ballistics.ipynb

{
 "metadata": {
  "name": ""
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Ballistic Trajectory Modelling"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Zero Drag"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "When drag is not considered, the equations become quite simple. Gravity acts on the mass of the object in the $z$ direction. In the $x$ direction, the object maintains its initial horizontal speed without slowing down. "
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The horizontal velocity component is found by:\n",
      "\n",
      "$$v_x = v\\cos\\theta$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Similarly, the vertical component of the velocity is found by:\n",
      "\n",
      "$$v_z = v\\sin\\theta$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "where $\\theta$ is the initial angle of the store relative to the horizontal plane."
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Zero initial height"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "At time $t$, the horizontal distance is found by:\n",
      "\n",
      "$$x(t) = vt\\cos\\theta$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Similarly, the vertical distance travelled is found by:\n",
      "\n",
      "$$z(t) = vt\\sin\\theta - \\frac{1}{2}gt^2$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Since we have assumed flat ground $\\left(z_0 = 0\\right)$, the starting and ending values will be the same. We can solve for $t$:\n",
      "\n",
      "$$\n",
      "\\begin{align}\n",
      "z & = vt\\sin\\theta - \\frac{1}{2}gt^2 \\\\\\\\\n",
      "z & = t\\left(v\\sin\\theta - \\frac{1}{2}gt\\right) \\\\\\\\\n",
      "\\end{align}\n",
      "$$\n",
      "\n",
      "So, $t=0$ or $t = \\frac{2v\\sin\\theta}{g}$\n",
      "\n",
      "Substituting $t$ into the equation for $x$ yields:\n",
      "\n",
      "$$x = \\frac{2v^2\\sin\\theta\\cos\\theta}{g}$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Simplifying:\n",
      "\n",
      "$$x = \\frac{v^2\\sin2\\theta}{g}$$"
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Trajectory from a starting height"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "If the start point from the trajectory is at a height other than zero, the equations of motion change slightly. $x(t)$ is unaffected, but for $z(t)$ an initial $z$ component is introduced:\n",
      "\n",
      "$$x(t) = vt\\cos\\theta$$\n",
      "$$z(t) = z_0 + vt\\sin\\theta - \\frac{1}{2}gt^2$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Setting $z=0$ we can solve for $t$:"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$$\n",
      "\\begin{align}\n",
      "0 & = z_0 + vt\\sin\\theta - \\frac{1}{2}gt^2 \\\\\\\\\n",
      "t & = \\frac{- v\\sin\\theta \\pm \\sqrt{v^2\\sin^2\\theta + 2gz_0}}{-g} \\\\\\\\\n",
      "t & = \\frac{v\\sin\\theta}{g} \\pm \\frac{\\sqrt{v^2\\sin^2\\theta + 2gz_0}}{g} \\\\\\\\\n",
      "t & = \\frac{v\\sin\\theta}{g} + \\frac{\\sqrt{v^2\\sin^2\\theta + 2gz_0}}{g} \\\\\\\\\n",
      "\\end{align}\n",
      "$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Solving for $x$:"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$$\n",
      "\\begin{align}\n",
      "x & = vt\\cos\\theta \\\\\\\\\n",
      "x & = v\\cos\\theta\\left[\\frac{v\\sin\\theta}{g} + \\frac{\\sqrt{v^2\\sin^2\\theta + 2gz_0}}{g}\\right] \\\\\\\\\n",
      "x & = \\frac{v\\cos\\theta}{g}\\left[v\\sin\\theta + \\sqrt{v^2\\sin^2\\theta + 2gz_0}\\right] \\\\\\\\\n",
      "\\end{align}\n",
      "$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let's try plotting something. First, we need functions that calculate time and position."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def calc_time(v, z, theta):\n",
      "    \"\"\"Calculate time of flight.\"\"\"\n",
      "    t = v * sin(deg2rad(theta)) / 9.81 + (sqrt(v ** 2 * sin(deg2rad(theta)) ** 2 + 2 * 9.81 * z) / 9.81)\n",
      "    return t"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def calc_distance(v, z, theta):\n",
      "    \"\"\"Calculate distance of flight.\"\"\"\n",
      "    x = v * cos(deg2rad(theta)) / 9.81 * (v * sin(deg2rad(theta)) + sqrt(v ** 2 * sin(deg2rad(theta)) ** 2 + 2 * 9.81 * z))\n",
      "    return x"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def calculate_x(v, t, theta):\n",
      "    \"\"\"Calculate x at time t.\"\"\"\n",
      "    x = v * t * cos(deg2rad(theta))\n",
      "    return x\n",
      "    "
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def calculate_z(v, t, theta, z0):\n",
      "    \"\"\"Calculate z at time t, given initial height z0.\"\"\"\n",
      "    z = z0 + v * t * sin(deg2rad(theta)) - 0.5 * 9.81 * t ** 2\n",
      "    return z"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 4
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "v = 250\n",
      "z0 = 5000\n",
      "theta = -10\n",
      "time_of_flight = calc_time(v, z0, theta)\n",
      "time = arange(0, time_of_flight, 0.1)\n",
      "x = []\n",
      "z = []\n",
      "\n",
      "for t in time:\n",
      "    x.append(calculate_x(v, t, theta))\n",
      "    z.append(calculate_z(v, t, theta, z0))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 5
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "axis(\"equal\")\n",
      "plot(x,z)\n",
      "axis()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 6,
       "text": [
        "(0.0, 7000.0, 0.0, 5000.0)"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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TCjTrRkTkG7p1sxufpaVBbCyMH+90opOj1o2IyHfYuBGysuyul4MGOZ3mu6l1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\nXkQkxP1/NJxAkOSFfbwAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7f07fc6db3d0>"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Introducing drag"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The drag of a body is defined by the following equation:\n",
      "\n",
      "$$\n",
      "D = \\frac12 C_d\\rho A v^2\n",
      "$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "where $C_d$ is the dimensionless coefficient of drag and $A$ is the cross-sectional area of the body."
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Terminal velocity"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "When a body is travelling downwards, gravity acts to increase the body's speed whilst drag acts to slow it down: \n",
      "\n",
      "$$\n",
      "F_{\\mathrm{net}} = D - mg\n",
      "$$\n",
      "\n",
      "The drag will increase until its force equals the weight of the body, making the net force zero. The velocity of the body at this point will be its terminal velocity.\n",
      "\n",
      "$$\n",
      "\\begin{align}\n",
      "F_{\\mathrm{net}} &= D - mg = 0 \\\\\\\\\n",
      "mg &= D \\\\\\\\\n",
      "mg &= \\frac{C_d \\rho A V_t^2}{2} \\\\\\\\\n",
      "V_t^2 &= \\frac{2mg}{C_d \\rho A} \\\\\\\\\n",
      "V_t &= \\left[\\frac{2mg}{C_d \\rho A}\\right]^\\frac{1}{2} \\\\\\\\\n",
      "\\end{align}\n",
      "$$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def calc_terminal_velocity(m, cd, a, rho=1.225):\n",
      "    \"\"\"Calculate the terminal velocity of a body with known drag coefficient\"\"\"\n",
      "    g = 9.81\n",
      "    vt = (2*m*g/(cd*rho*a))**0.5\n",
      "    return vt"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 7
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Motion of a falling body"
     ]
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "Calculating change in speed"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Two forces act on a falling body: gravity and drag. Gravity acts to accelerate the body towards the earth and its force is defined as:\n",
      "\n",
      "$$\n",
      "F_g = mg\n",
      "$$\n",
      "\n",
      "Drag acts on the body to slow it down. As described above it is a function of $v^2$, but to simplify things at first we will assume it is proportional but opposite to the velocity:\n",
      "\n",
      "$$\n",
      "F_d = -kv\n",
      "$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The net force acting on the body is the sum of the forces due to gravity and drag. We assume forces acting upwards to be positive and so:\n",
      "\n",
      "$$\n",
      "F_\\mathrm{net} = -F_g + F_d = -mg - kv\n",
      "$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The force acting on the body results in an acceleration $a$. Acceleration is equal to the rate of change of velocity, therefore:\n",
      "\n",
      "$$\n",
      "a = \\frac{\\mathrm{d}v}{\\mathrm{d}t}\n",
      "$$\n",
      "\n",
      "The net force is associated with a net acceleration, therefore we have a differential equation that relates speed and acceleration:\n",
      "\n",
      "$$\n",
      "\\begin{align}\n",
      "ma &= -mg - kv \\\\\n",
      "\\frac{\\mathrm d v}{\\mathrm d t} m &= -mg - kv\n",
      "\\end{align}\n",
      "$$\n",
      "\n",
      "Rearranging in terms of $\\frac{\\mathrm d v}{\\mathrm d t}$:\n",
      "\n",
      "$$\n",
      "\\begin{align}\n",
      "\\frac{\\mathrm d v}{\\mathrm d t} &= -g - \\frac{k}{m}v \\\\\n",
      "\\frac{\\mathrm d v}{\\mathrm d t} &= -\\frac k m \\left[ \\frac m k g + v \\right]\n",
      "\\end{align}\n",
      "$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let an auxilliary function $p$ be defined such that $p=\\frac k m$ and:\n",
      "\n",
      "$$\n",
      "\\frac{\\mathrm d v}{\\mathrm d t} = -g -pv\n",
      "$$\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This equation can be integrated in a fairly straightforward manner:\n",
      "\n",
      "$$\n",
      "\\begin{align}\n",
      "\\frac{\\mathrm d v}{\\mathrm d t} &= -g - pv \\\\\n",
      "\\int \\frac{1}{g + pv} \\mathrm d v &= -\\int \\mathrm d t \\\\\n",
      "\\end{align}\n",
      "$$\n",
      "\n",
      "Letting $u = g + pv$ so that $\\frac{\\mathrm d u}{p} = \\mathrm d v$:\n",
      "\n",
      "$$\n",
      "\\begin{align}\n",
      "\\int \\frac{1}{u} \\mathrm d v &= -\\int \\mathrm d t \\\\\n",
      "\\int \\frac{1}{u} \\frac 1 p \\mathrm d u &= -\\int \\mathrm d t \\\\\n",
      "\\frac 1 p \\int \\frac{1}{u} \\mathrm d u &= -\\int \\mathrm d t \\\\\n",
      "\\frac 1 p \\ln | u | &= -t + C \\\\\n",
      "\\frac 1 p \\ln | g + pv | &= -t + C \\\\\n",
      "\\ln | g + pv | &= -pt + C \\\\\n",
      "| g + pv | &= e^{-pt + C} \\\\\n",
      "| g + pv | &= e^Ce^{-pt} \\\\\n",
      "g + pv &= Ce^{-pt}\n",
      "\\end{align}\n",
      "$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "_TODO: Determine why we can disregard the modulus at this point._"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Rearranging in terms of $v$:\n",
      "\n",
      "$$\n",
      "\\begin{align}\n",
      "pv &= Ce^{-pt} - g\\\\\n",
      "v &= \\frac C p e^{-pt} - \\frac g p \\\\\n",
      "v &= Ce^{-pt} - \\frac g p\n",
      "\\end{align}\n",
      "$$\n",
      "\n",
      "Assuming the falling body has an initial vertical velocity $v_0$ we can solve for $C$. \n",
      "\n",
      "$v\\left(0\\right)=v_0$, so:\n",
      "\n",
      "$$\n",
      "\\begin{align}\n",
      "v_0 &= Ce^0 - \\frac g p \\\\\n",
      "C &= v_0 + \\frac g p\n",
      "\\end{align}\n",
      "$$\n",
      "\n",
      "Finally, the solution to the differential equation is:\n",
      "\n",
      "$$\n",
      "v = \\left(v_0 + \\frac g p \\right) e^{-pt} - \\frac g p \n",
      "$$\n",
      "\n",
      "And if we replace $p$ with $\\frac k m$ we have:\n",
      "\n",
      "$$\n",
      "v = \\left(v_0 + \\frac{mg}{k} \\right) e^{-\\frac k m t} - \\frac{mg}{k}\n",
      "$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We can now plot velocity (and speed) against time for some nominal falling object. First we will import some _nice_ colours from [colorbrewer2](http://colorbrewer2.org/) using `brewer2mpl`."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import brewer2mpl\n",
      "my_map = brewer2mpl.get_map(\"Paired\", \"Qualitative\", 4)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 8
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "m = 0.6\n",
      "g = 9.81\n",
      "k = 0.5\n",
      "v0 = 0\n",
      "\n",
      "t = arange(0, 10, 0.1)\n",
      "\n",
      "v = [(v0 + m*g/k)*exp(-k/m*i) - m*g/k for i in t]\n",
      "speed = [abs(i) for i in v]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 9
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "fig, (ax1, ax2) = plt.subplots(2,1, sharex=True)\n",
      "\n",
      "colour1 = my_map.hex_colors[0]\n",
      "colour2 = my_map.hex_colors[1]\n",
      "\n",
      "ax1.plot(t, v, lw=2, color=colour1)\n",
      "ax1.set_ylabel(\"Velocity [$ms^{-1}$]\")\n",
      "\n",
      "ax2.plot(t, speed, lw=2, color=colour2)\n",
      "ax2.set_ylabel(\"Speed [$ms^{-1}$]\")\n",
      "ax2.set_xlabel(\"Time [$s$]\")\n",
      "\n",
      "fig.tight_layout()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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swZo1a/DQQw9h8+bNSElJaXYcQcC5M/Pai0Im4p7oAAQVVuJUjg4XCypRXGXAqOgAeCs7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\nInJbTFJEROS2mKSIiMht/X+rBHzzJr+HLwAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7f07fc689d50>"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Note that the velocity becomes more negative and is asymptotic towards a value slightly less than $-12\\ ms^{-1}$. This is the terminal velocity of the object, although it cannot be compared to the terminal velocity equation derived earlier as we are using linear drag."
     ]
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "Calculating change in height"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We know that $v = \\frac{\\mathrm d z}{\\mathrm d t}$, so we could write:\n",
      "\n",
      "$$\n",
      "\\frac{\\mathrm d z}{\\mathrm d t} = \\left(v_0 + \\frac{mg}{k} \\right) e^{-\\frac k m t} - \\frac{mg}{k}\n",
      "$$\n",
      "\n",
      "Integration of this equation should yield $z$:\n",
      "\n",
      "$$\n",
      "\\begin{align}\n",
      "\\int \\mathrm d z &= \\int \\left[ \\left(v_0 + \\frac{mg}{k} \\right) e^{-\\frac k m t} - \\frac{mg}{k}\\right]\\mathrm d t \\\\\n",
      "z &= -\\frac m k \\left(v_0 + \\frac{mg}{k} \\right) e^{-\\frac k m t} -\\frac{mg}{k}t + C\n",
      "\\end{align}\n",
      "$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Assuming that $z(0)$ = $z_0$ we solve for $C$:\n",
      "\n",
      "$$\n",
      "\\begin{align}\n",
      "z_0 &= -\\frac m k \\left(v_0 + \\frac{mg}{k} \\right) e^{0} - 0 + C \\\\\n",
      "z_0 &= -\\frac m k \\left(v_0 + \\frac{mg}{k} \\right) + C \\\\\n",
      "C &= z_0 + \\frac{m}{k} \\left(v_0 + \\frac{mg}{k} \\right)\n",
      "\\end{align}\n",
      "$$\n",
      "\n",
      "And our final equation becomes:\n",
      "\n",
      "$$\n",
      "\\begin{align}\n",
      "z &= - \\frac m k \\left(v_0 + \\frac{mg}{k} \\right) e^{-\\frac k m t} - \\frac{mg}{k}t + z_0 + \\frac{m}{k} \\left(v_0 + \\frac{mg}{k} \\right) \\\\\n",
      "z &= z_0 - \\frac{mg}{k}t + \\frac m k \\left(v_0 + \\frac{mg}{k} \\right) \\left(1 - e^{-\\frac k m t} \\right)\n",
      "\\end{align}\n",
      "$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We can plot the change in height over time as follows."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "m = 0.6\n",
      "g = 9.81\n",
      "k = 0.5\n",
      "v0 = 0\n",
      "z0 = 100\n",
      "\n",
      "t = arange(0, 10, 0.1)\n",
      "\n",
      "z = [z0 - m*g*i/k + m/k*(v0 + m*g/k)*(1 - exp(-k/m*i)) for i in t]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 11
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "colour3 = my_map.hex_colors[2]\n",
      "fig, ax = plt.subplots(1)\n",
      "ax.plot(t, z, lw=2, color=colour3)\n",
      "ax.set_ylabel(\"Height [$m$]\")\n",
      "ax.set_xlabel(\"Time [$s$]\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 12,
       "text": [
        "<matplotlib.text.Text at 0x7f07fc497690>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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Mj51HcdpDJIVPw+3rY3/LO+w+/9/0ejrNjjfmXHc9jNHkdru55557+P73v8+q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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7f07fc4f7090>"
       ]
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "It can be seen that the exponential function only has influence at the very beginning of the trajectory. This suggests that the object reaches terminal velocity quite quickly in this case."
     ]
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "Calculating time of flight"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let us calculate the time of flight by setting $z=0$ and try to solve for $t$:\n",
      "\n",
      "$$\n",
      "\\begin{align}\n",
      "0 &= z_0 - \\frac{mg}{k}t + \\frac m k \\left(v_0 + \\frac{mg}{k} \\right) \\left(1 - e^{-\\frac k m t} \\right) \\\\\n",
      "\\end{align}\n",
      "$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "It doesn't look very friendly does it? Perhaps SymPy could help us:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import sympy\n",
      "from sympy.solvers import solve\n",
      "from sympy import symbols, exp\n",
      "sympy.init_printing()\n",
      "\n",
      "z0, m, g, k, t, v0 = symbols(\"z_0, m, g, k, t, v_0\")\n",
      "f = z0 - m*g*t/k + m/k*(v0 + m*g/k)*(1 - exp(-k*t/m))\n",
      "solution = solve(f,t)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 13
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "SymPy offers an algebraic solver, so what does our apparent solution look like?"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "solution"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\begin{bmatrix}\\frac{1}{g k m} \\left(g m^{2} \\left(\\operatorname{LambertW}{\\left (\\frac{1}{g m} \\left(- g m - k v_{0}\\right) e^{\\frac{1}{g m^{2}} \\left(- g m^{2} - k^{2} z_{0} - k m v_{0}\\right)} \\right )} + 1\\right) + k^{2} z_{0} + k m v_{0}\\right)\\end{bmatrix}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 14,
       "text": [
        "\u23a1     \u239b        \u239b                    2    2            \u239e    \u239e                 \u23a4\n",
        "\u23a2     \u239c        \u239c               - g\u22c5m  - k \u22c5z\u2080 - k\u22c5m\u22c5v\u2080\u239f    \u239f                 \u23a5\n",
        "\u23a2     \u239c        \u239c               \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f    \u239f                 \u23a5\n",
        "\u23a2     \u239c        \u239c                            2         \u239f    \u239f                 \u23a5\n",
        "\u23a2     \u239c        \u239c                         g\u22c5m          \u239f    \u239f                 \u23a5\n",
        "\u23a2   2 \u239c        \u239c(-g\u22c5m - k\u22c5v\u2080)\u22c5\u212f                       \u239f    \u239f    2            \u23a5\n",
        "\u23a2g\u22c5m \u22c5\u239c        \u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f + 1\u239f + k \u22c5z\u2080 + k\u22c5m\u22c5v\u2080\u23a5\n",
        "\u23a2     \u239c        \u239d                 g\u22c5m                  \u23a0    \u239f                 \u23a5\n",
        "\u23a2     \u239c                                                    \u239f                 \u23a5\n",
        "\u23a2     \u239dLambertW                                            \u23a0                 \u23a5\n",
        "\u23a2\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u23a5\n",
        "\u23a3                                   g\u22c5k\u22c5m                                    \u23a6"
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "I don't think I could have worked that out manually... What happens if we substitute our nominal values into the solution for $t$? We should be given time of flight in seconds:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "solution[0].subs([(g,9.81),(k,0.5),(m,0.6),(v0,0),(z0,100)])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$9.69436115228259$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAJ0AAAASCAYAAABBwNzbAAAABHNCSVQICAgIfAhkiAAABYFJREFU\naIHt2VuMXlUVB/DflML0AuVaWkwJLVAvCIZpqFEiBAw+cHkgjRGIhIAPCDFGNCUisbWxBSskhWAi\nARL5BEILJdAHSBRCEEykXoIXMAQhIlAUqBFRwAKh9WHtk9mz58z3nXNmAi/fPzmZOev8z1prz15n\n7bXWMMQQHzBGivsj8T3swvuYj2/jlYb6Pop1+B/eST/X4b8zYGM19sYPCvln8NVkay7m4Wr8saN/\nFeZhOz41wK9BvGewHo/gLZyAK/B1PJ3xPp3kc7EEv8FavFzoa8pbgu9gD0aTnxvxZEd9TdfRan+X\n4Z9YlcnOF5s3u+6FAsfgRZyY7hfjebGp07VxhFjoukI+hvsxJ5PdiP/g+A7+VViJ34oN64cmvD3F\n9S6+VnBW4EEckO73xWN4DUs78A7GNhyWyZaJwFneQV/TdbTe3234F2ZlsrkiYi+seyHD7LSgyzLZ\n4diJb8yAjZvFQtcV8uuS/JxMdlaS3dDBv0/gAfRE9poqmJry4IXk/324Jr1b4gEcXcjGkt4tHXiX\niwxU4kr8sIO+putotb/74D2RWks8I7JJP3xFRP4BfThdbazCueqD7gK8gS9ksoq7saV/JXoGZ7om\nvF800PGmyMKHFvLXReZoy7tRBEeJb2FTB30MXkfj/a0i8iCRDXbVvPCGOPf74Tw8h3/34XSxsS/O\nMPmrq3Ab9sdDmWyFqCU2t/Tvw8TzWCTqnxzviEzRlvcHnI07cWCSjYqjrtdBXxO03t+9RM1UF6U7\nxJc8Vc01IgryR3EyNuB6kWrHpmnjKhyVfq/LdCWW4VVc3MG/Ej0zk+kex5pk81rcKxqaHPNFjZnj\nI0nvIx14o8ZrzX+IYLsDp3e022QdnWLoJyKl5h3tYuxOL5QpuMIh6fnTuCSTnyK6wk92tHG86L4q\n9Au6s0Rn9SdRt+Q1RRv/cvTMTNA9J+rHCueLD6Pc7BIbRcY+sSNvP/zMeOF/v8hqgzCVvibraB1D\nC/EXMX4gonIDnkgvHDyFk4vS810mp+QdJtZqTW3Mwk9FnVChSaabLY7a7SLY2vqXo2dmgm5Wcb8X\n3jax0SlxtKi3Ngyw3Y93BW7Bmfir8ax3XEd9TdbRKYYOFDOWTYl8BH4njqdypldh76TwqZpn20V9\nMNrSxqU4tdDVJOiII3QP7unoX4WemQm6OryQrjqMiiNq0xTPm/C+KTrTCvNFp79b/fyyjd0cdevo\nEkOT8BIeHsB5Db+skT8qNuSwmmdT2ViMH9Vw6oLu4yYPZRck7m7RiHT1r2f6QffYFHZ3qC+4R0Tx\n//0BNvvxRkT3eWzNs0uSr+W4Y5DdtusoMSGGypRZYqGYUm8dwMuPsxyjIpPsbGHjNBFM27Kr+mrP\nTferRHD9XqTuozJ976efI+IImK5/08GY8cDPcYj6TLde1J5rM9kFLXkLxWjo2Zr3bhKD8zmFfJDd\ntuvI0TeGLhMZYUkmWy2iuXRyuYn10XnifM9lI2JEcXdHGzmWmpzp5oi50LOiXa9wQuL+uoN/OXqm\nn+m2mlzHVMPXNYX8IvWZ5uaWvBHxL6fP1XD2E3/rvJxoYrfpOlrv73fxN+PdyJjoTsou5mRxdP08\nk80SLfXqTPYlkUGWdrBRYrlY4FWF/Goxec9rhdtFV7qig385tiSb8wb41o+3UtSW1bMRMVv8lYkb\n/3nR9d1RXFtMnDc25Z2DPxdrWyCC54sd9DVdR+v9nSvmL7cmAw/hszW8jyVFPy7kB4mvfquoDzab\neOy1sVFhgZgX/d14B/q4GHxWuDDZuj3pu8vkOVhT/w4VH9NTxkcNO5MPX+7Ag5OSzdtEabDe5K/+\n9UxPea3vwCOSwzbx99iSfDilo92m62i7v0MMMcQQQwwxxMzg/+M4LeP5cAkLAAAAAElFTkSuQmCC\n",
       "prompt_number": 15,
       "text": [
        "9.69436115228259"
       ]
      }
     ],
     "prompt_number": 15
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Recall earlier the `calc_time` function for time of flight with no drag. If we enter the same parameters (in fact $z$ is the only required parameter), we get the following answer:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "calc_time(0, 100, 0)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$4.51523640986$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAH8AAAASCAYAAABy1wGXAAAABHNCSVQICAgIfAhkiAAABeVJREFU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\nVrxwW+A3NrzX6GQdwxjGMIYxjP9//A8dhbL+/z5pjAAAAABJRU5ErkJggg==\n",
       "prompt_number": 16,
       "text": [
        "4.51523640986"
       ]
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This suggests that drag is causing the object to take twice as long to reach the ground. Perhaps that is reasonable, but it is hard to tell at this stage. What we can do, however, is substitute the value for $t$ back into our original expression:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f.subs([(g, 9.81), (k, 0.5), (m, 0.6), (v0, 0), (t, 9.69436115228259)])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$z_{0} - 100.0$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAE8AAAAYCAYAAAC7v6DJAAAABHNCSVQICAgIfAhkiAAAAppJREFU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\n",
       "prompt_number": 17,
       "text": [
        "z\u2080 - 100.0"
       ]
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "So $z_0 = 100$, which is what we expect. I think we'll take our solution for $t$ as valid for now.\n",
      "\n",
      "Let's try to write a couple of functions to help with future work. First, the velocity equation:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def vz_drag(t, m, k, z0, vz0=0):\n",
      "    \"\"\"Calculate vertical velocity at time t.\n",
      "    \n",
      "    Parameters\n",
      "    ----------\n",
      "    t : float\n",
      "        Time in seconds\n",
      "    m : float\n",
      "        Mass in kilograms\n",
      "    k : float\n",
      "        Drag coefficient in Ns/m\n",
      "    z0 : float\n",
      "        Height above ground at t = 0\n",
      "    vz0 : float, optional\n",
      "        Initial vertical velocity\n",
      "    \n",
      "    Returns\n",
      "    -------\n",
      "    vz : float\n",
      "        Vertical velocity at time t\n",
      "    \n",
      "    \"\"\"\n",
      "    g = 9.81\n",
      "    vz = (vz0 + m*g/k) * exp(-k/m*t) - m*g/k\n",
      "    return vz\n",
      "    "
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 18
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Next, we will define a function to calculate the time of flight of an object. Let's get the equation printed as source code to minimise errors:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print(solution[0])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "(g*m**2*(LambertW((-g*m - k*v_0)*exp((-g*m**2 - k**2*z_0 - k*m*v_0)/(g*m**2))/(g*m)) + 1) + k**2*z_0 + k*m*v_0)/(g*k*m)\n"
       ]
      }
     ],
     "prompt_number": 19
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "And now define the function:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def calc_time_drag(m, k, z0, vz0=0):\n",
      "    \"\"\"Calculate time of flight from initial conditions.\n",
      "    \n",
      "    Parameters\n",
      "    ----------\n",
      "    m : float\n",
      "        Mass in kilograms\n",
      "    k : float\n",
      "        Drag coefficient in Ns/m\n",
      "    z0 : float\n",
      "        Height above ground at t = 0\n",
      "    vz0 : float, optional\n",
      "        Initial vertical velocity\n",
      "    \n",
      "    Returns\n",
      "    -------\n",
      "    t : float\n",
      "        Time of flight in seconds\n",
      "        \n",
      "    \"\"\"\n",
      "    from sympy import LambertW\n",
      "    g = 9.81\n",
      "    t = (g*m**2*(LambertW((-g*m - k*vz0)*exp((-g*m**2 - k**2*z0 - k*m*vz0)/\n",
      "        (g*m**2))/(g*m)) + 1) + k**2*z0 + k*m*vz0)/(g*k*m)\n",
      "    return float(t)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 20
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Checking it works as above:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "calc_time_drag(0.6, 0.5, 100)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$9.694361152282587$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAKgAAAASCAYAAADPGpWnAAAABHNCSVQICAgIfAhkiAAABeFJREFU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\nDznFGtOsF1BAAQUUUEAB/0/8A1lAVpbxdcsQAAAAAElFTkSuQmCC\n",
       "prompt_number": 21,
       "text": [
        "9.694361152282587"
       ]
      }
     ],
     "prompt_number": 21
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now we'll define a function for $z(t)$:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def z_drag(t, m, k, z0, vz0=0):\n",
      "    \"\"\"Calculate height z at time t.\n",
      "    \n",
      "    Parameters\n",
      "    ----------\n",
      "    t : float\n",
      "        Time in seconds\n",
      "    m : float\n",
      "        Mass in kilograms\n",
      "    k : float\n",
      "        Drag coefficient in Ns/m\n",
      "    z0 : float\n",
      "        Height above ground at t = 0\n",
      "    vz0 : float, optional\n",
      "        Initial vertical velocity\n",
      "        \n",
      "    Returns\n",
      "    -------\n",
      "    z : float\n",
      "        Height above ground in metres\n",
      "    \n",
      "    \"\"\"\n",
      "    g = 9.81\n",
      "    z = z0 - m*g*t/k + m/k*(vz0 + m*g/k)*(1 - exp(-k/m*t))\n",
      "    return z"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 22
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "Putting it all together"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We have a functions for the change in velocity and height with respect to time, and a means to calculate the time of flight. Can we now calculate the forward throw of an object with an initial horizontal velocity?"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Recall the original equation we had:\n",
      "\n",
      "$$\n",
      "x(t) = vt\\cos\\theta\n",
      "$$\n",
      "\n",
      "We know that $v$ changes as a function of time, due to drag. But we can't use our recently derived equation $\\left(v = \\left(v_0 + \\frac{mg}{k} \\right) e^{-\\frac k m t} - \\frac{mg}{k}\\right)$, since in the $x$ axis gravity has no effect. We must derive a new equation:\n",
      "\n",
      "$$\n",
      "\\begin{align}\n",
      "ma &= -F_d \\\\\n",
      "ma &= -kv \\\\\n",
      "\\frac{\\mathrm{d}v}{\\mathrm{d}t} &= -\\frac k m v \\\\\n",
      "\\int \\frac 1 v \\mathrm{d}v &= -\\frac k m \\int \\mathrm{d}t \\\\\n",
      "\\ln |v | &= -\\frac k m t + C \\\\\n",
      "|v| &= Ce^{-\\frac k m t}\n",
      "\\end{align}\n",
      "$$\n",
      "\n",
      "Setting our initial conditions $v(0) = v_0$:\n",
      "\n",
      "$$\n",
      "\\begin{align}\n",
      "v_0 &= Ce^0 \\\\\n",
      "C &= v_0\n",
      "\\end{align}\n",
      "$$\n",
      "\n",
      "Therefore our final equation for velocity in the $x$ axis, which we will denote as $v_x$ becomes:\n",
      "\n",
      "$$\n",
      "v_x = v_{x_0}e^{-\\frac k m t}\n",
      "$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let's visualise this to see if it is sensible:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "m = 0.6\n",
      "g = 9.81\n",
      "k = 0.5\n",
      "vx0 = 100\n",
      "\n",
      "t = arange(0, 10, 0.1)\n",
      "\n",
      "vx = [vx0 * exp(-k / m * i) for i in t]\n",
      "speed = [abs(i) for i in vx]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 23
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "fig, (ax1, ax2) = plt.subplots(2,1, sharex=True)\n",
      "\n",
      "colour1 = my_map.hex_colors[0]\n",
      "colour2 = my_map.hex_colors[1]\n",
      "\n",
      "ax1.plot(t, vx, lw=2, color=colour1)\n",
      "ax1.set_ylabel(\"Velocity [$ms^{-1}$]\")\n",
      "\n",
      "ax2.plot(t, speed, lw=2, color=colour2)\n",
      "ax2.set_ylabel(\"Speed [$ms^{-1}$]\")\n",
      "ax2.set_xlabel(\"Time [$s$]\")\n",
      "\n",
      "fig.tight_layout()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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lcd999/HGG2+0e/rvnd0F2Gwyg70QQvQGiq0X/EZXFIWxv9pKZW0z7y+YwF3x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\nLz4fHx/Atb34+gJFUVgw6Xb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9/kLd9shFvLf21VdfsWbNGkaNGkVycjLQ0iX2u9/9rpsr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\nSk51ExUbAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7f07f29f4950>"
       ]
      }
     ],
     "prompt_number": 24
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Our initial velocity is positive in the positive $x$ direction, so both speed and velocity tend toward zero. In other words, drag acts to slow the object down in the horizontal direction, which is what we expect.\n",
      "\n",
      "We can write a function to calculate this:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def vx_drag(t, m, k, z0, vx0=0):\n",
      "    \"\"\"Calculate horizontal velocity at time t.\n",
      "    \n",
      "    Parameters\n",
      "    ----------\n",
      "    t : float\n",
      "        Time in seconds\n",
      "    m : float\n",
      "        Mass in kilograms\n",
      "    k : float\n",
      "        Drag coefficient in Ns/m\n",
      "    z0 : float\n",
      "        Height above ground at t = 0\n",
      "    vx0 : float, optional\n",
      "        Initial horizontal velocity in m/s\n",
      "    \n",
      "    Returns\n",
      "    -------\n",
      "    v : float\n",
      "        Horizontal velocity at time t\n",
      "    \n",
      "    \"\"\"\n",
      "    g = 9.81\n",
      "    vx = vx0 * exp(-k / m * t)\n",
      "    return vx\n",
      "    "
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 25
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "To find position $x$ at time $t$, we need to integrate our equation for $v_x$:\n",
      "\n",
      "$$\n",
      "\\begin{align}\n",
      "v_x &= v_{x_0}e^{-\\frac k m t} \\\\\n",
      "\\frac{\\mathrm{d}x}{\\mathrm{d}t} &= v_{x_0}e^{-\\frac k m t} \\\\\n",
      "\\int \\mathrm{d}x &= v_{x_0} \\int e^{-\\frac k m t} \\mathrm{d}t \\\\\n",
      "x &= v_{x_0}\\left(-\\frac m k e^{-\\frac k m t} + C \\right) \\\\\n",
      "x &= -\\frac{m}{k}v_{x_0} e^{-\\frac k m t} + C \\\\\n",
      "\\end{align}\n",
      "$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Setting initial conditions of $x(0) = x_0$:\n",
      "\n",
      "$$\n",
      "\\begin{align}\n",
      "x_0 &= -\\frac{m}{k} v_{x_0}e^0 + C \\\\\n",
      "x_0 &= -\\frac{m}{k} v_{x_0} + C \\\\\n",
      "C &= x_0 + \\frac{m}{k} v_{x_0}\n",
      "\\end{align}\n",
      "$$\n",
      "\n",
      "Therefore,\n",
      "\n",
      "$$\n",
      "x = x_0 + \\frac{m}{k} v_{x_0} - \\frac m k v_{x_0}e^{-\\frac k m t}\n",
      "$$\n",
      "\n",
      "And we'll put this into a function as well:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def x_drag(t, m, k, x0=0, vx0=0):\n",
      "    \"\"\"Calculate forward throw x at time t.\n",
      "    \n",
      "    Parameters\n",
      "    ----------\n",
      "    t : float\n",
      "        Time in seconds\n",
      "    m : float\n",
      "        Mass in kilograms\n",
      "    k : float\n",
      "        Drag coefficient in Ns/m\n",
      "    x0 : float, optional\n",
      "        Initial x position at t = 0\n",
      "    vx0 : float, optional\n",
      "        Initial horizontal velocity\n",
      "        \n",
      "    Returns\n",
      "    -------\n",
      "    x : float\n",
      "        Forward throw in metres\n",
      "    \n",
      "    \"\"\"\n",
      "    g = 9.81\n",
      "    x = x0 + m*vx0/k - m/k*vx0*exp(-k/m*t)\n",
      "    return x"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 26
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let's assume our object has an initial velocity $v_0$ that has a direction at an angle $\\theta$ to the horizontal. If we decompose this vector we have two components:\n",
      "\n",
      "$$\n",
      "v_{x_0} = v \\cos \\theta \\\\\n",
      "v_{z_0} = v \\sin \\theta\n",
      "$$\n",
      "\n",
      "So we should have all we need to be able to define a function that calculates the trajectory of an object."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import brewer2mpl\n",
      "import numpy as np\n",
      "from matplotlib import pyplot as plt\n",
      "\n",
      "my_map = brewer2mpl.get_map(\"Paired\", \"Qualitative\", 4)\n",
      "\n",
      "def calc_traj(v0, theta, m, k, z0, x0=0):\n",
      "    \"\"\"Plot the trajectory of an object\n",
      "    \n",
      "    Parameters\n",
      "    ----------\n",
      "    v0: float\n",
      "        Initial speed in m/s\n",
      "    theta: float\n",
      "        Angle from horizontal in degrees of initial speed\n",
      "    m: float\n",
      "        Mass of object in kilograms\n",
      "    k: float\n",
      "        Drag coefficient in Ns/m\n",
      "    z0: float\n",
      "        Initial height above ground in m\n",
      "    \n",
      "    \"\"\"\n",
      "    g = 9.81\n",
      "    vx0 = v0 * np.cos(np.deg2rad(theta))\n",
      "    vz0 = v0 * np.sin(np.deg2rad(theta))\n",
      "    \n",
      "    # Calculate time of flight and generate timesteps.\n",
      "    time_of_flight = calc_time_drag(m, k, z0, vz0)\n",
      "    time = np.linspace(0, time_of_flight, 1000)\n",
      "    \n",
      "    x = [x_drag(i, m, k, x0, vx0) for i in time]\n",
      "    z = [z_drag(i, m, k, z0, vz0) for i in time]\n",
      "    vx = [abs(vx_drag(i, m, k, z0, vx0)) for i in time]\n",
      "    vz = [abs(vz_drag(i, m, k, z0, vz0)) for i in time]\n",
      "    \n",
      "    # Assign colours from ColorBrewer.\n",
      "    colour1 = my_map.hex_colors[0]\n",
      "    colour2 = my_map.hex_colors[1]\n",
      "    colour3 = my_map.hex_colors[2]\n",
      "    colour4 = my_map.hex_colors[3]\n",
      "    \n",
      "    fig, (ax1, ax2) = plt.subplots(1, 2, sharey=False)    \n",
      "    ax1.plot(x, z, lw=2, color=colour1)\n",
      "    ax1.set_ylabel(\"Height [$m$]\")\n",
      "    ax1.set_xlabel(\"Forward throw [$m$]\")\n",
      "    ax1.set_ylim(bottom=0)\n",
      "    \n",
      "    ax2.plot(time, vx, lw=2, label=\"$v_x$\", color=colour3)\n",
      "    ax2.plot(time, vz, lw=2, label=\"$v_z$\", color=colour4)\n",
      "    ax2.set_ylabel(\"Speed [$ms^{-1}$]\")\n",
      "    ax2.set_xlabel(\"Time [$s$]\")\n",
      "    ax2.legend()\n",
      "    \n",
      "    fig.tight_layout()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 27
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let's test the function with an initial velocity of 100 m/s at an angle of 5&deg; up from horizontal:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "calc_traj(100, 5, 0.5, 0.5, 100)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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cLl26hIyMDMyePRuMMXz11Vd2ubhsCwWvgoxTQGAGGE06KGTWuzZNSEe6vAr6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\nAMQ6Hq2GOCuyAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7f07fc4e1090>"
       ]
      }
     ],
     "prompt_number": 28
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "It all looks about right:\n",
      "\n",
      "* The trajectory continues upwards before rapidly dropping\n",
      "* The final trajectory is nearly vertical\n",
      "* The speed in the $x$ axis decays exponentially\n",
      "* The vertical speed initially reduces but then increases exponentially until it approaches terminal velocity"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Improvements"
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Using NumPy arrays"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The code so far has used loops in the form of list comprehensions, such as:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": [
      "x = [x_drag(i, m, k, x0, vx0) for i in time]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 29
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This is quick an easy to write, but it is also relatively slow to run. NumPy arrays can be operated on element-by-element directly. We already use arrays, such as when we generate a time array:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "time = np.linspace(0, time_of_flight, 1000)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Until now, however, we have been iterating over one time value at a time and substituting that into our equations of motion. With NumPy, we can operate directly on the array.\n",
      "\n",
      "We need to rewrite the two functions of position to use arrays. We change the time argument in the function from a specific time to the total time, and the array generated (with a default resolution of 1000 points) is passed directly to our functions for position:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def x_drag_array(t, m, k, x0=0, vx0=0, res=1000):\n",
      "    \"\"\"Calculate horizontal trajectory over t seconds.\n",
      "    \n",
      "    Parameters\n",
      "    ----------\n",
      "    t : float\n",
      "        Total time in seconds\n",
      "    m : float\n",
      "        Mass in kilograms\n",
      "    k : float\n",
      "        Drag coefficient in Ns/m\n",
      "    x0 : float, optional\n",
      "        Initial x position at t = 0\n",
      "    vx0 : float, optional\n",
      "        Initial horizontal velocity\n",
      "    res : integer, optional\n",
      "        Resolution of result\n",
      "        \n",
      "    Returns\n",
      "    -------\n",
      "    x : ndarray\n",
      "        Horizontal position array in metres\n",
      "    \n",
      "    \"\"\"\n",
      "    g = 9.81\n",
      "    t = linspace(0, t, res)\n",
      "    x = x0 + m*vx0/k - m/k*vx0*np.exp(-k/m*t)\n",
      "    return x"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 30
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def z_drag_array(t, m, k, z0, vz0=0, res=1000):\n",
      "    \"\"\"Calculate vertical trajectory over t seconds.\n",
      "    \n",
      "    Parameters\n",
      "    ----------\n",
      "    t : float\n",
      "        Total time in seconds\n",
      "    m : float\n",
      "        Mass in kilograms\n",
      "    k : float\n",
      "        Drag coefficient in Ns/m\n",
      "    z0 : float\n",
      "        Height above ground at t = 0\n",
      "    vz0 : float, optional\n",
      "        Initial vertical velocity\n",
      "    res : integer, optional\n",
      "        Resolution of result\n",
      "        \n",
      "    Returns\n",
      "    -------\n",
      "    z : ndarray\n",
      "        Vertical position array in metres\n",
      "    \n",
      "    \"\"\"\n",
      "    g = 9.81\n",
      "    t = linspace(0, t, res)\n",
      "    z = z0 - m*g*t/k + m/k*(vz0 + m*g/k)*(1 - np.exp(-k/m*t))\n",
      "    return z"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 31
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Improving the two functions for $v_x$ and $v_z$ in the same way gives us the following."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def vx_drag_array(t, m, k, z0, vx0=0, res=1000):\n",
      "    \"\"\"Calculate horizontal velocity at time t.\n",
      "    \n",
      "    Parameters\n",
      "    ----------\n",
      "    t : float\n",
      "        Total time in seconds\n",
      "    m : float\n",
      "        Mass in kilograms\n",
      "    k : float\n",
      "        Drag coefficient in Ns/m\n",
      "    z0 : float\n",
      "        Height above ground at t = 0\n",
      "    vx0 : float, optional\n",
      "        Initial horizontal velocity in m/s\n",
      "    res : integer, optional\n",
      "        Resolution of result\n",
      "    \n",
      "    Returns\n",
      "    -------\n",
      "    vx : ndarray\n",
      "        Horizontal velocity array in m/s\n",
      "    \n",
      "    \"\"\"\n",
      "    g = 9.81\n",
      "    t = linspace(0, t, res)\n",
      "    vx = vx0 * np.exp(-k / m * t)\n",
      "    return vx"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 32
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def vz_drag_array(t, m, k, z0, vz0=0, res=1000):\n",
      "    \"\"\"Calculate vertical velocity at time t.\n",
      "    \n",
      "    Parameters\n",
      "    ----------\n",
      "    t : float\n",
      "        Total time in seconds\n",
      "    m : float\n",
      "        Mass in kilograms\n",
      "    k : float\n",
      "        Drag coefficient in Ns/m\n",
      "    z0 : float\n",
      "        Height above ground at t = 0\n",
      "    vz0 : float, optional\n",
      "        Initial vertical velocity\n",
      "    res : integer, optional\n",
      "        Resolution of result\n",
      "    \n",
      "    Returns\n",
      "    -------\n",
      "    vz : ndarray\n",
      "        Vertical velocity array in m/s\n",
      "    \n",
      "    \"\"\"\n",
      "    g = 9.81\n",
      "    t = linspace(0, t, res)\n",
      "    vz = (vz0 + m*g/k) * np.exp(-k/m*t) - m*g/k\n",
      "    return vz\n",
      "    "
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 33
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Some modification to our `calc_traj` function is also required. We no longer need to use list comprehension as we will get an array back for both $x$ and $z$ trajectories, which we can then pass to the plot function."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import brewer2mpl\n",
      "import numpy as np\n",
      "from matplotlib import pyplot as plt\n",
      "\n",
      "my_map = brewer2mpl.get_map(\"Paired\", \"Qualitative\", 4)\n",
      "\n",
      "def calc_traj_array(v0, theta, m, k, z0, x0=0):\n",
      "    \"\"\"Plot the trajectory of an object\n",
      "    \n",
      "    Parameters\n",
      "    ----------\n",
      "    v0: float\n",
      "        Initial speed in m/s\n",
      "    theta: float\n",
      "        Angle from horizontal in degrees of initial speed\n",
      "    m: float\n",
      "        Mass of object in kilograms\n",
      "    k: float\n",
      "        Drag coefficient in Ns/m\n",
      "    z0: float\n",
      "        Initial height above ground in m\n",
      "    \n",
      "    \"\"\"\n",
      "    g = 9.81\n",
      "    vx0 = v0 * np.cos(np.deg2rad(theta))\n",
      "    vz0 = v0 * np.sin(np.deg2rad(theta))\n",
      "    \n",
      "    # Calculate time of flight and generate timesteps.\n",
      "    time_of_flight = calc_time_drag(m, k, z0, vz0)\n",
      "    time = np.linspace(0, time_of_flight, 1000)\n",
      "    \n",
      "    # time_of_flight is passed directly to functions, rather than using\n",
      "    # list comprehension.\n",
      "    x = x_drag_array(time_of_flight, m, k, x0, vx0)\n",
      "    z = z_drag_array(time_of_flight, m, k, z0, vz0)\n",
      "    vx = abs(vx_drag_array(time_of_flight, m, k, z0, vx0))\n",
      "    vz = abs(vz_drag_array(time_of_flight, m, k, z0, vz0))\n",
      "    \n",
      "    # These functions still use list comprehension with the time array.\n",
      "#     vx = [abs(vx_drag(i, m, k, z0, vx0)) for i in time]\n",
      "#     vz = [abs(vz_drag(i, m, k, z0, vz0)) for i in time]\n",
      "    \n",
      "    # Assign colours from ColorBrewer.\n",
      "    colour1 = my_map.hex_colors[0]\n",
      "    colour2 = my_map.hex_colors[1]\n",
      "    colour3 = my_map.hex_colors[2]\n",
      "    colour4 = my_map.hex_colors[3]\n",
      "    \n",
      "    fig, (ax1, ax2) = plt.subplots(1, 2, sharey=False)    \n",
      "    ax1.plot(x, z, lw=2, color=colour1)\n",
      "    ax1.set_ylabel(\"Height [$m$]\")\n",
      "    ax1.set_xlabel(\"Forward throw [$m$]\")\n",
      "    ax1.set_ylim(bottom=0)\n",
      "    \n",
      "    ax2.plot(time, vx, lw=2, label=\"$v_x$\", color=colour3)\n",
      "    ax2.plot(time, vz, lw=2, label=\"$v_z$\", color=colour4)\n",
      "    ax2.set_ylabel(\"Speed [$ms^{-1}$]\")\n",
      "    ax2.set_xlabel(\"Time [$s$]\")\n",
      "    ax2.legend()\n",
      "    \n",
      "    fig.tight_layout()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 34
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "calc_traj_array(100, 5, 0.5, 0.5, 100)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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cLl26hIyMDMyePRuMMXz11Vd2ubhsCwWvgoxTQGAGGE06KGTWuzZNSEe6vAr6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\nAMQ6Hq2GOCuyAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7f07fc609950>"
       ]
      }
     ],
     "prompt_number": 35
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "So it still works. Now let's time the two approaches to see if there is a noticeable difference."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "time = np.linspace(0, 10, 1000)\n",
      "m = 1\n",
      "k = 0.5\n",
      "x0 = 0\n",
      "vx0 = 100"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 36
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%timeit x = [x_drag(i, m, k, x0, vx0) for i in time]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "1 loops, best of 3: 51.8 ms per loop\n"
       ]
      }
     ],
     "prompt_number": 37
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%timeit x = x_drag_array(10, m, k, x0, vx0)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "10000 loops, best of 3: 47 \u00b5s per loop\n"
       ]
      }
     ],
     "prompt_number": 38
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "From these two results, it is quite clear that using arrays in our functions is much faster than looping. Something to be remembered!\n",
      "\n"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Animation"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Can we animate the trajectory?"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# This doesn't work in Windows so far; Notebook must not be started with\n",
      "# --inline set if pop-up windows are required.\n",
      "%pylab qt"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Populating the interactive namespace from numpy and matplotlib\n"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stderr",
       "text": [
        "WARNING: pylab import has clobbered these variables: ['exp', 'solve', 'f']\n",
        "`%pylab --no-import-all` prevents importing * from pylab and numpy\n"
       ]
      }
     ],
     "prompt_number": 36
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import numpy as np\n",
      "import brewer2mpl\n",
      "from matplotlib import pyplot as plt\n",
      "from matplotlib import animation\n",
      "\n",
      "# Initial conditions.\n",
      "v0 = 100\n",
      "theta = 5\n",
      "m = 0.5\n",
      "k = 0.5\n",
      "z0 = 100\n",
      "x0 = 0\n",
      "dt = 1/30\n",
      "interval = dt * 1000\n",
      "\n",
      "g = 9.81\n",
      "vx0 = v0 * np.cos(np.deg2rad(theta))\n",
      "vz0 = v0 * np.sin(np.deg2rad(theta))\n",
      "\n",
      "tof = calc_time_drag(m, k, z0, vz0)\n",
      "frames = int(tof / dt)\n",
      "\n",
      "my_map = brewer2mpl.get_map(\"Paired\", \"Qualitative\", 4)\n",
      "colour1 = my_map.hex_colors[0]\n",
      "colour2 = my_map.hex_colors[1]\n",
      "colour3 = my_map.hex_colors[2]\n",
      "colour4 = my_map.hex_colors[3]\n",
      "\n",
      "fig = plt.figure()\n",
      "ax = plt.axes(xlim=(0,100), ylim=(0,120))\n",
      "line, = ax.plot([], [], \"-\", lw=2, color=colour1)\n",
      "ax.set_xlabel(\"Forward throw [$m$]\")\n",
      "ax.set_ylabel(\"Height [$m$]\")\n",
      "\n",
      "def init():\n",
      "    line.set_data([], [])\n",
      "    return line,\n",
      "\n",
      "x = x_drag_array(tof, m, k, x0, vx0, frames)\n",
      "z = z_drag_array(tof, m, k, z0, vz0, frames)\n",
      "\n",
      "def animate(i, x, z):\n",
      "    line.set_data(x[:i], z[:i])\n",
      "    return line,\n",
      "\n",
      "anim = animation.FuncAnimation(fig, animate, fargs=(x, z), init_func=init,\n",
      "                               frames=frames, interval=interval, blit=True)\n",
      "\n",
      "plt.show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x7fa72c817910>"
       ]
      }
     ],
     "prompt_number": 38
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "After a bit of trial and error, the above code will produce an animation that illustrates the trajectory in real time. Note that this will not work if IPython Notebook is started with the `--pylab=inline` argument. Removing the `inline` parameter from the notebook start-up command (or setting an alternative backend) means that plots are opened in separate windows, and this allows the animation to be displayed.\n",
      "\n",
      "The code below uses the `JSAnimation` library located [here](https://github.com/jakevdp/JSAnimation) and described [here](http://jakevdp.github.io/blog/2013/05/19/a-javascript-viewer-for-matplotlib-animations/) is designed to display animations directly in the notebook using Javascript. Unfortunately it doesn't work properly with Python 3 at the moment."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from JSAnimation import IPython_display\n",
      "\n",
      "fig = plt.figure()\n",
      "ax = plt.axes(xlim=(0, 100), ylim=(0, 120))\n",
      "line, = ax.plot([], [], lw=2, color=colour2)\n",
      "\n",
      "def init():\n",
      "    line.set_data([], [])\n",
      "    return line,\n",
      "\n",
      "x = x_drag_array(tof, m, k, x0, vx0, frames)\n",
      "z = z_drag_array(tof, m, k, z0, vz0, frames)\n",
      "\n",
      "def animate(i, x, z):\n",
      "    line.set_data(x[:i], z[:i])\n",
      "    return line,\n",
      "\n",
      "animation.FuncAnimation(fig, animate, fargs=(x, z), init_func=init,\n",
      "                               frames=frames, interval=interval)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 39,
       "text": [
        "<matplotlib.animation.FuncAnimation at 0x7fa723d19a50>"
       ]
      }
     ],
     "prompt_number": 39
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "XKCD-style"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Just for fun of course..."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import brewer2mpl\n",
      "import numpy as np\n",
      "from matplotlib import pyplot as plt\n",
      "\n",
      "try:\n",
      "    import brewer2mpl\n",
      "    # Assign colours from ColorBrewer.\n",
      "    my_map = brewer2mpl.get_map(\"Paired\", \"Qualitative\", 4)\n",
      "    colour1 = my_map.hex_colors[0]\n",
      "    colour2 = my_map.hex_colors[1]\n",
      "    colour3 = my_map.hex_colors[2]\n",
      "    colour4 = my_map.hex_colors[3]\n",
      "\n",
      "except ImportError:\n",
      "    colour1 = \"#A6CEE3\"\n",
      "    colour2 = \"#1F78B4\"\n",
      "    colour3 = \"#B2DF8A\"\n",
      "    colour4 = \"#33A02C\"\n",
      "        \n",
      "def calc_traj_xkcd(v0, theta, m, k, z0, x0=0):\n",
      "    \"\"\"Plot the trajectory of an object in an XKCD style.\n",
      "    \n",
      "    Parameters\n",
      "    ----------\n",
      "    v0: float\n",
      "        Initial speed in m/s\n",
      "    theta: float\n",
      "        Angle from horizontal in degrees of initial speed\n",
      "    m: float\n",
      "        Mass of object in kilograms\n",
      "    k: float\n",
      "        Drag coefficient in Ns/m\n",
      "    z0: float\n",
      "        Initial height above ground in m\n",
      "    \n",
      "    \"\"\"\n",
      "    g = 9.81\n",
      "    vx0 = v0 * np.cos(np.deg2rad(theta))\n",
      "    vz0 = v0 * np.sin(np.deg2rad(theta))\n",
      "    \n",
      "    # Calculate time of flight and generate timesteps.\n",
      "    time_of_flight = calc_time_drag(m, k, z0, vz0)\n",
      "    time = np.linspace(0, time_of_flight, 1000)\n",
      "    \n",
      "    # time_of_flight is passed directly to functions, rather than using\n",
      "    # list comprehension.\n",
      "    x = x_drag_array(time_of_flight, m, k, x0, vx0)\n",
      "    z = z_drag_array(time_of_flight, m, k, z0, vz0)\n",
      "    vx = abs(vx_drag_array(time_of_flight, m, k, z0, vx0))\n",
      "    vz = abs(vz_drag_array(time_of_flight, m, k, z0, vz0))\n",
      "    \n",
      "    # These functions still use list comprehension with the time array.\n",
      "#     vx = [abs(vx_drag(i, m, k, z0, vx0)) for i in time]\n",
      "#     vz = [abs(vz_drag(i, m, k, z0, vz0)) for i in time]\n",
      "                  \n",
      "    # Turn on the XKCD filter.\n",
      "    with plt.xkcd():\n",
      "        fig, (ax1, ax2) = plt.subplots(1, 2, sharey=False)    \n",
      "        ax1.plot(x, z, lw=2, color=colour1)\n",
      "        ax1.set_ylabel(\"Height, m\")\n",
      "        ax1.set_xlabel(\"Forward throw, m\")\n",
      "        ax1.set_ylim(bottom=0)\n",
      "        \n",
      "        ax2.plot(time, vx, lw=2, label=\"horizontal speed\", color=colour3)\n",
      "        ax2.plot(time, vz, lw=2, label=\"vertical speed\", color=colour4)\n",
      "        ax2.set_ylabel(\"Speed, m/s\")\n",
      "        ax2.set_xlabel(\"Time, s\")\n",
      "        ax2.legend()\n",
      "        \n",
      "        fig.tight_layout()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 39
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "calc_traj_xkcd(100, 5, 0.5, 0.5, 100)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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i09pwu911xqhEq0tNbli6dOlCx44d2blzJ/v27eO9994Dwg2vqKjA4XDQt29f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\n0xgyZIg4M7OoqAhPPfUUPvjgg6CfecePH0ezZs2wfv16AL5Zn2+88Qb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vShgMBqSnp+M/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\nwgILg8FgMCTl/wObBkqEFll0LgAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7f07fc6d8bd0>"
       ]
      }
     ],
     "prompt_number": 40
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Realistic drag"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "In reality, drag is defined as:\n",
      "\n",
      "$$\n",
      "F_d = \\frac12 C_d \\rho A v^2\n",
      "$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The net force acting on the body is the sum of the forces due to gravity and drag, defining positive forces as acting upwards:\n",
      "\n",
      "$$\n",
      "F_\\mathrm{net} = F_d - F_g = \\frac12 C_d \\rho A v^2 - mg\n",
      "$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "To simplify the equation, we'll make the subsitution $b = \\frac 1 2 C_d \\rho A $.\n",
      "\n",
      "$$\n",
      "F = bv^2 - mg\n",
      "$$\n",
      "\n",
      "As we did for linear drag above, we can form a differential equation:\n",
      "\n",
      "$$\n",
      "\\begin{align}\n",
      "F &= bv^2 - mg \\\\\n",
      "ma &= bv^2 - mg \\\\\n",
      "a = &= \\frac b m v^2 - g \\\\\n",
      "\\frac{\\mathrm d v}{\\mathrm d t} &= \\frac b m v^2 - g\n",
      "\\end{align}\n",
      "$$\n",
      "\n",
      "Let's make one more substitution of $k = \\frac b m$ and start to solve the equation:\n",
      "\n",
      "$$\n",
      "\\begin{align}\n",
      "\\frac{\\mathrm d v}{\\mathrm d t} & = kv^2 - g \\\\\n",
      "\\int \\frac{1}{kv^2 - g} \\mathrm d v &= \\int \\mathrm d t \\\\\n",
      "\\end{align}\n",
      "$$"
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Analytical solution"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The left hand side of the differential equation above is difficult to solve. Can SymPy help?"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sympy import *\n",
      "init_printing()\n",
      "k, v, g, t, C = symbols(\"k, v, g, t, C\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 41
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f = 1 / (k * v ** 2 - g) - t\n",
      "f"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$- t + \\frac{1}{- g + k v^{2}}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAGMAAAAlCAYAAABWOlfkAAAABHNCSVQICAgIfAhkiAAAAxVJREFU\naIHt2m+IFVUYx/FPq5KiYaWoiFogBKXkC/sHubALYpD0JoooE4mEwDUSQfJVLUh/iOptkYibEEEU\nCEFlvZFehBKRRCAhUiiUFhSkUia6vXiO3evNdu6dPe3uvTNfGJhzzznPPHeeOc/8zjlDTc0U5zq8\nj2WT7UjV2YxhjOLmSfWk5h8mPBh9E3mxmrHJHYzl+BE3ZbZbCXIH4wHciNMl+t6O6Xnd6S5yB6Mf\nh/Fnib7bsSSvO91F7mCsweeZbVaGHMF4BJ+IEbEAg6k8lMH2ZLABb6Tzl7F1En0pzVM4j1kl+4+o\nuK7PmaYG8SX+yGizUjSrl1Xi6bymzb5f44mm8gD2ZPGqorR744tYgW9xHz4taPu2CHwry3AKf12l\n7kl8NR4Hq8RWXMCcVL4eSzu0MaK9d8Zorx65Jln9OIKzqfwM3spku5Vco3mi6RMK87LAeaW1Qa5g\nTMMP6fxO8RL/KZPtXmE99uMkPsBqLak3l5rahcV4TaiqVzPZ7SWW49F0flznaXxCGdHb84xrxaYV\nHBAP7xVMpYW535Vb0+oWzqejHwfF6nbNOFmAfUKwdMpcPJfXnd6naD1qM14sYXcIM9KxtrWy3um7\nOvML6tfhow5tPoaXhMr8WUxwK8m9eF2omV14s6D98Bh1fTgm5Px6oRzPpPJ0Mar24q7U/h1sacfJ\nKoyMW7Abz+Nd8dR/Nw57q3ECG8V7Y6fYpbyIB9M1ZmtsPX8odj8LmUpqqhOeFrr9vzgsbgqxJ7FP\nPL3Ejdvd1HYhtrlyZr8GM5vKZ/BCOl+HW8Ui6Wl8jO9T3QExQgawKf12FL8W/6VqcFasChA6/xfF\nGWF4jLqDuFtMbn9Lv61oqt+I95rKQ2KOUUgV0tQpEQB4CIdwqaSt2VgpljGOaiz5DDa1WSjSGNyA\nc2J+Uci0kk51E8fxMBaJ1PIZvijoMyBGQCt3YJ749PMc7kl292ukwRPperM09ohGy7vfuxzSSFlj\n8ez/7UgV2SE+mIDb8I0pnJq7VU21yyKhih4XkvR+5d8XNTU1NTU1NTX/5m90TrKHwcL3twAAAABJ\nRU5ErkJggg==\n",
       "prompt_number": 42,
       "text": [
        "         1    \n",
        "-t + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "             2\n",
        "     -g + k\u22c5v "
       ]
      }
     ],
     "prompt_number": 42
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "solution_1 = integrate(f, v)\n",
      "solution_1"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$- t v + \\frac{\\sqrt{\\frac{1}{g k}}}{2} \\log{\\left (- g \\sqrt{\\frac{1}{g k}} + v \\right )} - \\frac{\\sqrt{\\frac{1}{g k}}}{2} \\log{\\left (g \\sqrt{\\frac{1}{g k}} + v \\right )}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAhUAAAAhCAYAAABukFnsAAAABHNCSVQICAgIfAhkiAAACyVJREFU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\nGr1F4lnwrgAAAABJRU5ErkJggg==\n",
       "prompt_number": 43,
       "text": [
        "           _____    \u239b        _____    \u239e       _____    \u239b      _____    \u239e\n",
        "          \u2571  1      \u239c       \u2571  1      \u239f      \u2571  1      \u239c     \u2571  1      \u239f\n",
        "         \u2571  \u2500\u2500\u2500 \u22c5log\u239c- g\u22c5  \u2571  \u2500\u2500\u2500  + v\u239f     \u2571  \u2500\u2500\u2500 \u22c5log\u239cg\u22c5  \u2571  \u2500\u2500\u2500  + v\u239f\n",
        "       \u2572\u2571   g\u22c5k     \u239d    \u2572\u2571   g\u22c5k     \u23a0   \u2572\u2571   g\u22c5k     \u239d  \u2572\u2571   g\u22c5k     \u23a0\n",
        "-t\u22c5v + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "                      2                                 2               "
       ]
      }
     ],
     "prompt_number": 43
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This doesn't look like a nice equation. Let's try to confirm whether it is correct."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "According to [Wikipedia](https://en.wikipedia.org/wiki/List_of_integrals_of_rational_functions#Integrands_of_the_form_xm_.2F_.28a_x2_.2B_b_x_.2B_c.29n), integrands of the form $\\int \\frac{1}{ax^2 + bx + c} \\mathrm d x$ have a solution depending on the result of $4ac - b^2$.\n",
      "\n",
      "In the equation above, $a=k$, $b=0$ and $c=-g$, so the expression $4ac - b^2 = -4kg$. Since this is less than zero the solution is stated to be:\n",
      "\n",
      "$$\n",
      "\\frac{1}{\\sqrt{b^2 - 4ac}} \\ln \\left| \\frac{2ax + b - \\sqrt{b^2 - 4ac}}{2ax + b + \\sqrt{b^2 - 4ac}} \\right| + C\n",
      "$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Substituting the values of our equation:\n",
      "\n",
      "$$\n",
      "\\begin{align}\n",
      "\\frac{1}{\\sqrt{4kg}} \\ln \\left| \\frac{2kv - \\sqrt{4kg}}{2kv + \\sqrt{4kg}} \\right| + C\n",
      "\\end{align}\n",
      "$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The two potential solutions so far look very different to each other. [Wolfram|Alpha](http://www.wolframalpha.com) provides yet another [solution](http://www.wolframalpha.com/input/?i=integrate+1+%2F+%28k+*+v+%5E+2+-+g%29+-t):\n",
      "\n",
      "$$\n",
      "-\\frac{\\tanh^{-1}\\left(\\frac{\\sqrt{k}v}{\\sqrt{g}}\\right)}{\\sqrt{g}\\sqrt{k}} - t + C\n",
      "$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We could try to compare these three solutions by substituting arbitrary values for $g, k, t$ and $v$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "solution_2 = 1 / sqrt(4 * k * g) * ln((2*k*v - sqrt(4*k*g)) / (2*k*v + sqrt(4*k*g))) - t*v + C\n",
      "solution_2"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$C - t v + \\frac{1}{2 \\sqrt{g k}} \\log{\\left (\\frac{2 k v - 2 \\sqrt{g k}}{2 k v + 2 \\sqrt{g k}} \\right )}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAOQAAAAzCAYAAACHbKQaAAAABHNCSVQICAgIfAhkiAAACmVJREFU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\n",
       "prompt_number": 44,
       "text": [
        "             \u239b            _____\u239e\n",
        "             \u239c2\u22c5k\u22c5v - 2\u22c5\u2572\u2571 g\u22c5k \u239f\n",
        "          log\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\n",
        "             \u239c            _____\u239f\n",
        "             \u239d2\u22c5k\u22c5v + 2\u22c5\u2572\u2571 g\u22c5k \u23a0\n",
        "C - t\u22c5v + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "                    _____       \n",
        "                2\u22c5\u2572\u2571 g\u22c5k        "
       ]
      }
     ],
     "prompt_number": 44
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "solution_3 = - atanh(sqrt(k*v) / sqrt(g)) / (sqrt(g)*sqrt(k)) - t*v + C\n",
      "solution_3"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$C - t v - \\frac{1}{\\sqrt{g} \\sqrt{k}} \\operatorname{atanh}{\\left (\\frac{\\sqrt{k v}}{\\sqrt{g}} \\right )}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAASIAAAAgCAYAAAC2AXUJAAAABHNCSVQICAgIfAhkiAAACWBJREFU\neJztnH2QV1UZxz+7sMuLKIiso4C8JKCJtWlKoZGkaGSGKTZoGpGOIyWTNGYE42QvW6aGL2XqjtVW\nZC1JMSW9EdkqWOH7oDYW2jDlmC8ViKSVKP3xvZc9e3+/e++5v3v23t9vPZ+ZnZ17f+c893nuebnn\nec5zL3g8Ho/HUxBvBY4rW4kBxBHAnLKV8HgaiWOATqDJgaw9Bf/VMyuBU8pWomSOYeC3s8cBY4A7\ngeGO5P0cGOpIVqMzGNgATCxbkRLx/cFjRTdwkiNZc4FljmQNFN6CJqPXI74/eKw4HuhxKO+nwH4O\n5Q0Ufgi8v2wlSsD3B48VdwBnOZJ1PNDhSFa9MBE3sY1ZwO+KVNzAhQ21xGgGYn/w9APjgX8BQxzJ\nux1ocyTLBWc6kNEJHOJAThOwDTgyY716siErTvtDsytBnrrjfcBG4L8OZLUDTwPPO5DlgsOBhTll\nHAS0AH/Lrw57gF8DZ2SoU282ZKHw/jAduBp16A1oFrwVmIyeAt9BN8PTv7wTLf33IL/chjXApY6u\n/11gQsLvpn5rHV0zic8DC3LK6ACOdqBLyELgrgzl69EGWwrrD/uhCeevwHn0Xd63AT8GuoDH81zE\n4FA0wzbCNmhZurYA/wEusyy/DTc5LlOAb1iUC/VzNfkl8SAwLEf94eih6pKjgZ3Y52rVow02FNYf\nJgN/RDPaATFlDkez3Y21XiTCUqR0I+QjlKXrceiez7AoOywoO9XBdTtRe6eRRb88zABW5ZRxMfBe\nB7qYjEH2j7MoW6822FBIfxgN/AnYCuyfUvZxsvnESfyIbMvaMilL1xXAiyiJLo0pqBPYBBP3Tfht\nHPB9Cxkg/XZhp18ebgDek6N+M9pNdJFlbjIY3fO3W5StVxugTvrDD9DNtEmA2wiMquUiVXgW+IIj\nWf1NWbquB35lWfZY1I6tCWWGA7cBX08osxK9o2bDevo/sW8QsIV8k92ZwIVu1KlgF0ryS6Jebaib\n/vA21HnvtiyfN9C2APglsDm47qbg+GKjzFTk/q0DzjbOLwKeAUbm1MHkRGA1Cs53odjY/Rl0tZED\ncDrwNXSfJ6OG/XLwtwk4tYrMFtTJO4ArgGuA76Gs6Wou4izkPsYxGLgcTagvUj0pbTSKBdoQ6ndF\ncDwE+BJwFYo1NuGmLU8meaDYsJZ0t7oV+CwamNcDi9FKeHFKvadI9xKKsgHs7SijP8TSiQbZhy0v\n5oqL0BZzNHDXDNyCbtIl6CkScj72y2AbLkArnfHB8SQ0kKMrkDhdbeW0og4BcB+aeC6lt2E+jTYI\nooT+9j2G7MGo05xfpfxM4DXS0zNGAy+h+xvlc8C7UupH9TsR2fhF4EDgU8BuFGt00ZZdKJGuVmYC\nn0kpMwxlo682zi1HdqRNMv8ETkspU4QNUJsdRfaHWLYGlccnFeoHutGAjDIXrR4AfoG2o03+gGIh\neWkHXgHOiZz/B3pKmMTpaivnZJTp3IQ6bdSm5cC/q8heAfyPyoS5F6i+i/Ym1JY2q4xbUVzQfEqN\nAH5mUTeq3/6ow4bxxQ7g27hpy6HAwyQ/TVtJdkdXoaByEl8FnqPvqmAumtjT6r4CzE74vSgboHY7\niugPibyMllM2mLsx7cBD6Abb/HVFZD2DZswo49ASbzzwKkrQM7kJ3ay811+HMpBbjHNvRAP5BEtd\nbeUcjJ5Ubw7Oz4rI6Kb6qwLrgz+TwwIZ1bboDwh+S8r1CGkPypqxjU8C8yzqmvptA76CnnxRbNsy\nifnAlQm/t6G45Xkxvx9Kukt0CJpMro+c7wAeS6m7D7qP0xLKFGED5LOjiP6QyLMoPyaNVjTbumA6\n8YMpZDmwnb7BvSbiJ4QsjEJLxdWR8x9FLpjphyfpmkUOKAXgJfo++VqAHcheIud3VTm/DK2I4l7h\neAF9K8aGu+l94g1BrqTtjkyo35Uo9tNFvJuVpy3XEP8Kxb5o2X8b8SvWG0ieJEDtsgd4d+T8RuDm\nlLqTUB9Iyg0qwgbIZwcU1x+q8hM0i6Z9t+Yy3GVzLgmuOSI4HkXlezMbqMzOnEPlU7UWwo86LY2c\nX01vZ3iDha5Z5IDu9W8iZeehZfMkNEGFLnLob8+MlH8Y+FYV2SF3YR/v+0Bw7SkoDnauZT1Tv7BP\nLEAdMZx8zZ3VWttyJHCvhS4zA12mR86Pxm7b+ZagvqnzUBTni7rcUU4lebVRlA2Qzw4orj/sxQxm\nXoe2FhclXOR0NOs/mEGxJGahARW6hJcE8k3GofiVyWnog0x52Rn8NwPEw5CfH7pIYeAuSdcscppR\nCnxPRJeFwG/RkvYMet3f2cE17zPKHomW0KGbWS24uAn7ZLK1aDX8cRTD6rasF+q3E90b0GS9D735\nKNcaZWtty/nY7dj8HngEDR6TxWhwprEDbQDsMM6dgFYFabljx6LNhDiKsgHy2QHF9Ye9mBNRD1oa\nXoV8VPO3NrS1NwFNWK4YhAYeqCFfBv4eKbMFGGscX4BWFK86uP7WQP6k4LgFbTEPDfQag4LNabpm\nkXMUeir0RHSZht4ja0WrhPD32aiD746U3Y4mm/kolSDKOuw/iLYbLdmXoO3dLPd2dqDHa8Hx9uD/\nTrQ7ZKaD1NqWZ6McNxs6gQ/R6yK1ItfAJi1lDfIIwrjGRHRfniQ9bHESSjKMoygbIJ8dUFx/SOQd\nyJAHkG94O0pkcvG6QJR2NMhWokmw2nbzWDRD34wmQVfZ3CHTkD98HYp9TUWrwh60gxDuMKTpaitn\nHvAolTsjH0Q7SjfR19XaTGXwcgRyc26k+moI5NP/GQXGbWgDniD7qyub6bt0b0bvIq1CsR/zPtXS\nlgeRLZN9JNp5XBQcf4S+eUtpXIgm8atRDs799LrAcYxFAzxut6toG6A2O0yK6A+e1wlLcLexUBZL\ngY9lrPNN5OKABuOgGq89HG1DL0optwK9TR9HmTaAvR0eT7/QityhRv5Uyz3Y5c2YzEAB02XAJyzr\ntFG5RX0WinGMrSy+lxEoSJ30ydSibIDa7fB4+pU56PsxjcgUsiXSmTyE4hNJL3GadKNVQ7hrfDCK\nqaRNAteS7DYVaQPUbkep5FnueRqDv6DEygloN6aRWIJyX7akFaxCExqAtrurLWjAH4bejD8HxTSS\nPtdxCvokxjUJZYq0AWqzw+MpjMvRRkQjcQe9eVv1xhEo4J4WfK1nGzwej8fj8Xg8Ho/H4/F4PJ66\n4f8D+txHwE3ivQAAAABJRU5ErkJggg==\n",
       "prompt_number": 45,
       "text": [
        "               \u239b  _____\u239e\n",
        "               \u239c\u2572\u2571 k\u22c5v \u239f\n",
        "          atanh\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\n",
        "               \u239c   ___ \u239f\n",
        "               \u239d \u2572\u2571 g  \u23a0\n",
        "C - t\u22c5v - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "             ___   ___  \n",
        "           \u2572\u2571 g \u22c5\u2572\u2571 k   "
       ]
      }
     ],
     "prompt_number": 45
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "First, the SymPy solution (with the constant of integration also added):"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "solution_1.subs([(g, 9.81), (k, 0.5), (t, 10), (v, 10)]) + C"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$C - 100.214874840009$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAALsAAAASCAYAAAADg69MAAAABHNCSVQICAgIfAhkiAAABdhJREFU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\nxBBDDDHE/wL+AyULKO/UoAfXAAAAAElFTkSuQmCC\n",
       "prompt_number": 46,
       "text": [
        "C - 100.214874840009"
       ]
      }
     ],
     "prompt_number": 46
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Second, the Wikipedia solution:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "solution_2.subs([(g, 9.81), (k, 0.5), (t, 10), (v, 10)])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$C - 100.214874840009$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAALsAAAASCAYAAAADg69MAAAABHNCSVQICAgIfAhkiAAABdhJREFU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\nxBBDDDHE/wL+AyULKO/UoAfXAAAAAElFTkSuQmCC\n",
       "prompt_number": 47,
       "text": [
        "C - 100.214874840009"
       ]
      }
     ],
     "prompt_number": 47
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "That's quite surprising, but promising. Finally, the Wolfram|Alpha solution:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "N(solution_3.subs([(g, 9.81), (k, 0.5), (t, 10), (v, 10)]))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$C - 100.404175359577$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAALwAAAASCAYAAADhX7Q1AAAABHNCSVQICAgIfAhkiAAABftJREFU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\nSUVORK5CYII=\n",
       "prompt_number": 48,
       "text": [
        "C - 100.404175359577"
       ]
      }
     ],
     "prompt_number": 48
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This one doesn't quite match. Let's try to solve for $C$ using an initial velocity of zero at time zero: "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "solution_1.subs([(g, 9.81), (k, 0.5), (t, 0), (v, 0)]) + C"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$C + 0.225761820492865 i \\pi$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAN0AAAAUCAYAAAAJIzhmAAAABHNCSVQICAgIfAhkiAAAB/RJREFU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\n",
       "prompt_number": 49,
       "text": [
        "C + 0.225761820492865\u22c5\u2148\u22c5\u03c0"
       ]
      }
     ],
     "prompt_number": 49
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "solution_2.subs([(g, 9.81), (k, 0.5), (t, 0), (v, 0)])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$C + 5.3362023969869 \\cdot 10^{-126} + 0.225761820492865 i \\pi$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAcQAAAAZCAYAAACviF5aAAAABHNCSVQICAgIfAhkiAAADDBJREFU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\ngg==\n",
       "prompt_number": 50,
       "text": [
        "C + 5.3362023969869e-126 + 0.225761820492865\u22c5\u2148\u22c5\u03c0"
       ]
      }
     ],
     "prompt_number": 50
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "N(solution_3.subs([(g, 9.81), (k, 0.5), (t, 0), (v, 0)]))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$C$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAABIAAAASCAYAAABWzo5XAAAABHNCSVQICAgIfAhkiAAAAPJJREFU\nOI3l061KhEEYxfGfCgoG8YNtFsPCgs0igs0LEJNFWG9AMCh7AQtWy4YVgzab4C0oGI2ibJFNBkFR\nQURBwzvhRXZmXtDmgYFhzsP/PDPDwx9pKOPPo4klvOMRT9jDHY7Rwn0MMIFD9LGBsZJXwymOcJPq\nYg7XuMRMpKaBL3RikGncooepVFroZi1mnoSklQwELjA5yFgMkPMKEFiPGQcB1KwIiqoXQLO/Bb3h\ntWJt/efBcGn/HFZOo9hKFZzhA+MZ0C4WUh3tYwSbCcgqPnGVCbODF8VYlENqaEtcadDQLmNbMSoP\ninfro6v42f+mb/9DLEu0inQTAAAAAElFTkSuQmCC\n",
       "prompt_number": 51,
       "text": [
        "C"
       ]
      }
     ],
     "prompt_number": 51
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Three different results. However, closer inspection of the first two suggests that they could be similar except for some floating point imprecision.\n",
      "\n",
      "However, solving the first solution for $C$ yields a different answer of $C=0$."
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Numerical solution"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Returning to the original differential equation, we have:\n",
      "\n",
      "$$\n",
      "\\frac{\\mathrm d v}{\\mathrm d t} = \\frac b m v^2 - g\n",
      "$$\n",
      "\n",
      "If we replace $b$ with the original constants, and re-write slightly, we have:\n",
      "\n",
      "$$\n",
      "\\dot{v} = \\frac{C_d \\rho A}{2m}v^2 - g\n",
      "$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Using SciPy, we can solve differential equations. First, create a function for $\\dot v$. We also define some nominal values for the constants."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from scipy.integrate import odeint\n",
      "\n",
      "cd = 0.8\n",
      "rho = 1.225\n",
      "A = 0.0025\n",
      "m = 1\n",
      "g = 9.81\n",
      "\n",
      "def dv(v, t):\n",
      "    \"\"\"Return derivative of v.\"\"\"\n",
      "    dv = (cd * rho * A) / (2 * m) * v**2 - g\n",
      "    \n",
      "    return dv"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 57
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We need to set the initial state for $v$ and $t$. For now, assume that both are zero. We will also define an array of time values."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "v0 = 0\n",
      "t = np.linspace(0, 30, 500)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now, we can solve the differential equation."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "v = odeint(dv, v0, t)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let's plot $v$ against $t$:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import brewer2mpl\n",
      "from matplotlib import pyplot as plt\n",
      "\n",
      "my_map = brewer2mpl.get_map(\"Paired\", \"Qualitative\", 4)\n",
      "\n",
      "colour1 = my_map.hex_colors[0]\n",
      "colour2 = my_map.hex_colors[1]\n",
      "colour3 = my_map.hex_colors[2]\n",
      "colour4 = my_map.hex_colors[3]\n",
      "    \n",
      "fig, ax1 = plt.subplots(1)    \n",
      "ax1.plot(t, v, lw=2, color=colour1)\n",
      "ax1.set_ylabel(\"Speed [$ms^{-1}$]\")\n",
      "ax1.set_xlabel(\"Time [$s$]\")\n",
      "\n",
      "fig.tight_layout()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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34LPrfRgYcQIAFsXpscqYwIlqiSIEQ4ohpXgeSUJ9lw1ftN2A0yNBFID81FiY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\nijRZWVlYvnw5AGD58uUoLS0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       "text": [
        "<matplotlib.figure.Figure at 0x7f58ac73e890>"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now, what if we re-write the velocity differential equation in terms of displacement? We know that $v = \\frac{\\mathrm d s}{\\mathrm d t}$ (where $s$ is simple displacement), so by substitution it becomes a second-order differential equation:\n",
      "\n",
      "$$\n",
      "\\begin{align}\n",
      "\\frac{\\mathrm d}{\\mathrm d t}\\left(\\frac{\\mathrm d s}{\\mathrm d t}\\right) &= \\frac{C_d \\rho A}{2m}\\frac{\\mathrm d s}{\\mathrm d t}^2 - g \\\\\n",
      "\\frac{\\mathrm d^2 s}{\\mathrm d t} &= \\frac{C_d \\rho A}{2m}\\frac{\\mathrm d s}{\\mathrm d t}^2 - g \n",
      "\\end{align}\n",
      "$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "In order for `scipy.integrate.odeint` to be able to solve this equation, it needs to be re-written as a system of first-order differential equations.\n",
      "\n",
      "Let's define a new function $S$, such that:\n",
      "\n",
      "$$\n",
      "\\begin{align}\n",
      "S &= \\dot{s} \\\\\n",
      "\\dot{S} &= \\ddot{s}\n",
      "\\end{align}\n",
      "$$\n",
      "\n",
      "The second-order differential equation can now be re-written as first-order:\n",
      "\n",
      "$$\n",
      "\\dot{S} = \\frac{C_d \\rho A}{2m} S^2 - g\n",
      "$$\n",
      "\n",
      "As before, we can write a function to solve this:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def dS(S, t):\n",
      "    \"\"\"Return derivative of S.\"\"\"\n",
      "    dS = (cd * rho * A) / (2 * m) * S**2 - g\n",
      "    \n",
      "    return dS"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 57
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This is identical to the original equation we had, except that $S$ has replaced $v$. Is that correct? Quite possibly; all we have done really is substitute $S$ for $\\dot{s}$. To be more realistic, how about we consider the initial position $s_0$."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$$\n",
      "\\frac{\\mathrm d^2 s}{\\mathrm d t^2} = \\frac{C_d \\rho A}{2m}\\frac{\\mathrm d s}{\\mathrm d t}^2 + s_0 - g\n",
      "$$\n",
      "\n",
      "Re-writing again in terms of $\\dot{S}$:\n",
      "\n",
      "$$\n",
      "\\dot{S} = \\frac{C_d \\rho A}{2m}S^2 + s_0 - g\n",
      "$$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def dy(y, t):\n",
      "    \"\"\"Return derivative of S.\"\"\"\n",
      "    s, S = y[0], y[1]\n",
      "    ds = S\n",
      "    dS = (cd * rho * A) / (2 * m) * S**2 + s - g\n",
      "    \n",
      "    return [ds, dS]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 58
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "y0 = [100, 0]\n",
      "t = np.linspace(0, 50, 500)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 59
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "s = odeint(dy, y0, t)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "ename": "NameError",
       "evalue": "name 'full_output' is not defined",
       "output_type": "pyerr",
       "traceback": [
        "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m\n\u001b[1;31mNameError\u001b[0m                                 Traceback (most recent call last)",
        "\u001b[1;32m<ipython-input-68-6fb76febcf0d>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m()\u001b[0m\n\u001b[1;32m----> 1\u001b[1;33m \u001b[0ms\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0modeint\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mdy\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0my0\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mt\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mfull_output\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m",
        "\u001b[1;31mNameError\u001b[0m: name 'full_output' is not defined"
       ]
      }
     ],
     "prompt_number": 68
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import brewer2mpl\n",
      "from matplotlib import pyplot as plt\n",
      "\n",
      "my_map = brewer2mpl.get_map(\"Paired\", \"Qualitative\", 4)\n",
      "\n",
      "colour1 = my_map.hex_colors[0]\n",
      "colour2 = my_map.hex_colors[1]\n",
      "colour3 = my_map.hex_colors[2]\n",
      "colour4 = my_map.hex_colors[3]\n",
      "    \n",
      "fig, ax1 = plt.subplots(1)    \n",
      "ax1.plot(t, s, lw=2, color=colour1)\n",
      "ax1.set_ylabel(\"Height [$m$]\")\n",
      "ax1.set_xlabel(\"Time [$s$]\")\n",
      "\n",
      "fig.tight_layout()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "ename": "ValueError",
       "evalue": "x and y must have same first dimension",
       "output_type": "pyerr",
       "traceback": [
        "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m\n\u001b[1;31mValueError\u001b[0m                                Traceback (most recent call last)",
        "\u001b[1;32m<ipython-input-67-8df17e5899a9>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m()\u001b[0m\n\u001b[0;32m     10\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m     11\u001b[0m \u001b[0mfig\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0max1\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mplt\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0msubplots\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;36m1\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m---> 12\u001b[1;33m \u001b[0max1\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mplot\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mt\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0ms\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mlw\u001b[0m\u001b[1;33m=\u001b[0m\u001b[1;36m2\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mcolor\u001b[0m\u001b[1;33m=\u001b[0m\u001b[0mcolour1\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m     13\u001b[0m \u001b[0max1\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mset_ylabel\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;34m\"Height [$m$]\"\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m     14\u001b[0m \u001b[0max1\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mset_xlabel\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;34m\"Time [$s$]\"\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
        "\u001b[1;32m/home/paul/.virtualenvs/notebook/lib/python3.3/site-packages/matplotlib/axes.py\u001b[0m in \u001b[0;36mplot\u001b[1;34m(self, *args, **kwargs)\u001b[0m\n\u001b[0;32m   4137\u001b[0m         \u001b[0mlines\u001b[0m \u001b[1;33m=\u001b[0m \u001b[1;33m[\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   4138\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m-> 4139\u001b[1;33m         \u001b[1;32mfor\u001b[0m \u001b[0mline\u001b[0m \u001b[1;32min\u001b[0m \u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0m_get_lines\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m*\u001b[0m\u001b[0margs\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;33m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m   4140\u001b[0m             \u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0madd_line\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mline\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   4141\u001b[0m             \u001b[0mlines\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mline\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
        "\u001b[1;32m/home/paul/.virtualenvs/notebook/lib/python3.3/site-packages/matplotlib/axes.py\u001b[0m in \u001b[0;36m_grab_next_args\u001b[1;34m(self, *args, **kwargs)\u001b[0m\n\u001b[0;32m    317\u001b[0m                 \u001b[1;32mreturn\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    318\u001b[0m             \u001b[1;32mif\u001b[0m \u001b[0mlen\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mremaining\u001b[0m\u001b[1;33m)\u001b[0m \u001b[1;33m<=\u001b[0m \u001b[1;36m3\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m--> 319\u001b[1;33m                 \u001b[1;32mfor\u001b[0m \u001b[0mseg\u001b[0m \u001b[1;32min\u001b[0m \u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0m_plot_args\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mremaining\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mkwargs\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m    320\u001b[0m                     \u001b[1;32myield\u001b[0m \u001b[0mseg\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    321\u001b[0m                 \u001b[1;32mreturn\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
        "\u001b[1;32m/home/paul/.virtualenvs/notebook/lib/python3.3/site-packages/matplotlib/axes.py\u001b[0m in \u001b[0;36m_plot_args\u001b[1;34m(self, tup, kwargs)\u001b[0m\n\u001b[0;32m    295\u001b[0m             \u001b[0mx\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0marange\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0my\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mshape\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;36m0\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mdtype\u001b[0m\u001b[1;33m=\u001b[0m\u001b[0mfloat\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    296\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m--> 297\u001b[1;33m         \u001b[0mx\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0my\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0m_xy_from_xy\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mx\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0my\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m    298\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    299\u001b[0m         \u001b[1;32mif\u001b[0m \u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mcommand\u001b[0m \u001b[1;33m==\u001b[0m \u001b[1;34m'plot'\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
        "\u001b[1;32m/home/paul/.virtualenvs/notebook/lib/python3.3/site-packages/matplotlib/axes.py\u001b[0m in \u001b[0;36m_xy_from_xy\u001b[1;34m(self, x, y)\u001b[0m\n\u001b[0;32m    237\u001b[0m         \u001b[0my\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0matleast_1d\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0my\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    238\u001b[0m         \u001b[1;32mif\u001b[0m \u001b[0mx\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mshape\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;36m0\u001b[0m\u001b[1;33m]\u001b[0m \u001b[1;33m!=\u001b[0m \u001b[0my\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mshape\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;36m0\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m--> 239\u001b[1;33m             \u001b[1;32mraise\u001b[0m \u001b[0mValueError\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;34m\"x and y must have same first dimension\"\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m    240\u001b[0m         \u001b[1;32mif\u001b[0m \u001b[0mx\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mndim\u001b[0m \u001b[1;33m>\u001b[0m \u001b[1;36m2\u001b[0m \u001b[1;32mor\u001b[0m \u001b[0my\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mndim\u001b[0m \u001b[1;33m>\u001b[0m \u001b[1;36m2\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    241\u001b[0m             \u001b[1;32mraise\u001b[0m \u001b[0mValueError\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;34m\"x and y can be no greater than 2-D\"\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
        "\u001b[1;31mValueError\u001b[0m: x and y must have same first dimension"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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SsrKyZmLUaZfKXixevFiDg4OSpMHBQS1atEhud9KXCd9ba9eu1YIFC976+GS6mda486Gn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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7f58a3848610>"
       ]
      }
     ],
     "prompt_number": 67
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "s.shape"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 64,
       "text": [
        "(500, 2)"
       ]
      }
     ],
     "prompt_number": 64
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}