \n\\boldsymbol{v}_0 &=& -\\frac{1}{\\beta} \\boldsymbol{g} + \\boldsymbol{C} \\\\
    \n\\therefore\\ \\boldsymbol{C} &=& \\boldsymbol{v}_0 + \\frac{1}{\\beta} \\boldsymbol{g}
    \n\\end{eqnarray}<\/p>\n

    \u3053\u306e $\\boldsymbol{C}$ \u3092\u4ee3\u5165\u3059\u308b\u3068\uff0c<\/p>\n

    \\begin{eqnarray}
    \n\\boldsymbol{v} = \\frac{d\\boldsymbol{r}}{dt} &=&
    \n-\\frac{1}{\\beta} \\boldsymbol{g} +
    \n\\left( \\boldsymbol{v}_0 + \\frac{1}{\\beta} \\boldsymbol{g} \\right) e^{-\\beta t} \\\\
    \n&=& \\boldsymbol{v}_0\\, e^{-\\beta t} -\\frac{1}{\\beta} \\boldsymbol{g}\\,
    \n(1 -e^{-\\beta t})
    \n\\end{eqnarray}<\/p>\n

    \u4e21\u8fba\u3092\u3082\u3046\u4e00\u5ea6\uff0c$t$ \u3067\u7a4d\u5206\u3057\u3066…<\/p>\n

    \\begin{eqnarray}
    \n\\boldsymbol{r}(t) &=& -\\frac{1}{\\beta} \\boldsymbol{v}_0\\, e^{-\\beta t}
    \n-\\frac{1}{\\beta}\\boldsymbol{g}\\,\\left(t + \\frac{1}{\\beta} e^{-\\beta t}\\right) + \\boldsymbol{C}’ \\\\
    \n&=& -\\frac{1}{\\beta} \\boldsymbol{v}_0\\, e^{-\\beta t}
    \n-\\frac{1}{\\beta^2}\\boldsymbol{g}\\,\\left(\\beta t + \\ e^{-\\beta t}\\right) + \\boldsymbol{C}’
    \n\\end{eqnarray}<\/p>\n

    \u7a4d\u5206\u5b9a\u30d9\u30af\u30c8\u30eb $\\boldsymbol{C}’$ \u306f\u521d\u671f\u6761\u4ef6\u3092\u4f7f\u3063\u3066\u66f8\u3051\u308b\u3002$t = 0$ \u3067<\/p>\n

    \\begin{eqnarray}
    \n\\boldsymbol{r}_0 &=& -\\frac{1}{\\beta} \\boldsymbol{v}_0 -\\frac{1}{\\beta^2}\\boldsymbol{g} + \\boldsymbol{C}’ \\\\
    \n\\therefore\\ \\boldsymbol{C}’ &=& \\boldsymbol{r}_0 + \\frac{1}{\\beta} \\boldsymbol{v}_0 + \\frac{1}{\\beta^2}\\boldsymbol{g}
    \n\\end{eqnarray}<\/p>\n

    \u3053\u306e $\\boldsymbol{C}’$ \u3092\u4ee3\u5165\u3059\u308b\u3068\u6700\u7d42\u7684\u306a\u89e3\u304c\u5f97\u3089\u308c\u308b\u3002<\/p>\n

    \u904b\u52d5\u304c\u5e73\u9762\u5185\u306b\u9650\u3089\u308c\u308b\u7406\u7531<\/h3>\n

    \u659c\u65b9\u6295\u5c04\u306e\u554f\u984c\u306f\uff0c\u3057\u3070\u3057\u3070\u904b\u52d5\u304c $xy$ \u5e73\u9762\u5185\u306b\u9650\u3089\u308c\u308b\u3053\u3068\u3092\u81ea\u660e\u3068\u3057\u3066\u8aac\u660e\u3055\u308c\u308b\u304c\uff0c\u904b\u52d5\u304c\uff08\u5909\u76f4\u65b9\u6cd5\u306b\u5e73\u884c\u306a\uff09\u5e73\u9762\u5185\u306b\u9650\u3089\u308c\u308b\u3053\u3068\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8aac\u660e\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n

    \u4e0a\u8a18\u306e\u89e3\u304b\u3089\u76f4\u3061\u306b\u308f\u304b\u308b\u304c\uff0c\u7a7a\u6c17\u62b5\u6297\u304c\u3042\u308d\u3046\u304c\u306a\u304b\u308d\u3046\u304c\uff0c\u4f4d\u7f6e\u30d9\u30af\u30c8\u30eb\u306e\u5909\u5316\u5206 $\\boldsymbol{r}(t) -\\boldsymbol{r}_0$ \u306f2\u3064\u306e\u5b9a\u30d9\u30af\u30c8\u30eb $\\boldsymbol{v}_0$ \u3068 $\\boldsymbol{g}$ \u306e\u7dda\u5f62\u7d50\u5408\u3067\u66f8\u304b\u308c\u3066\u3044\u308b\u3002<\/p>\n

    \u3064\u307e\u308a\uff0c\u4f4d\u7f6e\u30d9\u30af\u30c8\u30eb $\\boldsymbol{r}(t)$ \u306e\u6642\u9593\u5909\u5316\u306f\u306f2\u3064\u306e\u30d9\u30af\u30c8\u30eb $\\boldsymbol{v}_0$ \u3068 $\\boldsymbol{g}$ \u3067\u5f35\u3089\u308c\u308b\u5e73\u9762\u5185\u306b\u9650\u3089\u308c\u308b\u3053\u3068\u306b\u306a\u308a\uff0c\u659c\u65b9\u6295\u5c04\u3055\u308c\u305f\u7269\u4f53\u306e\u904b\u52d5\u306f\u3053\u306e\u5e73\u9762\u5185\u306b\u9650\u3089\u308c\u308b\u3053\u3068\u306b\u306a\u308b\u3002<\/p>\n

    \u3053\u308c\u304c\u659c\u65b9\u6295\u5c04\u3055\u308c\u305f\u7269\u4f53\u306e\u904b\u52d5\u304c\u5e73\u9762\u5185\u306b\u9650\u3089\u308c\u308b\u7406\u7531\u3067\u3042\u308b\u3002<\/p>\n

    \u4ee5\u4e0b\u3067\u306f\u7c21\u5358\u306e\u305f\u3081\u306b\uff0c\u305d\u306e\u5e73\u9762\u3092 $xy$ \u5e73\u9762\u3068\u3057\uff0c<\/p>\n

    $$\\boldsymbol{r} = (x, y, 0), \\quad \\boldsymbol{r}_0 = (0, h, 0),
    \n\\quad \\boldsymbol{v}_0 = (v_0 \\cos\\theta, v_0 \\sin\\theta, 0)$$<\/p>\n

    \u3068\u3059\u308b\u3002<\/p>\n

    $x, y$ \u6210\u5206\u3067\u66f8\u3044\u305f\u89e3<\/h3>\n

    \u3042\u3089\u305f\u3081\u3066\u89e3\u3092\u66f8\u304f\u3068<\/p>\n

    \\begin{eqnarray}
    \nx &=& \\frac{1-e^{-\\beta t}}{\\beta} v_0 \\cos\\theta\u00a0 \\\\
    \ny&=& h + \\frac{1-e^{-\\beta t}}{\\beta} v_0 \\sin\\theta + \\frac{1 -\\beta t -e^{-\\beta t}}{\\beta^2} g\\\\
    \nv_x &=& e^{-\\beta t} v_0 \\cos\\theta\u00a0 \\\\
    \nv_y &=& e^{-\\beta t} v_0 \\sin\\theta \\\u00a0 -\\frac{1-e^{-\\beta t}}{\\beta} g
    \n\\end{eqnarray}<\/p>\n

    \u7121\u6b21\u5143\u5316<\/h3>\n

    \u3053\u306e\u7cfb\u306b\u7279\u5fb4\u7684\u306a\u6642\u9593 $\\displaystyle \\tau \\equiv \\frac{v_0}{g}$ \u304a\u3088\u3073\u9577\u3055 $\\displaystyle \\ell \\equiv v_0 \\tau = \\frac{v_0^2}{g}$ \u3067\u89e3\u3092\u7121\u6b21\u5143\u5316\u3057\u3066\u304a\u304f\u3002\u7a7a\u6c17\u62b5\u6297\u306e\u6bd4\u4f8b\u5b9a\u6570 $\\beta$ \u306f\u6642\u9593\u306e\u9006\u6570\u306e\u6b21\u5143\u3092\u3082\u3064\u306e\u3067\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u7121\u6b21\u5143\u5316\u3059\u308b\u3002<\/p>\n

    \\begin{eqnarray}
    \nb &\\equiv& \\beta \\, \\tau \\\\
    \nT &\\equiv& \\frac{t}{\\tau} \\\\
    \nH &\\equiv& \\frac{h}{\\ell} \\\\
    \nX &\\equiv& \\frac{x}{\\ell} = \\frac{1-e^{-b T}}{b}\u00a0 \\cos\\theta \\\\
    \nY&\\equiv& \\frac{y}{\\ell} = H+ \\frac{1-e^{-b T}}{b}\u00a0 \\sin\\theta\u00a0 + \\frac{1 -b T -e^{-b T}}{b^2}\u00a0\u00a0 \\\\
    \nV_x &\\equiv& \\frac{dX}{dT} =e^{-b T} \\cos\\theta \\\\
    \nV_y &\\equiv& \\frac{dY}{dT} = e^{-b T} \\sin\\theta \\\u00a0 – \\frac{1-e^{-b T}}{b}
    \n\\end{eqnarray}<\/p>\n

    SymPy \u3084 Maxima \u3092\u4f7f\u3063\u3066\u6700\u5927\u6c34\u5e73\u5230\u9054\u8ddd\u96e2\u3092\u6c42\u3081\u308b<\/h3>\n

    \u5225\u30da\u30fc\u30b8\u306b\u30e1\u30e2\u3068\u3057\u3066\u307e\u3068\u3081\u3066\u304a\u3044\u305f\u3002<\/p>\n