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    "<center><h2>Projectile Motion with Air Resistance</h2></center>\n",
    "<p>\n",
    "    The equations of motion are are solved for a projectile experience air resistance  proportional to n power of velocity\n",
    "    \n",
    " $$\n",
    "\\frac{d^2 x}{dt^2} = - k\\, v_x^n \\, \\frac{v_x}{|v|}, \\quad\n",
    "\\frac{d^2 y}{dt^2} = -g -k \\,v_y^n\\, \\frac{v_y}{|v|},\\quad |v|\n",
    "=\\sqrt{v_x^2+v_y^2}.\n",
    " $$"
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    "\"\"\" From \"COMPUTATIONAL PHYSICS\" & \"COMPUTER PROBLEMS in PHYSICS\"\n",
    "    by RH Landau, MJ Paez, and CC Bordeianu (deceased)\n",
    "    Copyright R Landau, Oregon State Unv, MJ Paez, Univ Antioquia, \n",
    "    C Bordeianu, Univ Bucharest, 2020. \n",
    "    Please respect copyright & acknowledge our work.\"\"\"\n",
    "\n",
    "# ProjectileAir.py: Order dt^2 projectile trajectory + drag\n",
    "\n",
    "from vpython import *\n",
    "import numpy as np\n",
    "\n",
    "v0 = 22.;   angle = 34.;   g = 9.8;   kf = 0.4;   N = 25\n",
    "v0x = v0*cos(angle*pi/180.);   v0y = v0*sin(angle*pi/180.)\n",
    "T = 2.*v0y/g;   H = v0y*v0y/2./g;   R = 2.*v0x*v0y/g\n",
    "escene = graph(width=500, height=200, \n",
    "               title='<b>Projectile with & without Drag</b>',\n",
    "  xtitle='x', ytitle='y', xmax=R, xmin=-R/20.,ymax=8,ymin=0.0)\n",
    "funct = gcurve(color=color.red)\n",
    "funct1 = gcurve(color=color.blue) "
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    "def plotNumeric(k):\n",
    " vx = v0*cos(angle*pi/180.)\n",
    " vy = v0*sin(angle*pi/180.)\n",
    " x = 0.0\n",
    " y = 0.0\n",
    " dt =  vy/g/N/2. \n",
    " for i in range(3*N):\n",
    "    vx = vx - k*vx*dt\n",
    "    vy = vy - g*dt - k*vy*dt\n",
    "    x = x + vx*dt\n",
    "    y = y + vy*dt\n",
    "    funct.plot(pos=(x,y)) \n",
    "\n",
    "def plotAnalytic():\n",
    "    v0x = v0*cos(angle*pi/180.)\n",
    "    v0y = v0*sin(angle*pi/180.)\n",
    "    dt = 2.*v0y/g/N)\n",
    "    for i in range(N):\n",
    "        rate(30)\n",
    "        t = i*dt\n",
    "        x = v0x*t\n",
    "        y = v0y*t -g*t*t/2.\n",
    "        funct1.plot(pos=(x,y))\n",
    "         \n",
    "kf = 0.3        \n",
    "plotNumeric(kf)\n",
    "plotAnalytic()"
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