{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "###### Content under Creative Commons Attribution license CC-BY 4.0, code under BSD 3-Clause License © 2017 L.A. Barba, N.C. Clementi"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Step to the future\n",
    "\n",
    "Welcome to Lesson 2 of the course module \"Tour the dynamics of change and motion,\" in _Engineering Computations_. The previous lesson, [Catch things in motion](http://go.gwu.edu/engcomp3lesson1), showed you how to compute velocity and acceleration of a moving body whose positions were known. \n",
    "\n",
    "Time history of position can be captured on a long-exposure photograph (using a strobe light), or on video. But digitizing the positions from images can be a bit tedious, and error-prone. Luckily, we found online a data set from a fancy motion-capture experiment of a falling ball, with high resolution [1]. You computed acceleration and found that it was not only smaller than the theoretical value of $9.8 \\rm{m/s}^2$, but it _decreased_ over time. The effect is due to air resistance and is what leads to objects reaching a _terminal velocity_ in freefall.\n",
    "\n",
    "In general, not only is [motion capture](https://en.wikipedia.org/wiki/Motion_capture) (a.k.a., _mo-cap_) expensive, but it's inappropriate for many physical scenarios. Take a roller-coaster ride, for example: during design of the ride, it's more likely that the engineers will use an _accelerometer_. It really is the acceleration that makes a roller-coaster ride exciting, and they only rarely go faster than highway speeds (say, 60 mph) [2].\n",
    "How would an engineer analyze data captured with an accelerometer?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## A roller-coaster ride\n",
    "\n",
    "Prof. Anders Malthe-Sorenssen has a file with accelerometer data for a roller-coaster ride called \"The Rocket\" (we don't know if it's real or made up!). He has kindly given permission to use his data. So let's load it and have a look. We'll first need our favorite numerical Python libraries, of course."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy\n",
    "from matplotlib import pyplot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "\n",
    "pyplot.rc('font', family='serif', size='14')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If you don't have the data file in the location we assume below, you can get it by adding a code cell and executing this code in it, then commenting or deleting the `filename` assignment before the call to `numpy.loadtxt()`.\n",
    "\n",
    "```Python\n",
    "from urllib.request import urlretrieve\n",
    "URL = 'http://go.gwu.edu/engcomp3data2?accessType=DOWNLOAD'\n",
    "urlretrieve(URL, 'therocket.txt')\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "filename = '../data/therocket.txt'\n",
    "t, a = numpy.loadtxt(filename, usecols=[0,1], unpack=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We'll take a peek at the data by printing the first five pairs of $(t, a)$ values, then plot the whole set below. Time is given in units of seconds, while acceleration is in $\\rm{m/s}^2$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.0 0.2731644\n",
      "0.1 1.4411079\n",
      "0.2 2.6693138\n",
      "0.3 4.2383806\n",
      "0.4 5.6499504\n"
     ]
    }
   ],
   "source": [
    "for i in range(0,5): print(t[i],a[i])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
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zNdC2kG1uHjLPdcA8fz/Pxjsp9biQ6c+F7ZtRRTje04En/Fjn+cu/qZB6F/rH0jL/tRvB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\nAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f0abf7c66d8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# plot the acceleration over time\n",
    "fig = pyplot.figure(figsize=(8, 3))\n",
    "\n",
    "pyplot.plot(t, a, color='#004065', linestyle='-', linewidth=1) \n",
    "pyplot.title('Acceleration of roller-coster ride data, from [1]. \\n')\n",
    "pyplot.xlabel('Time [s]')\n",
    "pyplot.ylabel('[m/s$^2$]');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Set things up to compute velocity and position\n",
    "\n",
    "Our challenge now is to find the motion description—the position $x(t)$—from the acceleration data. In the [previous lesson](http://go.gwu.edu/engcomp3lesson1), we did the opposite: with position data, get the velocity and acceleration, using _numerical derivatives_:\n",
    "\n",
    "\\begin{equation}\n",
    "v(t_i) = \\frac{dx}{dt} \\approx \\frac{x(t_i+\\Delta t)-x(t_i)}{\\Delta t}\n",
    "\\end{equation}\n",
    "\n",
    "\\begin{equation}\n",
    "a(t_i) = \\frac{dv}{dt} \\approx \\frac{v(t_i+\\Delta t)-v(t_i)}{\\Delta t}\n",
    "\\end{equation}\n",
    "\n",
    "Since this time we're dealing with horizontal acceleration, we swapped the position variable from $y$ to $x$ in the equation for velocity, above. \n",
    "\n",
    "The key to our problem is realizing that if we have the initial velocity, we can use the acceleration data to find the velocity after a short interval of time. And if we have the initial position, we can use the known velocity to find the new position after a short interval of time. Let's rearrange the equation for acceleration above, by solving for the velocity at $t_i + \\Delta t$:\n",
    "\n",
    "\\begin{equation}\n",
    " v(t_i+\\Delta t)   \\approx  v(t_i) + a(t_i) \\Delta t\n",
    "\\end{equation}\n",
    "\n",
    "We need to know the velocity and acceleration at some initial time, $t_0$, and then we can compute the velocity $v(t_i + \\Delta t)$. For the roller-coaster ride, it's natural to assume that the initial velocity is zero, and the initial position is zero with respect to a convenient reference system. We're actually ready to solve this!\n",
    "\n",
    "Let's save the time increment for our data set in a variable named `dt`, and compute the number of time increments in the data. Then, we'll initialize new arrays of velocity and position to all-zero values, with the intention of updating these to the computed values."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.10000000000000001"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#time increment\n",
    "dt = t[1]-t[0]\n",
    "dt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "151"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#number of time increments\n",
    "N = len(t)\n",
    "N"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "#initialize v and x arrays to zero\n",
    "v = numpy.zeros(N)\n",
    "x = numpy.zeros(N)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Step through time\n",
    "\n",
    "In the code cell below, we use a `for` statement to step through the sequence of acceleration values, each time computing the velocity and position at the subsequent time instant. We are applying the equation for $v(t_i + \\Delta t)$ above, and a similar equation for position:\n",
    "\n",
    "\\begin{equation}\n",
    " x(t_i+\\Delta t)   \\approx  x(t_i) + v(t_i) \\Delta t\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "for i in range(N-1):\n",
    "    v[i+1] = v[i] + a[i]*dt\n",
    "    x[i+1] = x[i] + v[i]*dt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And there you have it. You have computed the velocity and position over time from the acceleration data. We can now make plots of the computed variables. Note that we use the Matplotlib [`subplot()`](https://matplotlib.org/api/_as_gen/matplotlib.pyplot.subplot.html?highlight=matplotlib%20pyplot%20subplot#matplotlib.pyplot.subplot) function to get the two plots in one figure. The argument to `subplot()` is a set of three digits, corresponding to the number of rows, number of columns, and plot number in a matrix of sub-plots. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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H1RkEDG+eh5kNMrPrIbg85u4vELQRq2DbbbuerdtLrplNi4llV48vneFVgvag8etmFMGX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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f0abcd189b0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# plot velocity and position over time\n",
    "fig = pyplot.figure(figsize=(10,6))\n",
    "\n",
    "pyplot.subplot(211)\n",
    "pyplot.plot(t,  v, color='#0096d6', linestyle='-', linewidth=1) \n",
    "pyplot.title('Velocity and position of roller-coster ride (data from [1]). \\n')\n",
    "pyplot.ylabel('$v$ [m/s] ')\n",
    "\n",
    "pyplot.subplot(212)\n",
    "pyplot.plot(t,  x, color='#008367', linestyle='-', linewidth=1) \n",
    "pyplot.xlabel('Time [s]')\n",
    "pyplot.ylabel('$x$ [m]');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## Euler's method\n",
    "\n",
    "The method we used above to compute the velocity and position from acceleration data is known as _Euler's method_. The eminent Swiss mathematician Leonhard Euler presented it in his book _\"Institutionum calculi integralis,\"_ published around 1770 [3].\n",
    "\n",
    "You can understand why it works by writing out a Taylor expansion for $x(t)$:\n",
    "\n",
    "\\begin{equation}\n",
    "x(t+\\Delta t) = x(t) + \\frac{d x}{dt}\\Delta t + \\frac{d^2 x}{dt^2}\\frac{\\Delta t^2}{2} + \\frac{d^3 x}{dt^3}\\frac{\\Delta t^3}{3!}+\\cdots\n",
    "\\end{equation}\n",
    "\n",
    "With $v=dx/dt$, you can see that the first two terms on the right-hand side correspond to what we used in the code above. That means that Euler's method makes an approximation by throwing away the terms $\\frac{d^2 x}{dt^2}\\frac{\\Delta t^2}{2} + \\frac{d^3 x}{dt^3}\\frac{\\Delta t^3}{3!}+\\cdots$. So the error made in _one step_ of Euler's method is proportional to $\\Delta t^2$. Since we take $N=T/\\Delta t$ steps (for a final time instant $T$), we conclude that the error overall is proportional to $\\Delta t$. \n",
    "\n",
    "#### **Euler's method is a first-order method** because the error in the approximation goes with the first power of the time increment $\\Delta t$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Initial-value problems\n",
    "\n",
    "To get velocity and position from the acceleration data, we needed to know the _initial values_ of the velocity and position. Then we could apply Euler's method to _step in time_ starting at $t_0$, with time increment $\\Delta t$. This setting corresponds to the numerical solution of _initial-value problems_. \n",
    "(We follow here the presentation in [4], p.86.)\n",
    "\n",
    "Consider the differential equation corresponding to an object in free fall:\n",
    "\n",
    "\\begin{equation}\n",
    "\\ddot{y}=-g,\n",
    "\\end{equation}\n",
    "\n",
    "where the dot above a variable represents the time derivative, and $g$ is the acceleration of gravity. Introducing the velocity as intermediary variable, we can write:\n",
    "\n",
    "\\begin{eqnarray}\n",
    "\\dot{y} &=& v \\nonumber\\\\\n",
    "\\dot{v} &=& -g\n",
    "\\end{eqnarray}\n",
    "\n",
    "The above is a system of two ordinary differential equations, with time as the independent variable. For its numerical solution, we need two initial conditions, and Euler's method:\n",
    "\n",
    "\\begin{eqnarray}\n",
    "y(t_0) = y_0, \\qquad y_{i+1} &=& y_i + \\dot{y} \\Delta t \\nonumber\\\\\n",
    "v(t_0) = v_0, \\qquad v_{i+1} &=& v_i + \\dot{v} \\Delta t\n",
    "\\end{eqnarray}\n",
    "\n",
    "It's so neatly symmetrical that it's just asking for a vectorized equation! Combine the two dependent variables into a vector of unknowns, $\\mathbf{y}$:\n",
    "\n",
    "\\begin{equation}\n",
    "\\mathbf{y} = \\begin{bmatrix}\n",
    "y \\\\ v\n",
    "\\end{bmatrix},\n",
    "\\end{equation}\n",
    "\n",
    "and write the differential equation in vector form, as follows:\n",
    "\n",
    "\\begin{equation}\n",
    "\\dot{\\mathbf{y}} = \\begin{bmatrix}\n",
    "v \\\\ -g\n",
    "\\end{bmatrix}.\n",
    "\\end{equation}\n",
    "\n",
    "Equation (9) above represents the _state_ of the system, at any given instant in time. A code design for the numerical solution that generalizes to other changing systems (or _dynamical systems_) is to write one function that computes the right-hand side of the differential equation (the derivatives of the state variables), and another function that takes a state and applies the numerical method for each time increment. The solution is then computed in one `for` statement that calls these functions. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Study the code below: the function `freefall()` computes the right-hands side of the equation, and the function `eulerstep()` takes the state and applies Euler's method to update it one time increment."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def freefall(state):\n",
    "    '''Computes the right-hand side of the freefall differential \n",
    "    equation, in SI units.\n",
    "    \n",
    "    Arguments\n",
    "    ----------    \n",
    "    state : array of two dependent variables [y v]^T\n",
    "    \n",
    "    Returns\n",
    "    -------\n",
    "    derivs: array of two derivatives [v -g]^T\n",
    "    '''\n",
    "    \n",
    "    derivs = numpy.array([state[1], -9.8])\n",
    "    return derivs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def eulerstep(state, rhs, dt):\n",
    "    '''Uses Euler's method to update a state to the next one. \n",
    "    \n",
    "    Arguments\n",
    "    ---------\n",
    "    state: array of two dependent variables [y v]^T\n",
    "    rhs  : function that computes the right hand side of the \n",
    "           differential equation.\n",
    "    dt   : float, time increment. \n",
    "    \n",
    "    Returns\n",
    "    -------\n",
    "    next_state: array, updated state after one time increment.       \n",
    "    '''\n",
    "    \n",
    "    next_state = state + rhs(state) * dt\n",
    "    return next_state"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Numerical solution vs. experiment\n",
    "\n",
    "Here's an idea! Let's use the `freefall()` and `eulerstep()` functions to obtain a numerical solution with the same initial conditions as the falling-ball experiment from [Lesson 1](http://go.gwu.edu/engcomp3lesson1), and compare with the experimental data. \n",
    "\n",
    "You can grab the data from its location online running the following code in a new cell:\n",
    "```Python\n",
    "filename = 'fallingtennisball02.txt'\n",
    "url = 'http://go.gwu.edu/engcomp3data1'\n",
    "urlretrieve(url, filename)\n",
    "```\n",
    "\n",
    "You already imported `urlretrieve` above. Remember to then comment the assignment of the `filename` variable below. We'll load it from our local copy."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "filename = '../data/fallingtennisball02.txt'\n",
    "t, y = numpy.loadtxt(filename, usecols=[0,1], unpack=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We'll need to use the same time increment, so let's compute it from two time samples. The initial position is the first value of the `y` array, while the initial velocity is zero. And we'll only look at the section of data before the ball bounces from the ground, which gives us the number of time steps."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "#time increment\n",
    "dt = t[1]-t[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "y0 = y[0] #initial position\n",
    "v0 = 0    #initial velocity\n",
    "N = 576   #number of steps"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, let's create a new array, called `num_sol`, to hold the results of the numerical solution. The array has dimensions `Nx2`, with each two-element row holding the state variables, $(y,v)$, at a given time instant. After saving the initial conditions in the solution array, we are ready to start stepping in time in a `for` statement. Study the code below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "#initialize array\n",
    "num_sol = numpy.zeros([N,2])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "#Set intial conditions\n",
    "num_sol[0,0] = y0\n",
    "num_sol[0,1] = v0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "for i in range(N-1):\n",
    "    num_sol[i+1] = eulerstep(num_sol[i], freefall, dt)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Did it work? Exciting! Let's plot in the same figure both the numerical solution and the experimental data. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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+pKqRqhrZsmXLGg3OmLoqKnIgPa55l02BffHJOcSB+bfw56fPQH6+06GZOq6u\nJZktQCsREY9hrd3Pmx2Ix5h6o0ObVoy55VXWtLmAfAW/5a+R+9XdkJ3udGimDqtrSeYbXCf5e3sM\niwQygF8ciciYeiTI34+Lr72HxEH30qFNK3y2LYaP/gkHdjgdmqmjanWSEZGHRGS1iAQAqOoaYAFw\nm3u8L3Aj8KSqHnIuUmPqDxHhzLPOp9H4V6BpJ9ifwM45V7B73WKnQzN1kNN/xvQTkVhc/3UBmCsi\n8z2KBOA6oe95eGySu+5SYAmwGLivxoM1pqEJaQ/nPc+WoL4k70tm73vXsyn2LaejMnWMaAP6A1Zk\nZKTGxcU5HYYxdcqB9Cw+eXk6J6R8iQAacQ7Hj7sPvO1qTg2BiCxT1cijrV+rD5cZY5wX0sifS67/\nL6t6/otsfGDdJyx7/kpy0/c7HZqpAyzJGGPK5ePtxaWXTWVP9KMcIBjfPX/y5zMXczDpL6dDM7Vc\nufu7IjKikm1mqurvRxmPMaYWO/2UU1jWuj3bP5hG96wdBHx2HZx6H3Qe5nRoppYq95yMiFT231gJ\nqlorL/Fi52SMqR7b96QQvGQmoTt/AhEYch30vdDpsEwNOBbnZH5QVa+KPoCtRxuMMaZu6NCqGaFn\nPQiRk0GVXV/+j40fPgh2IzRTTEWSTFL5RapU3hhTF4nAgEmsDr+eXQdzyfzjQ1a/fh1qVwgwHspN\nMqp6SWUarGx5Y0zd1jv6AjZGzuCQBpK/9VdWPX8FuWm7nQ7L1BLV1rtMRN6rrraMMXWHiHDxOWdz\n6PSn2E0L2LeJtc9dSvrOdU6HZmqBSiUZEQkRkXtFZL6ILBKR7woewKgaitEYUwecNnQgzS97hc1e\nXeBwMvGvXUnK2linwzIOq+yezHvAaGAT8CPwg8cjtXpDM8bUNf17dqbfP15jpf9A/PKzafLjDFg9\nv9x6pv6q7HUhWqrqgJJGiEhaNcRjjKnjuoQ1JfTGZ5AVb+Cz9h34ZTYcTIKoa8HL/v/d0FT2E19R\ncEXkEuyqajDGmPqhaWN/QodfDSPvQr282fnTG2yedyfk5TgdmjnGKrsncwvwmIgk4Uoqnp3i7wDm\nVldgxph6oOdo4nZD/jf3EbTmKzamp9DzsifBr5HTkZljpLJJ5jrgX0AyR97uOKxaIjLG1CuRJ53G\nm/uFLksfIDRhKetfmkz45OeQRi2cDs0cA5U9XDYFOE5Vw1S1i+cD+KkG4jPG1HEiwhXnnEbKqFns\n0hZk79lQT2zkAAAgAElEQVTIuhcuJ3dfgtOhmWOgsklmjaqWdtnV8VUNxhhTf503YiB+FzzHZjqQ\ne2Anf718BZmJfzodlqlhlU0yL4rITSLSVkSk2Djrp2iMKdPIfj3pdMWLrPKKwCfnEL4Lb4WEX5wO\ny9SgSt0Z0+OKzCVWUlXv6giqpthVmI2pHTYlpdL6z2dpHP81iBcMvxUiznI6LFOCql6FubIn/v8A\nbiopDmDW0QZhjGlYurcOhbC7YFkbdNkb7P7sQXxTkmk+bJLToZlqVtkk87Cq/lDSCBG5uxriMcY0\nFCIQeSXfJeTQYscL+H37FHmHU2l16o2ucaZeKPecjIicVvBaVd8vrZyqflG8vDHGlGfQWVfyWfMr\nycqDvb+8wa7PHoL8yt4r0dRWFTnxf0cl26xseWNMAxYc4Mut//wnC9teR2a+N/vi5pP44R2Qm+10\naKYaVORwWRcRua8SbYYebTDGmIYpwNebaVMnMeutxkRveQJd/Q35mYfoePET4BvodHimCiqSZLYC\nIyvR5oajjMUY04D5ensx7fKLePr9Rpy45mECt/6Ofn4rMuYRCGjidHjmKFWqC3NdZ12Yjan9VJV5\ni37lvJ0z8cvYC826wBlPQKPmTofWIFW1C7Ndd9sYU6uICONOHYrf+c9DaEc0JZ7kd6dCml3ovS6y\nJGOMqZ0at4RznmZlZhg7t29h65wpkLrd6ahMJVmSMcbUXoGhbIl6kHXaiQPJO0mYMwVNiXc6KlMJ\nlmSMMbXaBYPDCTpvJqu0G2kpe0iYcxW6d6PTYZkKqlSSEZHjaioQY4wpzdkDutH8gpks13AOpu4j\n4Y1r0N1rnA7LVMDR3H55tog0rZFojDGmFGP6daLD+Cf4XXtzMG0/Se/+C3audDosU47KJplBQG/g\nLxG5XkRq9VWXjTH1yyl92tNjwv/YFDyIloH5sPDfkLjM6bBMGSqVZFR1laqeClwF3ACsEpHTayQy\nY4wpQXREW8be+jw+x50JuVnol3eQG/+z02GZUhzViX9V/QjXHs0bwFwR+cLO1xhjjhXx8oYRt5Hf\n6zwSkw+Q+N408jbFOh2WKUFVepcFActwJZrRwJ8i8pSIhFRLZMYYUxYvLxLCr+KdjMGkHc4k8YPb\nLdHUQpXtXXaTiLwjIhuBfcCnwEBgNq5DaMcBa0UkqtojNcaYYrq2CmbU5Ol8SgwHDme5Es3mWKfD\nMh4quydzK+ALPA+cBISo6hBVvUVV31TV04CngNeqOU5jjCnRgM7NGT353r8TzTxLNLVJZU/8d1DV\ncao6S1V/VdWSbvjwOq49GmOMOSYiuzTntMn38ok70eywRFNr1MQ//vcCJ9dAu8YYU6qBXZpz2qR7\n+ZiRZOfmwrczYEus02E1eNWeZNTlh+pu1xhjyjOoa3POvuo+2o2cijcK395vicZhdu0yY0y90q9j\nUwKHXgMnXoZqPikf302+9TpzjCUZY0z9IwIDr+JT75NJ3J/Ojg/vQO0cjSMsyRhj6icRWsT8kwU6\nkv3pmey0ROMISzLGmHpraPeWxFx2J/N1JPsOZbJr/l2Q8IvTYTUolmSMMfVadHgrho7/Nx/ljyD5\n4GGSPrwdtv/udFgNhiUZY0y9d1qfNpx44e18lj+EPQcOsW/+bbBjudNhNQiWZIwxDcLZ/drR49w7\nWR08nFB/ha/ugqRVTodV71mSMcY0GBcN7Mj4W57Eu+doyMmAhbfDnvVOh1WvWZIxxjQo3t7eEHMH\n+V2iSdydTPK8GyB5k9Nh1VuOJxkROUdElorIjyLyi4hEllN+vYjEFnvceKziNcbUA17efBU2lYWH\nurFr9x72vX8dpMQ7HVW95GiSEZEBwLvAFao6AngY+EpEWpdRLUlVY4o9Zh+TgI0x9caY49uzf8hd\nxOX3ZGfSbvbPux5StzsdVr3j9J7MncBXqroWQFU/A3YD/3I0KmNMvSci3HZGH7YNuIM/87uwY+cO\n0j64DtJ2OR1aveJ0kjkViCs2bCkwyoFYjDENjIhw77knsqLXbazJ60hi4jYOzb8B0vc5HVq94ViS\nEZFmQAhQfLMhCehaRtVGIvKa+xxOrIjcIyIBNRaoMaZe8/ISHh4fRWzXW1mf15bknQnwxa2QmeZ0\naPWCk3syjdzPWcWGZwFBZdTbADzvPodzIXAG8EFphUXkahGJE5G4vXv3ViVeY0w95evtxczLhrF1\n4H207xLu6gTw5R2Qfdjp0Oo8J5NMuvvZv9hwf6DUT1ZVL1PVpe7XycC9wJki0q+U8i+paqSqRrZs\n2bIawjbG1EeBft7cfE4UPmfPhMZh6O415Hx5N+SWdANgU1GOJRlVTQFSgeI9yVoDmyvRVEHZ7tUR\nlzGmgWvcipzT/8eGAz5sW/UTud8+APl5TkdVZzl94v9boPj/YiLdw48gIn1F5Kpig9u5n7dVc2zG\nmAZqn3cY9+ZMYm+WD4nLvyLvh/+BqtNh1UlOJ5lHgNEiEgEgImcAbYBn3e8fEpHVHif2mwP/FpHm\n7vH+uLpBLwGWHevgjTH1U+uQAP571XnM9L6S5EzYseRD8n991hLNUXA0yajqMuBS4E0R+RG4Gxit\nqknuIgG4OgGI+/2fuE7yfyEiscAvuHqjnaOqtj9rjKk2PcKCuevKcTzBRPYdzifp57fQ5W86HVad\nI9qAMnNkZKTGxRX/W44xxpTul03JvDTnVW72mku7EH9ajv439Dnf6bCOGRFZpqplXu6rLE4fLjPG\nmFptWPcWXHDRRJ7LPZe0jFz0l9nw1zdOh1Vn+DgdgDHG1HbnnNCWQN9/0CGjJxL3IsQ+DAEh0GGQ\n06HVerYnY4wxFTCqVxh+AybACZeg+XlkLbzb7kVTAZZkjDGmEjL7T+HzzD5s3plMxie32pWby2FJ\nxhhjKsHLy5t5jS5lSW4Ptu3cRdant8LhFKfDqrUsyRhjTCX4+Xjx7MRBfNZqKqtz2rBt62ZyP5sG\n2enlV26ALMkYY0wlNfb34fnJw3ml8TVszmrK9k1/kvflXXadsxJYkjHGmKPQKjiA56acwkzfqWzP\nCGDn2l/R7/8D+flOh1arWJIxxpij1KVFIx6bPJqHuZJD+X7o5lhY/JRdfsaDJRljjKmCfh1CefSa\n8+k2cTZePr6wZgGseNvpsGoNSzLGGFNFx7cPxa/DADj5HvIRMn59EdZ/4XRYtYIlGWOMqSYZ7Yfz\nXPaZbNmTTsaiR2D7UqdDcpwlGWOMqSb+Pl6sa3YK7+cOJyH5ENlf3gPJm5wOy1F27TK3nJwcEhMT\nyczMdDoUY6qdt7c3oaGhtGjRAi8v27asKV5ewhPjTuCyl8fx3c79+CStpcsXt+Nz/gvQuGHe/t2S\njFtiYiLBwcF07twZESm/gjF1hKqSk5PD7t27SUxMpGPHjk6HVK8F+Hrz0hWDGPfsYZqlPYdv4jY6\nfnEbXuc+A/6NnQ7vmLNNGrfMzEyaN29uCcbUOyKCn58f7dq1Iz3d/pV+LDRr5MfLVw7jWd9JrMsI\nYVf8WvSb+yAv1+nQjjlLMh4swZj6zA6THVtdWjTiyStG8B+dzH5thO5YBj893uD+Q2PfOmOMqSGR\nnZvxzDVnctykZ/Hy8YcNC2H5G06HdUxZkjHGmBrUr0MoPq17wSnTyRcvMn97FTZ86XRYx4wlmTrq\nyy+/JCYmBhHhiiuuOGL8KaecQuvWrenXrx+PPPKIAxFWXP/+/Zk/f361tLVhw4bC5RIbG1stbRY3\ne/ZsjjvuODp37lzpujNmzCAhIaHIsPnz59O/f//qCc7UWgdbD2L24TFs2ZtO9nePQOIyp0M6JizJ\n1FFjxowp/BF98803+eCDD4qMX7RoEWPGjOHJJ5/kjjvucCDCiuvZsyfNmjWrlrbCw8NrLLkUuPHG\nG496md5///1HJJlmzZrRs2fPaojM1GYBvt7ENY5mXs5QEpIPkvfVPbBvs9Nh1ThLMnVcp06dOOOM\nM7jmmmvYuXOn0+Eclblz5xITE+N0GI6JiYlh7ty5TodhapivtxfPTRjAj6Hnsygrgu27k9GFd9T7\nG55ZkqkHXn/9dXx9fZk8eTJaSs+Vzz//nH79+hXpQTd58mRCQ0OZMWMGUPRQ08svv8y4ceOIiIjg\noosuIiMjg/vvv58RI0bQt29fVqxYUaT9uLg4oqOjGTp0KMOGDWP69Onk5rq6a3oeXnrzzTc588wz\nadmyJZMmTeLSSy+ldevWTJo0qUh7M2fO5Pjjj2f48OH079+fO++8k4yMDAA+/PBDhg4dysiRI4mK\niuLmm28mKyurUstsy5YtjBkzhhEjRjB8+HDGjRvHhg0bCscvXbqU6OhoBg4cSJ8+fbjzzjsL56e4\nHTt2HHGIbvbs2XTu3Lkwee7du7fw9U033URMTAzTp0/nm2++YfDgwYhIkT2cv/76izPOOIMBAwbQ\nt29frrnmmsLuxz/88ENhnffee4+xY8cSERHBJZdcUunlYI6tkCBfXpkcxSs+l/D74TYk7dwGX90F\nufX4c1PVBvMYMGCAlmbt2rVHDOt0+2elPt75bWthuXd+21pmWU9nPvVjhcpVVKdOnVRVdeHChSoi\nOnv27MJxV1xxhX7//feF77///nt1feR/i46O1unTpxcZBuh5552nubm5mpmZqV26dNHTTjtN//rr\nL1VVvf322zUmJqaw/N69ezU0NFQ/+eQTVVU9fPiwDh48WO+9997CMq+//roGBgbq008/raqqv/32\nm15zzTWFcV5xxRWFZV988UVt166d7tq1S1VVExISNCQkROPj41VVdfz48frxxx+rqmp2draOHj1a\n77///iPmwXPeizv99NOLxHf55Zfr66+/rqqqe/bs0ZCQEH3jjTdUVTUtLU2PP/54veuuu4rMT8Gy\nL22a06dP1+jo6HLjio+PV6Bw/gqWecE8ZWdn66hRo3TChAlH1PnHP/6hqq5l3q5dO33ttddKnWfV\nkr/n5tj7ZdNe7X/ne/r1PTG6b+Zg1a/vU83LczqsEgFxWoXfXduTqSfGjBnDjTfeyO23387atWur\n3N4FF1yAt7c3/v7+REZGkpeXR/fu3QEYPnx4kT2ZZ555hrCwMM4++2wAAgMDueyyy3jmmWeKtJmb\nm8vVV18NQFRUFC+88EKJ0/7Pf/7DxIkTad26NeA6JHj//ffTuLHr39KPP/544bR8fX0ZO3YsCxcu\nrNT87dixg+3bt5OXl1c4zVGjRhXOT1BQEBMnTgQgODiYf/zjH8ycObNwb6omvfvuuyQmJnLLLbcA\nrnm85ZZb+L//+z/i4+OLlL3kkksA1zIfNGgQK1eurPH4TNUN7daCW88dzAO5E0nJ9kG3xMKy15wO\nq0bYZWXKkPDImRUqNyGqIxOiKnapjs+uH16VkMr0yCOPEBsby2WXXcaSJUuq1FabNm0KXwcFBeHv\n71/4vlGjRhw4cKDw/apVq9izZ0+R8yrp6ek0adKEtLQ0mjRpAkBYWBh+fn5lTvfgwYNs27atMKEV\nuPHGGwtfHzhwgNtuu42tW7fi5+dHUlJSpQ8T3X///UycOJHvv/+eiy++mCuvvLLw5Pvq1avp1q1b\nkUOL3bt3JzMzk02bNtG3b99KTauyVq9eTVhYWGFSLZi+qrJ69Wq6dOlSOLxt27aFr4ODg0lLS6vR\n2Ez1mRDVkbAmZ9I1qBfy1Z2w/C0I6Qg9T3M6tGplezL1iL+/P++++y7r16/nvvvuO2J8SVc0KNiS\nL87b27vM98VFREQQGxtb+Fi6dCkJCQmFCaYibVREeno6J598Mk2bNuWnn34iNjaWO+64o9RzUaU5\n77zzSExM5M4772TRokX07t2bjz76qMrxeSpt2VYnz2UqIpVeDsZZp0SE4dVpMAy9nnyFvNhHYdef\nTodVrSzJ1DMRERHMnDmTxx57jN9//73IuODgYMC1t1Bgx44dVZ5m37592bx5c5Ef1f379zN16tRK\ntxUcHEzHjh3ZtKno5dHfeusttmzZwvr169mzZw8XXXRR4Q9sdnZ2pafzwQcfEBISwjXXXMPSpUsZ\nO3Ysr776KgB9+vRhy5YtRX6wN2/eTEBAwBF7WMVjL2/ZeiZ6z7Ke+vTpw+7duzl06FCR6YsIffr0\nqfhMmjpjT6czePNgf7YlHyD/q7vhQNXXy9rCkkw9dO2113LOOeewbt26IsN79OhBo0aNWLx4MeD6\nL82ePXuqPL3rrruO7OxsXnzxxcJh//nPf2jevPlRtXf33Xfz1ltvkZSUBLh6vc2YMYPWrVvTuXNn\nAgMDWbRoEeDaW/j0008rPY3i565ycnIKD5ddd911pKen88477wBw6NAhnn/+eW655RYCAwNLbbNf\nv36Fy3b37t18//33R5Rp1aoVKSkp5OTk0K9fvxLbmTBhAu3bt+fJJ58EXOeyZs2axSWXXFLkUJmp\nP7Jy8nkmYzSxh7uStGcPfHkHZJW8EVLnVKXXQF17VLZ3WW22cOFCjY6OVn9/f42OjtYVK1YUGZ+c\nnKzt2rU7oifTa6+9pt27d9eTTz5ZH374YY2OjtZOnTrp3XffrYmJiRodHa2AnnDCCbpo0SK97bbb\nNCwsTMPCwvS2227TRYsW6QknnKCARkdHa2JioqqqxsXF6YgRI7Rfv3560kkn6bRp0zQnJ0dVVV95\n5RUNDw8vjHXRokWF8UyYMKGw/SlTphQOf+KJJ7RPnz46fPhwHTlypMbFxRWOW7BggYaHh+vAgQP1\nvPPO08mTJxe2vX79+iLzMG/evBKX3+zZs3Xw4ME6cuRIHTRokE6ePFkPHjxYOH7JkiU6fPhwjYyM\n1N69e+vtt99eOD9PPvlkkfkpqLd06VLt06ePDh06VKdOnarTpk3TkJAQPfPMMwvbffbZZzU8PFyj\noqL0+eef16+//lqjoqIU0KioKP3pp59UVXXjxo06ZswY7d+/v/bu3VunTp1aOJ24uLgiddasWaN3\n3HFH4XK88cYbS/3e1LXveUPye/w+Pf6u+frpPadp8hODVT+9WTU3x+mwqty7TLQBHcONjIzUuLi4\nEsetW7eOiIiIYxyRMceWfc9rtw+WJfLovFhm+b3AiS2h0QnnwfBbwcErxIvIMlWNPNr6drjMGGNq\niQsHtOfck/rzQM5lbNqXRfbqj2HVB+VXrMUsyRhjTC1yx+nHEdb9RP6XfQEp6dnw23Ow/ffyK9ZS\nlmSMMaYW8fH24ulLTuSUM8cTFnMNaD58ez+kbnM6tKNiScYYY2qZ0CA/Jg/rggyYBF2Go9mH4Ms7\n62SPM0syxhhTW3l5sfPEm/lpXxPSdifAogcgP9/pqCrFkowxxtRin65NZVraeNamCJnxv8GSkq/5\nV1tZkjHGmFps6vCuDOgTwYPZFxO/L4PcP+bWqds3W5IxxphazMtLePyiE8gLO54nM89ie8ph9MfH\nIWm106FViCUZY4yp5Rr5+/Dy5ZEsCRjK3PRIklIPwtf3wKG9TodWLksydVR8fDwxMTEEBAQU3oHR\n8xEQEOB0iIWWLl1Khw4dyMzMdDoUUlNTmTFjBqmpqRWu43nH0II7X1bEPffcU+TumBU1Z86cSk3H\nNAwdmgXx3IT+vK5nsuhAO9IPJMPXd0OO8+tVWSzJ1FFdunQhNja28NbFnpfZLxheWwQHBxMeHo6v\nr6/ToZCamsr9999fqSQTHh5+VD/6Dz300BG3la4ISzKmNEO7t+Ces/uSe/J0glp0gL0b4IdHoRZf\nHsxuWlZPvfHGG06HUOi4447j22+/dToMY+qFy4d0dr1I+S989E/Y/B007wYnXuZoXKWxPZl6JiEh\ngZiYGKKjowGYPXs2bdu2JSAggOnTp5OUlMSJJ55ImzZtmD59Ou+99x79+vVDRHj77bc57bTT6Nmz\nJ2eccUbhpfYLzJw5k379+hEdHc2IESP47rvvCsedfvrphIaGcvvtt/PPf/6T6OhovLy8ePrpp484\n1HTnnXcWHkZ67LHHGDlyJD169OCLL77gjz/+YNy4cYSHh3PDDTcUmX5ubi533nkn/fr1IyYmhlNP\nPbXwdsMHDhwoPEz42GOPMXHiRAYOHMiQIUMKb1m8atUqLr74YgAuvvhiYmJieP755wH48MMPGTp0\nKCNHjiQqKoqbb7650nfbBHj55Zfp2rUrJ510ElOmTClyTxiAbdu2MW7cOIYMGcKIESM49dRTi9xy\n4NJLL2XlypXMmTOHmJgYzj333ArVMw1Qsy7sGTiNnWlZ6NJXYOtipyMqWVUu4VzXHpW61P8LI5x5\nVFKnTp10+vTphe/j4+M1Ojq6SJklS5aot7e3fvLJJ6qqevHFF+vixYsLx3///fcK6A033KCqqrm5\nuXrmmWfq6NGjC8u88sor2r17d01JSVFV12Xt/f39dcOGDYVloqOjtX379pqQkKCqqtdff73++OOP\nqqoKFLntwPTp07Vx48b6ww8/qKrqSy+9pGFhYfrYY4+pqutWBUFBQRobG1tY55577tFhw4ZpRkaG\nqqrOnz9fQ0NDdd++fUWWx8CBAwsviz927Fi9/PLLiywfQOPj44sso/Hjx+vHH3+sqqrZ2dk6evRo\nvf/++4uUKT4PxS1evFi9vb11yZIlqqq6adMmDQsLK/J5LFy4UC+44ALNz89XVdU333xTe/bsWXgb\ngYLl6PmZVrReRdil/uuPnNw8HfHYdzrtrmma9L8o1ddOV92/rdqnQxUv9W97MvVAwVZvTExM4Za6\np0GDBnHTTTcxdepUnnvuOcLCwhgyZMgR5Qr2HLy9vbnhhhv46quv2LhxIwAPPvggU6ZMoWnTpgBE\nRkbSt29fXnih6B/DTjnlFDp16gTAU089xfDhw0uNOywsjBEjRgAwbNgwdu/eXRhX8+bN6dWrFytW\nrAAgIyODJ554guuuu66wU8PYsWPx8fHh7bffLtLu2WefTePGjQGIiYkp3Nspy+OPP87ZZ58NgK+v\nL2PHjmXhwoXl1vP09NNPM2TIEAYNGgRAt27dGDVqVJEyJ510Ei+99FLhHTLHjRvHxo0b2bx5c5lt\nH209U3/5eHsx45zefJAfzYLU7qQdPODqCJCd7nRoRdg5mdJc84PTEVTYpEmTmDFjBuA6XFbSyeYH\nH3yQjz76iAceeKDUH6aC5ACuH0hw3X+kTZs2bN26lTfffJMvv/z7T2CHDh064nBQhw4dKhx3mzZt\nCl8HBQUdMaxRo0YcOHAAgE2bNpGRkcFjjz1WJLGFhoYecRK/bdu2ha+Dg4NJS0srN5YDBw5w2223\nsXXrVvz8/EhKSqr04bJ169bRt2/fIsM6duzI9u3bC997e3sze/ZsvvvuO7y8vAqTRlJSEuHh4aW2\nfbT1TP02MrwVt5wazpPfnE+XlBeJ8YnH//v/wqgHwat27ENYkqlnOnfuXGLPpMDAQHr37s3nn3/O\n8uXLy9zDKM3NN9/M1KlTyyzj7e1d4fZKKlt8mBbrNfPoo48esXdQVhsickQbxaWnp3PyySdzwQUX\n8Pbbb+Pt7c2cOXMKE3dVSLGbTU2bNo3PP/+c3377rbAHYEViPNp6pv7718ju/JGYyox1l/JCykv0\n9foZr5VvQ//LnQ4NsBP/9dbnn39eZC9j7ty59O3bl6uvvpopU6aQkZFxRJ1t2/6+lHjB3k5ERATB\nwcF06tSJDRs2FCm/YMEC3nnnnRqag6K6d+9OQEDAETG8+OKLfPPNNxVux6vY1t3BgwdZv349e/bs\n4aKLLipMUNnZ2ZWOMSIigi1bthQZ5rlMAX788Ueio6MLE0VJ0/GM8fDhw+Tl5VWonmmYvLyEmeP7\nEdCiIw+kn09iagYa9xps/dXp0IBakGRE5BwRWSoiP4rILyJS5m0+RaSJiMxx11kuIo+KiO2RFfO/\n//2P5ORkAJKTk3nhhRe49957efTRR8nIyGD69OlH1Hn55ZcByMvL46mnnmL06NH07NkTgHvvvZe3\n3nqLhIQEAFJSUrj33nuPODxUUwIDA5k2bRrPPvss+/btA1yHBh9//PFKxdCiRQu8vLxISUkhKSmJ\nk08+mc6dOxMYGMiiRYsA1/x/+umnlY7x+uuv59dff+X33103mIqPj+fzzz8vUqZXr1789ttvpKe7\njpsvWLDgiHZatWpFSkoKAOeffz7r16+vUD3TcDUJ8OWFywbwl39vtnW9xPW/me8egtTt5VeuaVXp\nNVDVBzAAOAT0cr8/C9gHtC6jznzgLfdrP2Ax8N+KTK9Svctquc2bN2tUVJT6+flpu3btNCoqqsgj\nODhY4+Pjdd68edqzZ0/t2rWrxsbG6sKFC7VDhw7q7e2to0aNUtW/e5d98sknOmbMGO3Ro4eOGTNG\nd+7cWWSas2bN0oiICB02bJiOGDFCP/vss8JxF154oYaEhGinTp0K21VV/fPPPzU6OloBPeGEE3Te\nvHn60EMPaadOnTQkJEQnTpyoa9as0aioKAU0KipK16xZoxMnTixs76GHHlJVV6+3u+++W8PDw3XE\niBF6yimnFPaSy83N1ejoaPX399fw8HB95513dO7cuRoeHq7+/v5Fenjddddd2rt3bx00aJB+9NFH\nqqq6YMECDQ8P14EDB+p5552nkydPLqy3fv36I+ahNC+//LJ26dJFhw4dqhMmTNDrr79eQ0JCdMyY\nMaqqmpiYqKeffrp26dJFzznnHJ0xY0ZhuwsXLlRV1V9++UUjIiJ0+PDhOmnSpArXq4i69j03lZN6\nOFs1P1/1q3tcvVXfu1w1K71KbVLF3mWiDh7TFZEPAFHVCzyGrQU+VNV7SyjfB1gFHK+qq9zDxgFz\ngFaqeqh4HU+RkZEaFxdX4rh169YRERFxtLNSp8XGxjJy5Eg7vt8ANOTveYOSfZj096/G7+A2fLtF\nw6gHoNj5wYoSkWWqWuYRprI4fbjsVKD4r/5SoLQzu6cCmYDn5UeXAoHASdUenTHG1EFLd2Zy3qbT\n2ZiST378j7Di7fIr1RDHkoyINANCgF3FRiUBXUup1hXYrUU3uZM8xplKeu+997jpppsA139KNm3a\n5GW0M/gAAAqeSURBVHBExpiq6tKiEQf9WjP90FiSDmRB3KuQWPJRnJrm5J5MI/dz8T8jZAFBZdQp\nqTyl1RGRq0UkTkTi9u6t/ZfFPtbGjx/PypUrUVViY2Pp3r270yEZY6qoRWN/nr30RFZKBE8eOIn4\noOOh5XGOxOJkkin4W6p/seH+wOEy6pRUntLqqOpLqhqpqpEtW7Y8qkCNMaauGdCpGXefGcG8vGhe\nDrwS/Bs7EodjXX9VNUVEUoHi16RvDZR2rYwtQCsREY9DZgX1q3x9DVU94s9zxtQX1rGj4Zk0tDOd\nWzQipqdzG9hOn/j/FijeayHSPbwk3+A6yd+7WPkM4JeqBOLt7U1OTk5VmjCmVsvIyKgV9/Qxx46I\nMDK8laMbz04nmUeA0SISASAiZwBtgGfd7x8SkdUiEgCgqmuABcBt7vG+wI3Ak+V1Xy5PaGgou3fv\nJj8/vyrNGFPrqCqHDx9mx44dtGrVyulwTAPj6D/lVXWZiFwKvCkiGYA3MFpVC3qMBeA6oe+ZhicB\nT4vIUnf5b4H7qhpLixYtSExMPOKyJcbUB76+voSF/X979x4rR1mHcfz70LTFtBQBsdWKRWIU0SoI\nSlBTGm1TREOEGEJUsGKjGGmqJKSQFIOtoTVeoKLBQMCDipeCBFGIIsRTBMT0QgNCKgophkuxQMEK\nLZfm5x/zrqzTc/acmZ3Z2dPzfJLJnrmdeZ/M7L4778zOO51p06Y1XRQbZxr9MWavdfoxppmZ7Wms\n/xjTzMz2Yq5kzMysNq5kzMysNq5kzMysNq5kzMysNuPq7jJJ24BHSq7+OuCpCovTD5xpbHCm/re3\n5YFXM82KiNKPDBhXlUw3JK3v5ja+fuRMY4Mz9b+9LQ9Ul8nNZWZmVhtXMmZmVhtXMqN3edMFqIEz\njQ3O1P/2tjxQUSZfkzEzs9r4TMbMzGrjSgaQdJKkdZJul3SnpI53VEiaJmkgrbNR0jclNfpE67yi\nmdI675P0gKSBHhSxsCKZJE2XdJGkOyQNSrpH0vljeT9JmixpRcp0W8p0g6S+6jO7zLGX1psiaYuk\nwZqLWFiJz4jN6bhrH5b0qryjUfIz4ixJa9M6/5B01YgbiohxPQBHA/8BjkjjHweeBmZ0WOd64Cfp\n70nAXcBFTWfpMtNSsm4T7gcGms7QbSbgbGADsF8aPwTYBixvOksXmWYAjwPT0/g+wBpgfdNZujn2\n2tb9DrAdGGw6R7eZ+i1DRZmWAr8BJqfx9wD/GnFbTYdtegCuA36Vm/YAsGKY5d8FBDC7bdqpwAvA\n1KbzlMmU5p9E1m/PYJ9WMkX306nAablpPwAeajpLF5kmAUflpi0Gnm06S9lMbcscBdwOXN1vH9Al\n3099laHbTMBBZD0Qvy03fc5I23JzGcwD8p3MrAPmd1h+F/DX3PKvAT5UeenKKZqJiLgx0lHTpwpl\niog1EfGL3OSdwOQaylZW0UwvRcQ9rXFJM4HPAqtrK2FxhY89SfuQfQH4MtkXuH5TONMYUDTTicBz\nEfFg+8SIuH2kDY3rSkbSgcD+wBO5WVuBw4ZZ7TDgydwH8ta2eY0qmamvVZjpOLLmpcZ1k0nSTEkb\ngIfImji/XkshC+oi09nAnyLivrrKVlYXmaZIuipduxiUtKzVjXzTSmaaDTwuaZGkP0q6S9IPJY34\nuJlxXckAU9Lri7npL5J1+zzcOkMtT4d1eqlMpn7XdSZJ84A3A8srLFc3SmeKiMci4mhgFvBB4Nrq\ni1dK4UyS3gQsok8qyiGU3U9/Ay6LiDnAJ8nOBK6rvnillMl0ANmlguPJznaOB14LDEqa2Glj472S\neT695ptQJpNdYxlunaGWp8M6vVQmU7/rKpOkWcBlwEkR8WzFZSur6/0UEU8CXwFOkfThCstWVplM\n3wPOj4h+PTZL7aeI+ExErEt/PwVcAHxM0pG1lLKYMpl2AxOBCyPilYh4GfgacASwoNPGxnUlExHP\nAM+S3bXTbgZZU8RQHgZeL0m55emwTs+UzNTXuskkaTrwa2BR+/WMppXJJGmCpAm5yQ+k13dWW8Li\nimaStB9wJHBu6zZf4ATgyDS+suYij6jC91Nr2cZvNy+Z6dHcK7z6RPu3dNreuK5kkluB/P3hx6Tp\nQ/kD2UX+9jf1MWQXle+svHTlFM00FhTOJOkA4Ldk35TXpmlfqK2ExRXNdDrw1dy0N6bXxyosVzdG\nnSkidkTEYRExtzUAvwM2pfHz6y/uqBTaT5JmS1qUmzwzvf6z4rKVVfTYW5te39A2bXp67Zyp6Vvp\nmh7I7hffAbwjjZ8IPEO6Xxz4BtmdZPu2rXM9cHX6eyJwB/33O5lCmdrWHaQ/b2EulAmYCtwNrExv\nntawoeksXWRaCGwGDm479n5G9o1yWtN5uj320vwB+uz23xL7aS7wIHBQGp9M9mXnbmBC03nK7qf0\nObe6bfxismtPQ+7L1tBXv35uQkRskPRp4MeSdgITgAUR0bpjbF+yi2HtzWMLgUslrUvL30rWPtkX\nymRK3/A/RdZ8cXhquvhuRNzY08IPo0SmJcCxaTiv1+UdjRKZbgPeC9wiaQfZBdyHgHkR8e/eln5o\nJd9PpGsVlwCHA/um4+9bEXFTzwo/jBKZ7iW7yH9zWn4qsAk4MyJ297b0Qyu5n04m+9zbSNZy8xgw\nPyJ2ddqWH5BpZma18TUZMzOrjSsZMzOrjSsZMzOrjSsZMzOrjSsZMzOrjSsZMzOrjSsZMzOrjSsZ\ns5xWF8BtQ+S6090qaW565P6TqV+XXpdxsK2cJ4xi+dbzwDZL2tKDIpoB+Bf/ZkOJ7DlaAEgKYFVE\nDKTxgTRrF9ljNXb2uHgtAxFx4WgWjIhNwFxJC4FRrWNWBVcyZnu6ZIT5NwBbIuJpYE4PymM2Zrm5\nzCwnIjpWMhFxA/B8an7alc4OkLSk1RwlaaGk30t6WNLnJB0i6RpJ90v6uaT/68tD0jmSNklam3pT\nLNw/jKSDJF2Xei0clHSTpGOL/h+zKvlMxqyEiNhG1vy0pW3aaknPkXWQ9nJELJA0n+wJvKuAM8ie\nnLwZOA24GkDS54EvAe+PiO2SjgHukPTuyPWpPoIVwAsR8YH0f5cDHwX+0l1as/J8JmNWPQG/TH/f\nCUwC/h4Ru9MTa9cBR7UtfwFwZURsB4iI9cB9wFkFtzsTmNHWl/xq4KflIphVw2cyZtXbFhGvAETE\nC6kT1Sfa5j8P7A//6x1yFnBG7i6xqWkoYhXZ9aJHJK0BfhQRG8tFMKuGKxmz6g3VZ0h+mnLjF0fE\nFd1sNCL+LOlQ4BTgTGCDpMUR8f1u/q9ZN9xcZtagiNhB1rPl29unSzo5dSo1apJOBl6KiGsi4iPA\nt4EvVlZYsxJcyZg1bwVwejoLQdKBadp9Bf/PEmBe2/hEsm6AzRrj5jKzYUg6DliZRs+T9NaIWJbm\nHQxcC8xI86aS/SjzXLKL77eQ3UF2fVr/EknnACekAUmXRsTiiLgyXZu5WdIzZE1rSyPi3oJFvhxY\nJmkpWfe5TwBnlwpvVhF3v2w2BkkaBAZH+4v/tvUWAhdGxKHVl8psT24uMxubtgKfKPrsMrIzm0fr\nLpxZi89kzMysNj6TMTOz2riSMTOz2riSMTOz2riSMTOz2riSMTOz2riSMTOz2vwX9U0N0qOvlyoA\nAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f0abcbc3278>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = pyplot.figure(figsize=(6,4))\n",
    "pyplot.plot(t[:N], num_sol[:,0], linewidth=2, linestyle='--', label='Numerical solution')\n",
    "pyplot.plot(t[:N], y[:N], linewidth=2, alpha=0.8, label='Experimental data')\n",
    "pyplot.xlabel('Time [s]')\n",
    "pyplot.ylabel('$y$ [m]')\n",
    "pyplot.title('Free fall tennis ball (no air resistance) \\n')\n",
    "pyplot.legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The two lines look very close… but let's plot the difference to get an idea of the error."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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N/59O1As6Va9twP1EHyr/iC3zd6Kd8uQwPy7bbecSY1dZRHQr7hnhYDOinu4tXRfPi7s/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\nTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f0abcbd7ef0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = pyplot.figure(figsize=(6,4))\n",
    "pyplot.plot(t[:N], y[:N]-num_sol[:,0])\n",
    "pyplot.title('Difference between numerical solution and experimental data.\\n')\n",
    "pyplot.xlabel('Time [s]')\n",
    "pyplot.ylabel('$y$ [m]');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Air resistance\n",
    "\n",
    "In [Lesson 1](http://go.gwu.edu/engcomp3lesson1) of this module, we computed the acceleration of gravity and got a value less than the theoretical $9.8 \\rm{m/s}^2$, even when using high-resolution experimental data. Did you figure out why?\n",
    "\n",
    "We were missing the effect of air resistance! When an object moves in a fluid, like air, it applies a force on the fluid, and consequently the fluid applies an equal and opposite force on the object (Newton's third law).\n",
    "\n",
    "This force is the *drag* of the fuid, and it opposes the direction of travel. The drag force depends on the object's geometry, and its velocity: for a sphere, its magnitude is given by:\n",
    "\n",
    "\\begin{equation}\n",
    "    F_d = \\frac{1}{2} \\pi R^2 \\rho C_d v^2,\n",
    "\\end{equation}\n",
    "\n",
    "where $R$ is the radius of the sphere, $\\rho$ the density of the fluid, $C_d$ the drag coefficient of a sphere, and $v$ is the velocity."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since we have another force involved, we'll have to rethink the problem formulation. The state variables are still the same (position and velocity):\n",
    "\n",
    "\\begin{equation}\n",
    "\\mathbf{y} = \\begin{bmatrix}\n",
    "y \\\\ v\n",
    "\\end{bmatrix}.\n",
    "\\end{equation}\n",
    "\n",
    "But we'll adjust the differential equation to add the effect of air resistance. In vector form, we can write it as follows:\n",
    "\n",
    "\\begin{equation}\n",
    "\\dot{\\mathbf{y}} = \\begin{bmatrix}\n",
    "v \\\\ a_y\n",
    "\\end{bmatrix},\n",
    "\\end{equation}\n",
    "\n",
    "where $a_y$ now includes the acceleration due to the drag force:\n",
    "\n",
    "\\begin{equation}\n",
    " a_y = -g + a_{\\text{drag}} \n",
    "\\end{equation}\n",
    "\n",
    "With $F_{\\text{drag}} = m a_{\\text{drag}}$:\n",
    "\n",
    "\\begin{equation}\n",
    "    a_{\\text{drag}} = \\frac{1}{2m} \\pi R^2 \\rho C_d v^2\n",
    "\\end{equation}\n",
    "\n",
    "Finally, we can write our differential equation as:\n",
    "\n",
    "\n",
    "\\begin{equation}\n",
    "\\dot{\\mathbf{y}} = \\begin{bmatrix}\n",
    "v \\\\ -g + a_{\\text{drag}}\n",
    "\\end{bmatrix}.\n",
    "\\end{equation}\n",
    "\n",
    "Let's write a new function for this modified right-hand side of a falling tennis ball with air resistance."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### Note:\n",
    "\n",
    "According to the International Tennis Federation, [ITF](http://www.itftennis.com/home.aspx), the diameter of a tennis ball has to be in the range of $6.54$–$6.86 \\rm{cm}$, and its mass in the range of $56.0$–$59.4 \\rm{g}$. We chose a value in the middle of the range for each quantity."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def fall_drag(state):\n",
    "    '''Computes the right-hand side of the differential equation\n",
    "    for the fall of a ball, with drag, in SI units.\n",
    "    \n",
    "    Arguments\n",
    "    ----------    \n",
    "    state : array of two dependent variables [y v]^T\n",
    "    \n",
    "    Returns\n",
    "    -------\n",
    "    derivs: array of two derivatives [v (-g+a_drag)]^T\n",
    "    '''\n",
    "    R = 0.0661/2 # radius in meters\n",
    "    m = 0.0577   # mass in kilograms\n",
    "    rho = 1.22   # air density kg/m^3\n",
    "    C_d = 0.47   # drag coefficient for a sphere\n",
    "    pi = numpy.pi\n",
    "    \n",
    "    a_drag = 1/(2*m) * pi * R**2 * rho * C_d * (state[1])**2\n",
    "    \n",
    "    derivs = numpy.array([state[1], -9.8 + a_drag])\n",
    "    return derivs"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Assume the same initial conditions as before:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "y0 = y[0] # initial position\n",
    "v0 = 0    # initial velocity\n",
    "N = 576   # number of steps"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# initialize array\n",
    "num_sol_drag = numpy.zeros([N,2])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Set intial conditions\n",
    "num_sol_drag[0,0] = y0\n",
    "num_sol_drag[0,1] = v0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "for i in range(N-1):\n",
    "    num_sol_drag[i+1] = eulerstep(num_sol_drag[i], fall_drag, dt)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Time to plot and see how it looks! Would you expect the results to be better than in the previous case? Let's plot the three cases and check the differences."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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56DpjUUqVYc6ODrw9ZAwvf1WBNbFvk5QQi+Pm+dBuLLh4WB2euk7XTDIiEnwD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\n4qWXXspxexUrVqRmzZr22A8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qGHNwMLx1fzNucnoQz8v+XCGF8VvfICpsl9Wh5YkmGaWU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\nQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f0abc7d8668>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = pyplot.figure(figsize=(6,4))\n",
    "pyplot.plot(t[:N], num_sol[:,0], linewidth=2, linestyle='--', label='Num-solution no drag')\n",
    "pyplot.plot(t[:N], y[:N], linewidth=2, alpha=0.6, label='Experimental data')\n",
    "pyplot.plot(t[:N], num_sol_drag[:,0], linewidth=2, linestyle='--', label='Num-solution drag')\n",
    "\n",
    "pyplot.title('Free fall tennis ball \\n')\n",
    "\n",
    "pyplot.xlabel('Time [s]')\n",
    "pyplot.ylabel('$y$ [m]')\n",
    "pyplot.legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "All the lines look very close… but let's plot the differences with the experimental data in both cases, to get an idea of the error."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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MMStXrjQ333yz2bx5swHMvHnzjDHGvPXWW2bGjBkVlu3UqZOZNm1a+evJkyeb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\nCINugLAopyNTTYwmI0qpJqewpJQ3vt3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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f0abca42d68>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = pyplot.figure(figsize=(6,4))\n",
    "pyplot.plot(t[:N], y[:N]-num_sol[:,0], label='No drag')\n",
    "pyplot.plot(t[:N], y[:N]-num_sol_drag[:,0], label='With drag')\n",
    "pyplot.title('Difference between numerical solution and experimental data.\\n')\n",
    "pyplot.xlabel('Time [s]')\n",
    "pyplot.ylabel('$y$ [m]')\n",
    "pyplot.legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### Discuss with your neighbor\n",
    "\n",
    "* What do you see in the plot of the difference between the numerical solution and the experimental data?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## What we've learned\n",
    "\n",
    "* Integrating an equation of motion numerically.\n",
    "* Drawing multiple plots in one figure,\n",
    "* Solving initial-value problems numerically\n",
    "* Using Euler's method.\n",
    "* Euler's method is a first-order method.\n",
    "* Freefall with air resistance is a more realistic model."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## References\n",
    "\n",
    "1. _Elementary Mechanics Using Python_ (2015), Anders Malthe-Sorenssen, Undergraduate Lecture Notes in Physics, Springer. Data at http://folk.uio.no/malthe/mechbook/\n",
    "\n",
    "2. _The Physics Hyptertextbook_ (n/a), Glenn Elert, [Acceleration](https://physics.info/acceleration/)\n",
    "\n",
    "3. Euler method. (2017, October 13). In Wikipedia, The Free Encyclopedia. Retrieved 01:21, November 10, 2017, from https://en.wikipedia.org/w/index.php?title=Euler_method&oldid=805120184\n",
    "\n",
    "4. _Computational Physics with Python_, lecture notes by Eric Ayars, California State University, Chico. Available online on the author's website: https://physics.csuchico.edu/ayars/312/handouts/comp-phys-python.pdf"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
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       "}\t\n",
       "\n",
       ".text_cell_render h3 {\n",
       "    font-family: 'Merriweather', serif;\n",
       "\tfont-size: 150%;\n",
       "    margin-top:12px;\n",
       "    margin-bottom: 3px;\n",
       "    font-style: regular;\n",
       "    color: #008367;\n",
       "}\n",
       "\n",
       ".text_cell_render h4 {    /*Use this for captions*/\n",
       "    font-family: 'Merriweather', serif;\n",
       "    font-weight: 300; \n",
       "    font-size: 100%;\n",
       "    line-height: 120%;\n",
       "    text-align: left;\n",
       "    width:500px;\n",
       "    margin-top: 1em;\n",
       "    margin-bottom: 2em;\n",
       "    margin-left: 80pt;\n",
       "    font-style: regular;\n",
       "}\n",
       "\n",
       ".text_cell_render h5 {  /*Use this for small titles*/\n",
       "    font-family: 'Source Sans Pro', sans-serif;\n",
       "    font-weight: regular;\n",
       "    font-size: 130%;\n",
       "    color: #e31937;\n",
       "    font-style: italic;\n",
       "    margin-bottom: .5em;\n",
       "    margin-top: 1em;\n",
       "    display: block;\n",
       "}\n",
       "\n",
       ".text_cell_render h6 { /*use this for copyright note*/\n",
       "    font-family: 'Source Code Pro', sans-serif;\n",
       "    font-weight: 300;\n",
       "    font-size: 9pt;\n",
       "    line-height: 100%;\n",
       "    color: grey;\n",
       "    margin-bottom: 1px;\n",
       "    margin-top: 1px;\n",
       "}\n",
       "\n",
       "    .CodeMirror{\n",
       "            font-family: \"Source Code Pro\";\n",
       "\t\t\tfont-size: 90%;\n",
       "    }\n",
       "/*    .prompt{\n",
       "        display: None;\n",
       "    }*/\n",
       "\t\n",
       "    \n",
       "    .warning{\n",
       "        color: rgb( 240, 20, 20 )\n",
       "        }  \n",
       "</style>\n",
       "<script>\n",
       "    MathJax.Hub.Config({\n",
       "                        TeX: {\n",
       "                           extensions: [\"AMSmath.js\"], \n",
       "                           equationNumbers: { autoNumber: \"AMS\", useLabelIds: true}\n",
       "                           },\n",
       "                tex2jax: {\n",
       "                    inlineMath: [ ['$','$'], [\"\\\\(\",\"\\\\)\"] ],\n",
       "                    displayMath: [ ['$$','$$'], [\"\\\\[\",\"\\\\]\"] ]\n",
       "                },\n",
       "                displayAlign: 'center', // Change this to 'center' to center equations.\n",
       "                \"HTML-CSS\": {\n",
       "                    styles: {'.MathJax_Display': {\"margin\": 4}}\n",
       "                }\n",
       "        });\n",
       "</script>\n"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Execute this cell to load the notebook's style sheet, then ignore it\n",
    "from IPython.core.display import HTML\n",
    "css_file = '../style/custom.css'\n",
    "HTML(open(css_file, \"r\").read())"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}