notebooks/newtonLawForParticles.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "ZhJwhjmOM8ie",
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Motion of a particle - Newtonian approach\n",
    "\n",
    "> Renato Naville Watanabe, Marcos Duarte  \n",
    "> [Laboratory of Biomechanics and Motor Control](http://pesquisa.ufabc.edu.br/bmclab)  \n",
    "> Federal University of ABC, Brazil"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h1>Contents<span class=\"tocSkip\"></span></h1><br>\n",
    "<div class=\"toc\"><ul class=\"toc-item\"><li><span><a href=\"#Python-setup\" data-toc-modified-id=\"Python-setup-1\"><span class=\"toc-item-num\">1&nbsp;&nbsp;</span>Python setup</a></span></li><li><span><a href=\"#Study-of-motion\" data-toc-modified-id=\"Study-of-motion-2\"><span class=\"toc-item-num\">2&nbsp;&nbsp;</span>Study of motion</a></span></li><li><span><a href=\"#The-development-of-the-laws-of-motion-of-bodies\" data-toc-modified-id=\"The-development-of-the-laws-of-motion-of-bodies-3\"><span class=\"toc-item-num\">3&nbsp;&nbsp;</span>The development of the laws of motion of bodies</a></span></li><li><span><a href=\"#Newton's-laws-of-motion\" data-toc-modified-id=\"Newton's-laws-of-motion-4\"><span class=\"toc-item-num\">4&nbsp;&nbsp;</span>Newton's laws of motion</a></span></li><li><span><a href=\"#Steps-to-find-the-motion-of-a-particle\" data-toc-modified-id=\"Steps-to-find-the-motion-of-a-particle-5\"><span class=\"toc-item-num\">5&nbsp;&nbsp;</span>Steps to find the motion of a particle</a></span><ul class=\"toc-item\"><li><span><a href=\"#Example-1:-Ball-kicked-into-the-air\" data-toc-modified-id=\"Example-1:-Ball-kicked-into-the-air-5.1\"><span class=\"toc-item-num\">5.1&nbsp;&nbsp;</span>Example 1: Ball kicked into the air</a></span><ul class=\"toc-item\"><li><span><a href=\"#Analytical-solution\" data-toc-modified-id=\"Analytical-solution-5.1.1\"><span class=\"toc-item-num\">5.1.1&nbsp;&nbsp;</span>Analytical solution</a></span><ul class=\"toc-item\"><li><span><a href=\"#Plot\" data-toc-modified-id=\"Plot-5.1.1.1\"><span class=\"toc-item-num\">5.1.1.1&nbsp;&nbsp;</span>Plot</a></span></li></ul></li><li><span><a href=\"#Numerical-solution\" data-toc-modified-id=\"Numerical-solution-5.1.2\"><span class=\"toc-item-num\">5.1.2&nbsp;&nbsp;</span>Numerical solution</a></span><ul class=\"toc-item\"><li><span><a href=\"#Plot\" data-toc-modified-id=\"Plot-5.1.2.1\"><span class=\"toc-item-num\">5.1.2.1&nbsp;&nbsp;</span>Plot</a></span></li></ul></li></ul></li><li><span><a href=\"#Example-2:-Ball-kicked-into-the-air-considering-the-air-drag\" data-toc-modified-id=\"Example-2:-Ball-kicked-into-the-air-considering-the-air-drag-5.2\"><span class=\"toc-item-num\">5.2&nbsp;&nbsp;</span>Example 2: Ball kicked into the air considering the air drag</a></span><ul class=\"toc-item\"><li><span><a href=\"#Analytical-solution\" data-toc-modified-id=\"Analytical-solution-5.2.1\"><span class=\"toc-item-num\">5.2.1&nbsp;&nbsp;</span>Analytical solution</a></span><ul class=\"toc-item\"><li><span><a href=\"#Plot\" data-toc-modified-id=\"Plot-5.2.1.1\"><span class=\"toc-item-num\">5.2.1.1&nbsp;&nbsp;</span>Plot</a></span></li></ul></li><li><span><a href=\"#Numerical-solution\" data-toc-modified-id=\"Numerical-solution-5.2.2\"><span class=\"toc-item-num\">5.2.2&nbsp;&nbsp;</span>Numerical solution</a></span><ul class=\"toc-item\"><li><span><a href=\"#Plot\" data-toc-modified-id=\"Plot-5.2.2.1\"><span class=\"toc-item-num\">5.2.2.1&nbsp;&nbsp;</span>Plot</a></span></li></ul></li></ul></li><li><span><a href=\"#Example-3:-Ball-kicked-into-the-air-considering-the-air-drag-proportional-to-square-of-speed\" data-toc-modified-id=\"Example-3:-Ball-kicked-into-the-air-considering-the-air-drag-proportional-to-square-of-speed-5.3\"><span class=\"toc-item-num\">5.3&nbsp;&nbsp;</span>Example 3: Ball kicked into the air considering the air drag proportional to square of speed</a></span><ul class=\"toc-item\"><li><span><a href=\"#All-numerical-solutions-plotted-together\" data-toc-modified-id=\"All-numerical-solutions-plotted-together-5.3.1\"><span class=\"toc-item-num\">5.3.1&nbsp;&nbsp;</span>All numerical solutions plotted together</a></span></li></ul></li></ul></li><li><span><a href=\"#Further-reading\" data-toc-modified-id=\"Further-reading-6\"><span class=\"toc-item-num\">6&nbsp;&nbsp;</span>Further reading</a></span></li><li><span><a href=\"#Video-lectures-on-the-internet\" data-toc-modified-id=\"Video-lectures-on-the-internet-7\"><span class=\"toc-item-num\">7&nbsp;&nbsp;</span>Video lectures on the internet</a></span></li><li><span><a href=\"#Problems\" data-toc-modified-id=\"Problems-8\"><span class=\"toc-item-num\">8&nbsp;&nbsp;</span>Problems</a></span></li><li><span><a href=\"#References\" data-toc-modified-id=\"References-9\"><span class=\"toc-item-num\">9&nbsp;&nbsp;</span>References</a></span></li></ul></div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Study of motion\n",
    "\n",
    "In Mechanics we are interested in the study of motion (including deformation) and forces (and the relation between them) of anything in nature.  \n",
    "\n",
    "As a good rule of thumb, we model the phenomenon of interest as simple as possible, with just enough complexity to understand the phenomenon. \n",
    "\n",
    "For example, we could model a person jumping as a particle (the center of gravity, with no size) moving in one direction (the vertical) if all we want is to estimate the jump height and relate that to the external forces to the human body. So, mechanics of a particle might be all we need.  \n",
    "However, if the person jumps and performs a somersault, to understand this last part of the motion we have to model the human body as one of more objects which displaces and rotates in two or three dimensions. In this case, we would need what is called mechanics of rigid bodies.\n",
    "\n",
    "If, besides the gross motions of the segments of the body, we are interested in understanding the deformation in the the human body segments and tissues, now we would have to describe the mechanical behavior of the body (e.g., how it deforms) under the action of forces. In this case we would have to include some constitutive laws describing the mechanical properties of the body."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The development of the laws of motion of bodies  \n",
    "\n",
    "\"The theoretical development of the laws of motion of bodies is a problem of such interest and importance that it has engaged the attention of all the most eminent mathematicians since the invention of dynamics as a mathematical science by Galileo, and especially since the wonderful extension which was given to that science by Newton.\"\n",
    "\n",
    "&#8212; Hamilton, 1834 (apud Taylor, 2005).  \n",
    "\n",
    "**Let's start with the study of the forces and motion in Mechanics looking at the motion of a particle using the Newtonian approach.**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Newton's laws of motion\n",
    "\n",
    "The Newton's laws of motion describe the relationship between the forces acting on a body and the resultant linear motion due to those forces:\n",
    "\n",
    "- **First law**: An object will remain at rest or in uniform motion in a straight line unless an external force acts on the body.\n",
    "- **Second law**: The acceleration of an object is directly proportional to the net force acting on the object and inversely proportional to the mass of the object: $\\vec{\\bf{F}} = m \\vec{\\bf{a}}$.\n",
    "- **Third law**: Whenever an object exerts a force $\\vec{\\bf{F}}_1$ (action) on a second object, this second object simultaneously exerts a force $\\vec{\\bf{F}}_2$ on the first object with the same magnitude but opposite direction (reaction): $\\vec{\\bf{F}}_2 = −\\vec{\\bf{F}}_1.$\n",
    "\n",
    "These three statements are astonishing in their simplicity and how much of knowledge they empower.   \n",
    "Isaac Newton was born in 1943 and his works that resulted in these equations and other discoveries were mostly done in the years of 1666 and 1667, when he was only 24 years old!  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "rnbQefDbM8iq",
    "lang": "en",
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Steps to find the motion of a particle\n",
    "\n",
    "1. Draw a free body diagram of the particle. Draw all the forces being applied to the particle.  \n",
    "2. Write the expression of each force applied to the particle. For external forces (for example gravity and air friction) write the constitutive laws of the phenomena.  \n",
    "3. Write the Newton's second Law $\\vec{\\bf{F}} = m \\vec{\\bf{a}}$, where $\\vec{\\bf{F}}$ is the sum of all forces applied to the particle and $\\vec{\\bf{a}}$ is the particle acceleration.  \n",
    "4. Separate the equation into the 3 cartesian components (or 2 components if the movement is bidimensional).  \n",
    "5. Solve the differential equations\n",
    " 1. If possible, solve the differential equations analytically.  \n",
    " 2. If not possible to solve the differential equations analytically, separate each equation into 2 first order differential equations and use some numerical method (e.g. Euler, Runge-Kutta) to solve the first order differential equations with the aid of a computer.   \n",
    "6. Use the solution to interpret the situation, or to find some error on your approach.  \n",
    "\n",
    "**Later, we will study in details how to draw a free-body diagram.**  \n",
    "Let's see now some examples on how to find the motion of a particle"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "HlV8bOoJM8ir",
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Example 1: Ball kicked into the air\n",
    "\n",
    "#### Analytical solution\n",
    "\n",
    "A football ball is kicked with an angle of 30 degrees with the ground (horizontal).  \n",
    "The mass of the ball is 0.43 kg. The initial speed of the ball is 20 m/s and the initial height is 0 m. Consider the gravitational acceleration as 9.81 $m/s^2$.  \n",
    "Find the motion of the ball. \n",
    "\n",
    "Solution:  \n",
    "We know that:  \n",
    "<span class=\"notranslate\">\n",
    "$$ x_0 = 0 m \\\\ y_0 = 0 m $$  \n",
    "</span>\n",
    "As the angle of the initial velocity of the ball with the ground is 30 degrees:  \n",
    "<span class=\"notranslate\">\n",
    "$$ v_{x0} = 20 \\cos(30^\\circ) = 20\\frac{\\sqrt{3}}{2} = 10\\sqrt{3} m/s \\\\\n",
    "v_{y0} = 20 \\sin(30^\\circ) = 20 \\frac{1}{2} = 10 m/s $$  \n",
    "</span>\n",
    "The free-body diagram of the ball is depicted below:\n",
    "   \n",
    "<figure><center><img src=\"../images/ballGrav.png\" alt=\"free-body diagram of a ball\" width=\"500\"/><figcaption><i>Figure. Free-body diagram of a ball under the influence of gravity.</i></figcaption></center></figure>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "nOkZSRWyM8it",
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "The only force acting on the ball is the gravitational force:\n",
    "<span class=\"notranslate\">\n",
    "$$ \\vec{\\bf{F}}_g = -mg \\; \\hat{\\bf{j}} $$ \n",
    "</span>\n",
    "So, we apply the Newton's second law:\n",
    "<span class=\"notranslate\">\n",
    "$$ \\vec{\\bf{F}}_g = m \\frac{d^2\\vec{\\bf{r}}}{dt^2} \\quad \\rightarrow \\quad - mg \\; \\hat{\\bf{j}} = m \\frac{d^2\\vec{\\bf{r}}}{dt^2} \\quad \\rightarrow \\quad - g \\; \\hat{\\bf{j}} = \\frac{d^2\\vec{\\bf{r}}}{dt^2} $$\n",
    "</span>\n",
    "Now, we can separate the equation in two components (x and y):\n",
    "<span class=\"notranslate\">\n",
    "$$ 0 = \\frac{d^2x}{dt^2} $$\n",
    "</span>\n",
    "and\n",
    "<span class=\"notranslate\">\n",
    "$$ -g = \\frac{d^2y}{dt^2} $$\n",
    "</span>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import seaborn as sns\n",
    "%matplotlib inline\n",
    "sns.set_context('notebook', font_scale=1.2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "colab": {},
    "colab_type": "code",
    "id": "h_GGqUuXM8it",
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [],
   "source": [
    "m   = 0.43           # [kg]\n",
    "x0  = 0              # [m]\n",
    "y0  = 0              # [m]\n",
    "vx0 = 10*np.sqrt(3)  # [m/s]\n",
    "vy0 = 10             # [m/s]\n",
    "g   = 9.81           # [m/s^2]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "KlXsUdmkM8iy",
    "lang": "en",
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "These equations can be easily solved  by integrating both sides of each equation:\n",
    "<span class=\"notranslate\">\n",
    "$$ 0 = \\frac{d^2x}{dt^2} \\quad \\rightarrow v_{x0} = \\frac{dx}{dt} \\quad \\rightarrow \\quad v_{x0}t + x_{0} = x(t) $$\n",
    "</span>\n",
    "and\n",
    "<span class=\"notranslate\">\n",
    "$$ -g = \\frac{d^2y}{dt^2} \\quad \\rightarrow - gt + v_{y0} = \\frac{dy}{dt} \\quad \\rightarrow \\quad -\\frac{g}{2}t^2 +v_{y0}t + y_{0} = y(t) $$\n",
    "</span>\n",
    "So the trajectory of the ball along time is:\n",
    "<span class=\"notranslate\">\n",
    "$$ x(t) = v_{x0}t $$\n",
    "</span>\n",
    "and \n",
    "<span class=\"notranslate\">\n",
    "$$ y(t) =  -\\frac{g}{2} t^2 + v_{y0}t $$\n",
    "</span>\n",
    "or \n",
    "<span class=\"notranslate\">\n",
    "$$ \\vec{\\bf{r}}(t) =  v_{x0}t \\; \\hat{\\bf{i}} + \\left(-\\frac{g}{2} t^2 + v_{y0}t \\right) \\; \\hat{\\bf{j}} $$\n",
    "</span>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "colab": {},
    "colab_type": "code",
    "id": "03OWHJ_lM8iy",
    "outputId": "1f99c1f4-d870-4738-b25a-0d167b3eac84",
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 576x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "dt = 0.001  # [s]\n",
    "t  = np.arange(0, 2.05, dt)\n",
    "x1a = vx0*t\n",
    "y1a = -g/2*t**2 + vy0*t\n",
    "plt.figure(figsize=(8, 4))\n",
    "plt.plot(x1a, y1a, lw=4)\n",
    "plt.xlim(0, 36)\n",
    "plt.ylim(0, 6)\n",
    "plt.title('Ball trajectory')\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('y')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "XSH7-IM-M8i2",
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "#### Numerical solution\n",
    "\n",
    "We can solve this problem numerically and compare both solutions.\n",
    "\n",
    "We start from the differential equations of each coordinate, and then break them into two first-order differential equations:\n",
    "<span class=\"notranslate\">\n",
    "$$ 0 = \\frac{d^2x}{dt^2} $$\n",
    "</span>\n",
    "and\n",
    "<span class=\"notranslate\">\n",
    "$$ - g = \\frac{d^2y}{dt^2} $$\n",
    "</span>\n",
    "The first equation can be broken as:\n",
    "<span class=\"notranslate\">\n",
    "$$ \\frac{dv_x}{dt} = 0 $$\n",
    "</span>\n",
    "<span class=\"notranslate\">\n",
    "$$ \\frac{dx}{dt} = v_x(t) $$\n",
    "</span>\n",
    "And the second equation can be broken as:\n",
    "<span class=\"notranslate\">\n",
    "$$ \\frac{dv_y}{dt} = -g $$\n",
    "</span>\n",
    "<span class=\"notranslate\">\n",
    "$$ \\frac{dy}{dt} = v_y(t) $$\n",
    "</span>\n",
    "You can use any numerical integration method you want (Euler, Runge-Kutta, etc), but here we will use the Euler method. Let's see the solution for the $x(t)$ variable.\n",
    "\n",
    "The derivative of $x(t)$ is given by:\n",
    "<span class=\"notranslate\">\n",
    "$$ \\frac{dx}{dt} = \\lim\\limits_{\\Delta t \\rightarrow 0} \\frac{x(t+\\Delta t) - x(t)}{\\Delta t} $$\n",
    "</span>\n",
    "Whcih can be approximated by: \n",
    "<span class=\"notranslate\">\n",
    "$$ \\frac{dx}{dt} \\approx \\frac{x(t+\\Delta t) - x(t)}{\\Delta t} \\quad \\rightarrow \\quad x(t+\\Delta t) \\approx x(t) + \\Delta t \\frac{dx}{dt} $$\n",
    "</span>\n",
    "So, with the initial conditions of all the variables, we can apply the equation above to find the values of the variables along time (for a revision of Ordinary Differential Equations, see the notebook [Ordinary Differential Equation](https://nbviewer.jupyter.org/github/BMClab/bmc/blob/master/notebooks/OrdinaryDifferentialEquation.ipynb)).  \n",
    "In the cell below, we apply the Euler method for the four first-order differential equations."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "colab": {},
    "colab_type": "code",
    "id": "dMNLst17M8i4",
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [],
   "source": [
    "x  = x0\n",
    "y  = y0\n",
    "vx = vx0\n",
    "vy = vy0\n",
    "r  = np.array([x, y])\n",
    "while y >= 0:\n",
    "    dxdt = vx\n",
    "    x = x + dt * dxdt\n",
    "    dydt = vy\n",
    "    y = y + dt * dydt\n",
    "    dvxdt = 0\n",
    "    vx = vx + dt * dvxdt\n",
    "    dvydt = -g\n",
    "    vy = vy + dt * dvydt\n",
    "    r = np.vstack((r, np.array([x,y])))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "colab": {},
    "colab_type": "code",
    "id": "RpDwAX9dM8i7",
    "outputId": "1b690a23-0e59-4ba0-9cc3-cda7f5b4e967",
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 576x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(8, 4))\n",
    "x1n = r[:, 0]\n",
    "y1n = r[:, 1]\n",
    "plt.plot(x1n, y1n, lw=4)\n",
    "plt.xlim(0, 36)\n",
    "plt.ylim(0, 6)\n",
    "plt.title('Ball trajectory')\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('y')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "nfEL9e-tM8jA",
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Example 2: Ball kicked into the air considering the air drag\n",
    "\n",
    "Now, besides the gravity, we consider the drag due to the air resistance ($b$ = 0.006 Ns/m). First we will consider the drag force  proportional to the speed and opposite direction to the velocity vector.\n",
    "       \n",
    "<figure><center><img src=\"../images/ballGravLinearRes.png\" alt=\"free-body diagram of the ball\" width=\"500\"/><figcaption><i>Figure. Free-body diagram of a ball under the influence of gravity and drag.</i></figcaption></center></figure>\n",
    "\n",
    "So the forces being applied on the ball are:\n",
    "<span class=\"notranslate\">\n",
    "$$ \\vec{\\bf{F}} = -mg \\; \\hat{\\bf{j}} - b\\vec{\\bf{v}} = -mg \\; \\hat{\\bf{j}} - b\\frac{d\\vec{\\bf{r}}}{dt} = -mg \\;  \\hat{\\bf{j}} - b\\left(\\frac{dx}{dt} \\; \\hat{\\bf{i}}+\\frac{dy}{dt} \\; \\hat{\\bf{j}}\\right) = - b\\frac{dx}{dt} \\; \\hat{\\bf{i}} - \\left(mg + b\\frac{dy}{dt}\\right) \\; \\hat{\\bf{j}} $$\n",
    "</span>\n",
    "Writing down the Newton's second law:\n",
    "<span class=\"notranslate\">\n",
    "$$ \\vec{\\bf{F}} = m \\frac{d^2\\vec{\\bf{r}}}{dt^2} \\quad \\rightarrow \\quad - b\\frac{dx}{dt} \\; \\hat{\\bf{i}} - \\left(mg + b\\frac{dy}{dt}\\right) \\; \\hat{\\bf{j}} = m\\left(\\frac{d^2x}{dt^2} \\; \\hat{\\bf{i}}+\\frac{d^2y}{dt^2} \\; \\hat{\\bf{j}}\\right) $$\n",
    "</span>\n",
    "Now, we can separate into one equation for each coordinate:\n",
    "<span class=\"notranslate\">\n",
    "$$ - b\\frac{dx}{dt} = m\\frac{d^2x}{dt^2} \\quad \\rightarrow \\quad \\frac{d^2x}{dt^2} = -\\frac{b}{m} \\frac{dx}{dt} $$\n",
    "</span>\n",
    "<span class=\"notranslate\">\n",
    "$$ -mg - b\\frac{dy}{dt} = m\\frac{d^2y}{dt^2} \\quad \\rightarrow \\quad \\frac{d^2y}{dt^2} = -\\frac{b}{m}\\frac{dy}{dt} - g $$ \n",
    "</span>\n",
    "We can solve these equations analytically, for example, by using Laplace Transform or classical methods to solve linear differential equations."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "x41TfRSQM8jB",
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "#### Analytical solution\n",
    "\n",
    "The solution of a linear differential equation can be found by finding the natural (or homogeneous) solution and the forced (or non-homogeneous) solution and then adding both solutions.\n",
    "\n",
    "First, we solve the first differential equation ($x$ coordinate). The characteristic polynomial of the equation is:\n",
    "<span class=\"notranslate\">\n",
    "$$ \\lambda^2 + \\frac{b}{m}\\lambda = 0 $$ \n",
    "</span>\n",
    "The roots of this equation are $\\lambda = 0$ and $\\lambda = -\\frac{b}{m}$, and consequently, its natural modes are:\n",
    "<span class=\"notranslate\">\n",
    "$$ x_{n_1}(t) = Ae^{0t} = A \\\\\n",
    "x_{n_2}(t) = B e^{-\\frac{b}{m}t} $$ \n",
    "</span>\n",
    "As there is no external forces in the $x$ direction, there is no forced solution. So, the motion of the ball in the $x$ coordinate is:\n",
    "<span class=\"notranslate\">\n",
    "$$ x(t) = A + Be^{-\\frac{b}{m}t} $$\n",
    "</span>\n",
    "To find the values of the $A$ and $B$ constants, we must use the initial conditions $x(0)$ and $v_x(0)$.\n",
    "<span class=\"notranslate\">\n",
    "$$ x(0) = 0 = A + B $$\n",
    "</span>\n",
    "<span class=\"notranslate\">\n",
    "$$ v_x(0) = v_{x0} = \\frac{dx(0)}{dt} = -\\frac{Bb}{m}e^{-\\frac{b}{m}0} \\quad \\rightarrow \\quad B = -\\frac{v_{x0}m}{b} \\quad \\rightarrow \\quad A = \\frac{v_{x0}m}{b} $$\n",
    "</span>\n",
    "So:\n",
    "<span class=\"notranslate\">\n",
    "$$ x(t) = \\frac{v_{x0}m}{b} - \\frac{v_{x0}m}{b}e^{-\\frac{b}{m}t} = \\frac{v_{x0}m}{b}\\left(1-e^{-\\frac{b}{m}t} \\right) $$\n",
    "</span>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "RqJr2O4NM8jC",
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Now, we solve the differential equation for the $y$ coordinate.  \n",
    "First, we find the natural solution, which comprises the solution from the equation without the gravitational force (the force indepedent of y(t) and its derivatives), in this case:\n",
    "<span class=\"notranslate\">\n",
    "$$ \\frac{d^2y}{dt^2} = -\\frac{b}{m}\\frac{dy}{dt} $$\n",
    "</span>\n",
    "The solution of this equation is the same from the $x$ coordinate:\n",
    "<span class=\"notranslate\">\n",
    "$$ y_n(t) = A + Be^{-\\frac{b}{m}t} $$\n",
    "</span>\n",
    "The forced solution (including the gravitational force, which is constant) happens when every derivative, with the exception of the derivative with the lowest order (in this case order 1 but it could be order 0, i.e. no derivative) goes to zero. \n",
    "<span class=\"notranslate\">\n",
    "$$\\frac{d^2y}{dt^2} \\quad = \\quad 0 \\quad = \\quad -\\frac{b}{m}\\frac{dy_f}{dt} - g \\quad \\rightarrow \\quad \\frac{dy_f}{dt} = -\\frac{mg}{b} \\quad \\rightarrow \\quad y_f(t) = -\\frac{mg}{b}t $$\n",
    "</span>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "5EzIiZbUM8jE",
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "The complete solution of the motion of the ball in the $y$ coordinate is the sum of the natural and forced solutions:\n",
    "<span class=\"notranslate\">\n",
    "$$ y(t) = A + Be^{-\\frac{b}{m}t} - \\frac{mg}{b}t $$\n",
    "</span>\n",
    "To find the values of the constants A and B, we must use the initial conditions $y(0)$ and $v_y(0)$.\n",
    "<span class=\"notranslate\">\n",
    "$$ y(0) = 0 = A + B $$\n",
    "</span>\n",
    "<span class=\"notranslate\">\n",
    "$$ v_y(0) = v_{y0} = \\frac{dy(0)}{dt} = - \\frac{Bb}{m} - \\frac{mg}{b} \\\\\n",
    "B = -\\frac{m^2g}{b^2} -\\frac{v_{y0}m}{b} \\\\\n",
    "A = +\\frac{m^2g}{b^2} + \\frac{v_{y0}m}{b} $$\n",
    "</span>\n",
    "So, the motion of the ball in the y coordinate is:\n",
    "<span class=\"notranslate\">\n",
    "$$ y(t) = \\left(\\frac{m^2g}{b^2} + \\frac{v_{y0}m}{b}\\right) - \\left(\\frac{m^2g}{b^2} + \\frac{v_{y0}m}{b}\\right)e^{-\\frac{b}{m}t} -\\frac{mg}{b}t \\\\\n",
    "y(t) = \\left(\\frac{m^2g}{b^2} + \\frac{v_{y0}m}{b}\\right)\\left(1 - e^{-\\frac{b}{m}t}\\right) - \\frac{mg}{b}t $$\n",
    "</span>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "colab": {},
    "colab_type": "code",
    "id": "04ZSdYNWM8jF",
    "outputId": "9ea68054-1836-4455-f488-59e61bb184db",
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAfAAAAEiCAYAAAAGWoDCAAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADh0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uMy4xLjMsIGh0dHA6Ly9tYXRwbG90bGliLm9yZy+AADFEAAAgAElEQVR4nO3dd3hUZdrH8e+dTiAktNAh9I4UBQFR17IqKhbsZe3o6mvdYnfFsrZdRV1X196xYseOCgqKNOmd0Ak9BEJCyvP+MYPkpECAZM7M5Pe5rnNB7jln5j4c4J7znKeYcw4RERGJLDF+JyAiIiL7TgVcREQkAqmAi4iIRCAVcBERkQikAi4iIhKBVMBFREQikAq4SAQxM2dmF5T4OdPM7ghxDiH/TBEpSwVcpJqZ2cvBwrtryzaziWY2JASffVjwMzOq8G0PAR6rqjczsxbBHI+sqvcUqQlUwEVCYzzQNLgdCkwFPjSzdr5mFWRm8WZmldnXObfeObe9unPaH2aW4HcOIqGiAi4SGjudc2uD21zgFiAe6LlrBzM7z8x+Cd6hbzCzz8ys4/5+YPCue3zwx6XBu9zvg6+9bGbfmNm1ZpYJ5AO1zexYM/vezDYF8/jBzPqVel9PE7qZxZnZ3Wa21MzyzGy2mV1Z6pg6ZjbSzFaYWX7wPW4Lvrwi+Ot3wRwzSxx3kZnNCR6z0szuM7O4Eq9/b2YvmNm9ZrYGWGVmI8xsfjl/Hi/tOn+RaBC3911EpCoF7xKvIFA0p5Z4KRG4F5gL1AVGAJ+ZWTfn3M79+KgVwCnAR0C/4M8l36cfkAOcChQBeUAd4CngNwJfMG4EvjCzDs65jRV8zvNAH+BKYGHwff9nZoXOuReCd/afAq2Aa4EZQAugU/D4PsE/h2HAhGAumNmJwIvAHcD7QG/gGcABd5b4/LOAN4CjgVhgM3C7mR3hnPsh+F4pwJnBHEWiggq4SGgcaWbbgr9PBnKBPznnlu3awTn3UskDzOxiYCOBZ84/7esHOueKzGxT8Mf1zrm1pXYpBi50zm0rEfugVA7DCRTW4wkUSUq93gb4E9DVOTcvGF5qZp0IFOsXgKOAI4BDnHOTg/ssAcbtyi3466ZSOd4CvO+ceyD48wIzawI8aGb3lvhSswa42jlXXCKvMQS+JP0QDJ1H4AvT+6XPQSRSqQldJDR+AXoFtz7APcArZnbcrh3MrJeZfRBsis4Blgdfal1NOc0tVbwxszZm9pqZLTKzrcBWIHUPORwMGDDZzLbt2oDbgA7BffoCm0sU78rqxu4iv8sPQBJQsu/AlJLFO+h/wDAzqxf8+QrgNedc3j7mIBK2dAcuEho7nHOLSvw83cyOBm4HvjSzZOAr4EfgUmDXnehsoLo6ZpXXEe1TYANwDbub3H/cQw67bgIGEmhVKMlV8Pt9Ufo4Kyde3nl8DmQBF5rZOAJfIi7azxxEwpIKuIh/Cgk0pwN0ARoBtwc7uWFmA9ldsPbXrmbm2L3taGYNgK7AEOfcl8FYCyB9D4dNCf7ayjn36R72qW9mB1dwF15RjrMJNL0/VSJ2OLCDQBN8hZxzxWb2PIE7707ABOfc7D0dIxJp1IQuEhoJZtYkuLUzs6uB49j9zHkZgWe01wZfPxp4nP2/c91lGYFn3UPMLN3MUvew72YCz6OvMLOOZjYAGEWgYJYr2KrwIvCcmV1oZu3N7CAzu9TMbg7uNpZAb/i3zeyUYDP9IDO7PPj6BmAb8Mfgn8+uZu8HCDSD3xLM5yzgbuDflezU9wLQGbgceLYS+4tEFBVwkdAYTKCz1RpgJoEm6lsIFCmccxuAC4BjCdx5/gv4K4Hiu9+cc1nArcHPWkOgR3pF+xYT6KndjkBP8ZeBkcHj9mQ4gYldbgfmAN8SaK5eEnxfB5wIjCHQi3w+8DrQsMTnXkOgN/kKYFowPobA44SLgFnBz/gvgd75lTn3NQQeCewA3qnMMSKRxAL/tkREKic43voh59xIv3PZGzObBPzinLvW71xEqpqegYtIpZhZHQKd1RoTuEMPW2aWTmAMfB/gXJ/TEakWIW9CN7M+ZvatmeWa2WYzU9OWSGT4K4Fn4k8458b6ncxeZAEPAdc75xb7nYxIdQhpE7qZdQEmEniW9R6B53tdnHOjQ5aEiIhIFAh1AX+fwIQOl+91ZxEREalQyAq4mcUSmNXpQQJzFncm8BztJufcrFL7pgFppd4iAWhLYK7lompPWERExF+xBFYw/NU5l1/6xVAW8CYEhqNsI/As7Vfg/4AhQAfnXE6Jfe8G/hGSxERERMLbYOfcj6WDoSzgzYBVwMvOuUuCsfhg7Hrn3KgS+5Z3B94a+H78+PG0aNEiJDmLiIj4ZeXKlQwePBigfXmdMUM5jGwDgabv39fpdc4VmNkSoGXJHZ1zW4AtJWOBFQmhRYsWZGRkVHeuIiIi4aLcx8YhG0YWnPpwGtB+V8zM4oAMdq+6JCIiIpUQ6olcHgNeMLPvCDwDv47AN4uKFkEQERGRcoS0gDvn3gzOkPQggWfck4BjS69JLCIiInsW8qlUg/Mnh/0cyiIiIuFMq5GJiIhEIBVwERGRCKQCLiIiEoFUwEVERCKQCriIiEgEUgEXERGJQCrgIiIiEUgFXEREJAKpgIuIiEQgFXAREZEIpAIuIiISgVTARUREIpAKuIiISARSARcREYlAKuAiIiIRSAVcREQkAqmAi4iIRCAVcBERkQikAi4iIhKBVMBFREQikAq4iIhIBFIBFxERiUAq4CIiIhFIBVxERCQCqYCLiIhEoLhQfZCZvQxcVCp8o3NuZKhyEKkp8gqKWLc1n/Xb8li3NZ+teQVszy9iR0ER2/MLyd1ZRO7OQvIKigEwAwNigr8xjOSEWOokxVEnMY6U4K91EuNIS06gSd0k0usmkhQf6++JitRgISvgQe8A15f4eWuIP18kKuQVFLF8Uy5L1m8nc+N2MjdsZ9nGXLJy8lifk09OXmFI8qiXHE/jukk0SU2iSd0k2jSsTbtGdWiXXoeW9WoRF6tGPpHqEuoCvsM5tzbEnykSsZxzrM7OY+bKbGavzmbWqmwWZG1jdfYOnPM7O9icW8Dm3ALmrc0p81p8rJHRIFDQOzVJ4aCWqfRonkajlEQfMhWJPqEu4EPNbD2wBhgFPOKcK3OrYGZpQFqpcIsQ5Cfiq235hUzO3MSkpZuYuSpQsDfnFvid1n4pKHIsXLeNheu28cXs3d/bm6Um0bNFGj1bptKrRRq9W9WjVoKa4kX2VSgL+BjgLWAV0Bd4BEgBbitn3xuAf4QuNRF/bM0rYHLmJn5Zsomfl2xk1uqtFBUf+K11XIzRKCWRRimJpKckkpacQO2EWJIT46idEEuthMCvu55hOxzOQbEL3PU7BzsKisjJKyAnv5BteYVsyy8kJ6+Qjdt3kpWdx7qcPPYn1dXZeazOXvt7UU+IjaF3qzQGtGvAwHYN6dUyjYQ4Nb2L7I05n9rhzOxC4GkgxZVKYg934OOXLl1KRkZGaJIUqWLOOeZn5fDt3HWMnbeOacs371cRNINmqbVo07A2bRrWJqNhbdo0TKZZWi3SU5JIqxVPTIxV/QmUUFTs2LAtn7XZeazdmsfKzTtYsn4bi9dvY/H67azPyd+v902Kj+GQjPoc0bERx3ZtTOsGtas4c5HIkJmZSZs2bQDaOOcyS7/uZwHvBswC0p1z6yuxfwawVAVcIk1eQRETl2xkbLBor9qyY5+OT06IpWvTunRvnkr35ql0a1aXNg1rh30P8OwdBSxZv42FWduYsWoLM1ZmM3fNVgqK9u3/nI6N63Bs18Yc27UJPZunVvsXE5FwEc4F/GzgRaBO6TvwCvbPQAVcIkRBUTHjF67n4+mr+XpOFtt3FlX62HaNatO/bQMOyahHj+ZptGlYm9goKVr5hUXMX5vDbyuz+W3FFn5ZupEVmyr/hSY9JZFjuzbm1N7N6duqnoq5RLWwKeBm9ijwNpAF9AGeBN52zt1UyeMzUAGXMFZU7Ji0dBMf/7aaz2etYUslO591bFyH/m0a0L9tffq1qU96SlI1ZxpeVmzKZeKSjUxcvJEJizeQtbVyTe8t6tXilF7NOK13c9qnp1RzliKhF04F/AsCndfqAsuB14CHnHOV+teqAi7havnGXN76dTnvT11ZqeKTGBfDYe0bclSXdI7qnE7T1FohyDIyOOdYsmE7389fz9dz1vJr5uZKderr1qwup/Vuzmm9m9OgjoapSXQImwJ+oFTAJZzsLCzmm7lZjJq0nPELN+x1/11Nv0d3SWdA24YaNlVJW3J38t38dXw9J4sf5q/f66OIhNgYTujRhAsObc3BrethpiZ2iVwq4CJVaNnG7bw5aTnvTV7Jxu0797hvWnI8J3RvytCDmtGvTf2oeY7tl/zCIsYv2MCH01fx9Zws8guL97h/p8YpnH9oK07t3Zy6SfEhylKk6qiAixwg5xyTl23muXFL+Hpu1h5nQEtOiOWPXRsztFczDmvfSOOZq0lOXgFfzs7iw2mrmLB4wx6H4iUnxHJa7+ZcMbgtGQ01JE0ihwq4yH4qLCrm81lreX78En5bmb3HfXu1TOO8fq04sWdTaieGeoLDmi1rax4fTlvFqEnLydyYW+F+ZnB8tyYMP7wtvVvVC2GGIvtHBVxkH23PL2TUpOW89FPmHsdspyTFcXrv5pzTrxVdmtYNYYZSnuJix0+LN/DGz8v5em7WHju/9WtTnysPb8sfOqVrKJqErb0VcN0qiARtzy/k1YnLeG78Ejbt4fl216Z1uWRQBif1bKbOaGEkJsYY3KERgzs0Ym12Hm//uoJRk5azdmtemX0nLQ3MN98hvQ7XH9OBId2bqpBLxNEduNR42/ILeXViJs+NW7LHhUOO7pzOZYPbMKBtA/VujhAFRcV8NmMN/xu3hLlrKl69uHOTFG44piPHdWusaythQ03oIhXYll/IKxMyeW78kgonXUmMi2FY3xZcOqgN7dPrhDhDqSrOOcYv3MCz45bw46KKh/11a1aXm47tyFGd01XIxXcq4CKlFBQV89ak5Yz8ZmGFQ8HqJMZxyaAMLh6YoYlBosysVdk8O24Jn85YXWHv9YNapnHz8Z0Y2K5haJMTKUEFXCTIOceXs9fy0BfzWbphe7n7pAQL96WHtSEtOSHEGUooLV6/jce/WcgnM1ZXODTwmC7p3DqkC+0aqfVFQk8FXASYsmwT/xwzjynLNpf7ekpiHJcc1obLBrUhNVmTftQkC7JyGPnNAsbMXFvu63Exxvn9W3H9MR2pX1tf6iR0VMClRluxKZd/jpnL57PK/8+5Vnwslw9uw+WHtVXhruHmrN7KyG8W8NWcrHJfT0mK49qj2nPRwAwS4zT6QKqfCrjUSHkFRTw7bglPfbeo3Ck3YwzOPqQlNx7TkfS6NWv1L9mzqcs3c9+nc5i6fEu5r7dukMyIod04slN6iDOTmkYFXGqcsfOyGPHJHJZVMCvXMV3Sufn4znRorCUopXzOOT6buYaHvphX4XrlQ3o04c6Tumo1Oak2mshFaowVm3IZ8ckcvplbfhNozxap3HpCFwa0axDizCTSmBkn9WzGsV0b88qETJ4cu4icvELPPmNmruWH+eu58diOXDQwg/hYzXsvoaU7cIl4BUXFPDtuCU98u7Dc5vL6tRO45YTOnNGnhWbbkv2yaftOHv16Pm/8srzcHuudm6Rw36ndOTijfuiTk6ilJnSJarNWZfO392aUO8tWjMEFh7bmL8d2Ugc1qRIzVm7hjg9nMaOCxW3+NKA1Nx/fWQvaSJVQAZeolFdQxMhvFvLc+CXlLlrRp1Ua95zSne7NU33ITqJZUbHjzUnLefiLeWWa1QFa1KvFQ8N6Mqi9JoGRA6MCLlHnlyUbuWX0zHInY6lfO4FbT+jMMDWXSzVbn5PPA2PmMnraqnJfP7dfK24b0pmUJLX+yP5RAZeokbuzkAfGzOO1n5eV+/qpvZpx18ndNNmGhNTExRu5ZfSMckc9NEtN4oFhPTmiYyMfMpNIt7cCrm6TEhGmLd/MiU/8WG7xbpqaxIsXH8zIc3qreEvIDWjXgC+uP5xLB7Wh9Ponq7PzuOjFSdw6eia5O8s2t4scCBVwCWuFRcU89vUCznhmYrlN5hcc2oqvbjycozo39iE7kYBaCbHcdXJX3r1yAG0b1i7z+qhJyznpyR+Ztar8zm8i+0MFXMLW0g3bGfbMRB7/dmGZjmoZDZJ5e/ih3HdqDz1jlLBxcEZ9xlw/mOGHt6V0F4wl67dz2n9/4pkfFlNc0TJoIvtABVzCjnOON35ZxpDHx/PbirLTWZ7fvxVjrh9M/7aakEXCT1J8LLcN6cJ7fx5Im1J34wVFjgc/n8f5z//CmuzyZ3gTqSwVcAkr2TsKuOr1Kdz+wSx2FBR5XmtYJ5EXLz6Y+0/rQXKCxtlKeOvTqh6fXnsY5xzSssxrE5ds5PiR4/l85hofMpNooQIuYWP6ii2c+MR4vpxddirUY7s25ssbButZt0SU2olxPDisJ89c0Ie0UpMJZe8o4M9vTOXuj2ezs5wZBEX2xrcCbmYfmJkzsyP9ykHCg3OOF35cypnPTGDlZm+zYu2EWB4e1pNnL+xLgzqJPmUocmCO796UL64/nEHtyz72eXlCJmf9byKrtqhJXfaNLwXczC4EynbVlBonO7eA4a9N4d5P51BQ5O3Y06N5KmOuH8xZh7TESo/PEYkwTVKTeO3S/tw+pAvxsd6/z7tan76bv86n7CQShbyAm1lz4D7g8lB/toSXacs3M+SJ8Xw9p2yT+cUDM3jvzwNo3UDf8yR6xMQYVxzelnevGkjzNO8ypFtyC7jkpV/591fzy50eWKQ0P+7Anwf+6ZxbXtEOZpZmZhklN6BFqBKU6jdq0vJymw1TkuJ45oI+3D20G4lxsT5lJ1K9erVM47PrDuOozullXnty7CIufOEXNm7L9yEziSQhLeBmdiUQ75z73152vQFYWmobX83pSQjkFxZx2wczuXX0zDJN5j1bpPLZtYM5vntTn7ITCZ205ASe/9PB/P34TmXGjE9YvJGh//mJ2as18YtULGQF3MxaAf+gck3nI4E2pbbB1ZedhMK6rXmc++zPvPlL2caXSwZl8O5VA2jVINmHzET8ERNjXH1ke9684lAapXg7aa7asoNhT0/g0xmrfcpOwl0o78D7AE2ARWZWaGa7Jgb+1sxeKLmjc26Lcy6z5AasDGGuUsWmLNvESU/+yNTl3olZasXH8uS5vfnHyWoyl5rr0LYN+Oy6wzi0bX1PPK+gmP97cxr/+nK+Zm+TMkJZwL8FegK9SmwQuCO/K4R5SIi9+ctyznn2Z9bleJ/ptaqfzOirB3LyQc18ykwkfKSnJPHaZf25eGBGmdf+890ihr82mZy8gtAnJmErZAXcOZfjnJtVcgu+tNQ5V/6CuhLRCouK+cdHs7jtg7LPuwd3aMjH/zeILk3r+pSdSPiJj43h7qHdeGhYjzJDzb6Zu47T/juh3EV9pGbSTGxSLXLyCrj81cm8MrHs8p9/PrIdL1/Sj7RkLf0pUp6zD2nFW8MPpWGpyYsWrdvG6f/9iV8zN/mUmYQTXwu4c86cc9/7mYNUvZWbcznj6Yl8P3+9J56cEMtT5/Xh5uM7E1u6262IePRtXZ9Prh1Ezxapnvjm3ALOf+4XPv5NndtqOt2BS5Watnwzpz41gflZOZ5409Qk3rtqICf21BAxkcpqmlqLd64cwKm9vP1EdhYVc92oaTz13SKcU+e2mkoFXKrMZzPWcM6zP7Oh1AQUPVuk8tE1g+jaTM+7RfZVUnwsj53dixuP6VjmtUe+nM8t78+koEiLodREKuBywJxz/Pf7RVzz5lTyS62qdEL3Jrw9fADpdZN8yk4k8pkZ1x/TgUfPOqhM57a3J6/gkpd+Zat6qNc4KuByQIqKHXd/PJuHv5hf5rWrj2zHU+f1oVaCxneLVIXT+7Tg1Uv7UzcpzhP/cdEGznx6Imuz83zKTPygAi77La+giGtHTS3T0zw+1njkjJ78/fjOxKizmkiVGtCuAaOvHkTL+t7FUOZn5TDs6QksWb/Np8wk1FTAZb9k7yjgohcnMWbmWk88JSmOVy/tz5kHt/QpM5Ho1z69Dh9cPYheLdM88VVbdnDGMxOZuVJzqNcEKuCyz9Zm53H2/ybyy1LvWNQmdZN496oBDGjXwKfMRGqOhnUSeWv4oRzTpbEnvmn7Ts55diITFm3wKTMJFRVw2SeL1gWa6eat9Q4Ta59eh/evHkjnJuppLhIqSfGxPHNBH87o611tefvOIi5+6VfGzFzjU2YSCirgUmm/rdjCGc+UXcO7b+t6vHfVAJqn1argSBGpLnGxMTxyRk+uPKKtJ76zqJhr3pzK6z+XnQ1RooMKuFTKL0s2cv7zv7Al1ztU5ZgujXnj8v6aFlXER2bGrSd04bYhnT1x5+COD2fxzA+LfcpMqpMKuOzVDwvWc9FLk9iWX+iJn9uvFc9c0IekeA0TEwkHww9vx7/OPKjMVMUPfj6Px79ZqFnboowKuOzRF7PWcPkrv5JX4J2g5eoj2/HP07oTF6u/QiLh5Iy+LXj2wr4kxnn/bT72zQIe/nK+ingU0f++UqHRU1dyzZvTyiwF+vfjO/H34ztjpjHeIuHo6C6NeeXSfiSXmkTp6e8Xc8+nc1TEo4QKuJTr9Z+XcdM7v1FU7P2HPmJoN64+sr1PWYlIZR3atgGvXdaPlETvrG0v/ZTJ7R/OorhYRTzSqYBLGc+PX8IdH87yxGIMHjmjJxcNzPAnKRHZZ31b1+fNKw4lLTneE3/zl+X8/f0ZZb6gS2RRAReP58cv4b7P5npi8bHGf87ro9nVRCJQjxapjLriUBrU9o4UeW/KSv7yznQV8QimAi6/e25c2eKdGBfDsxcezJAeWsdbJFJ1aVqXt688lPSURE/8w+mr+dt7ZR+VSWRQARcgULzvH+Mt3knxMbx08SH8oXO6T1mJSFVpn57CO1eWnXBp9NRV3Dp6hp6JRyAVcOHZcYvLLd4vXnwIA9s39CkrEalqGQ1r89bwQ2mWmuSJvzN5Jbd/OFNFPMKogNdwz45bzD/HzPPEfi/e7VS8RaJNy/rJjBp+KE3qeov4qEkruOvjWRpiFkFUwGuw58YtUfEWqYFaN6jNqOFln4m//vNyRnyiceKRQgW8hnplQmYFz7z7qXiL1ABtGgaKeMM63iL+8oRM7vtsrop4BFABr4HembyCf3w82xPbVby1lrdIzdGuUR1GXdG/zBCzF35cyuPfLvQpK6ksFfAa5pPfVnPL+zM8sV3N5ireIjVPh8YpvHnFodQvVcRHfrOQF35c6lNWUhkq4DXIN3OyuPHt6ZTsaBofa/zvwoPVbC5Sg3VqksLrl/WnbpJ32tV7P53DO5NX+JSV7E1IC7iZjTCzBWa2w8yyzOwNM2sSyhxqqh8XbuDqN6dSWKJ6x8YYT57bhyM6NvIxMxEJB12b1eWlS/pRq9TywLe8P4PPZ67xKSvZk1Dfgc8DrgS6ACcDrYBXQpxDjTM5cxNXvDqZnYW7lwQ1g3+feRDHd9f3JxEJ6Nu6Hs/+qS8JJZYJLnZw3VvTGLdgvY+ZSXkqXcDNbLGZ3WJmjff3w5xzo5xz3znnMp1zk4CHgUH7+36yd7NWZXPJS7+yo6DIE//naT04tXdzn7ISkXA1uEMjnji3NzElVgsuKHJc+doUpizb5F9iUsa+3IG/SuDuebmZjTaz4w7kg80sFTgP+OlA3kcqlrlhOxe/NImc/EJP/M6TunJuv1Y+ZSUi4e747k14+IyDPLEdBUVc/NKvzFu71aespLRKF3Dn3AigLTAUKAI+NrNMM7vDzJpV9n3M7Hwz2wZsAdoA55azT5qZZZTcgBaV/QyBdVvzuPDFX9iwbacn/tc/duSyw9r4lJWIRIoz+rbgHyd39cRy8gq56MVJrNqyw6espKR9egbuAr50zp1JoKA+B9wOLDOzj8xscCXe5mOgN3A0UAD8r5x9bgCWltrG70uuNVn2jgIueulXVmzy/iMbfnhbrvlDe5+yEpFIc8mgNtx0bEdPLGtrPhe9OIktuTsrOEpCZb86sZlZZ+BmAoV2G/AkkAd8Y2b37OlY51yOc26hc24scA5whpl1LbXbSAJ35yW3ynw5qPHyCoq44tXJzF3jbeYa1qcFt57QGTOr4EgRkbKuPao9Fw/M8MQWrdvG5a9MJq9U3xoJrX3pxJZkZhea2ThgNtCPQAFv4Zy7yTl3NnAmcP0+fP6uauL5W+Cc2xLs6Pb7Bqzch/etkQqLirlu1DQmLfV2NDmqczoPDuuh4i0i+8zMuPOkrgzp4R2xMnnZZq4bNU1riftoX+7A1xC4M54G9HDOHe6ce8M5l19in3FAud0UzSzezO43s0PMrLWZHQa8BkwFNGffAXLOcceHs/hqTpYn3rd1PZ46rw/xsZqzR0T2T2yM8ehZvejfpr4n/tWcLP6hFcx8sy//q98INHfOXe+cm1PeDsE754p6SDkC478/BhYAbwCLgJOcc8UVHCOV9OjXC3jrV++MSR0b1+GFiw6mVkJsBUeJiFROUnwsz/7pYDo1TvHEX/95OU99t8inrGq2femF/rJzLm9/P8g5V+icO90519Q5l+ica+2cu8I5pyl+DtCoSct5cqz3H1Cz1CReubQfackJFRwlIrJvUmvF8/Klh9A01buW+L++WsB7U/SUM9TUrhrhvp+/jjs+nOWJ1UuO59XL+tM0tZZPWYlItGqaWotXLu1XZt70W0fPYOLijT5lVTOpgEew2auzueaNqZ5OJLtWFmufXsfHzEQkmnVsnMILFx9CQtzuElJQ5Ljq9SksXr/Nx8xqFhXwCLV6yw4ufflXtu/c3YHfDB4/pze9W9XzMTMRqQkOyajPY2f18sSydxRw6cu/smm7xoiHggp4BNqaF/hHkrU13xO/88SuHNdNi5OISGic2LMpfzuukye2bGMuw1/VGPFQUAGPMAVFxVzzxlTmrc3xxC8ZlMGlmiJVRELs6iPbcdbB3pmuJy/bzM3vz9DwsmqmAh5BnHPcNnom4xdu8MSP69aYO04sPZmdiEj1MzPuO7UHA9o28MQ/mr6axwenH5YAABkNSURBVL7RFB/VSQU8gjz9w2LeLTVUo1fLNEae3ZvYGM2yJiL+SIiL4ZkL+tKuUW1P/IlvFzJ6qoaXVRcV8Ajx5ey1PPLlfE+sVf1kntdELSISBlKT43np4n7Ur+2de+KW92cyZdlmn7KKbirgEWD26mxufHs6JR8npdaK56VLDqFhnUT/EhMRKaFVg2Se+1Nfz/CynUXFXPnaFNZkawnSqqYCHubW5eRxxSuTyS0xXCw2xnj6/D60a6Sx3iISXvq2rs8jZ/T0xDZsy2f4q1PYsVM906uSCngYyyso4srXprA62zuD7Yih3RjYvqFPWYmI7NkpvZpz9ZHtPLGZq7L5u3qmVykV8DDlnOOW92cwbfkWT/yiAa254NDWPmUlIlI5f/1jJ47pku6JffLbav77/WKfMoo+KuBh6r/fL+bD6as9scEdGnLnSRouJiLhLybGeOzsXnQoNa3zv76az9ellj2W/aMCHoa+mFW2x3nbRrX5z3l9iNO63iISIVKS4nn+ooNJrRX/e8w5uOGtaSzIytnDkVIZqgZhZkFWDje9M90TS60VzwsXHeL5RyAiEglaN6jNf8/v45mrYvvOIi5/ZTJbcjVn+oFQAQ8j2bkFDH/V2+M8LtjjvE3D2ns4UkQkfA1q35C7Sj3+W74pl+vfmu5ZTVH2jQp4mCgqdlz/9jQyN+Z64v84uat6nItIxPvTgNac26+lJ/bDgvU8/s0CnzKKfCrgYeLRr+fz/fz1ntjZB7dUj3MRiQpmxoih3enTKs0Tf2LsIr5Rp7b9ogIeBsbMXMNT33mHVvRqmcY9p3bDTHOci0h0SIiL4b/n96VhHe90qze+PZ2lG7b7lFXkUgH32fy1Ofz13d88sUYpiTxzQV8S4zTHuYhElyapSTx1nrdTW05+IVe9NoXt+YU+ZhZ5VMB9lJ1bwPDXvJ3W4mMDndaapCb5mJmISPXp37YBtw3p4onNz8rRGuL7SAXcJ0XFjuvemsayUp3W7h7ajYMz6vuUlYhIaFw6KIOTD2rmiX06Yw0v/LjUp4wijwq4Tx7/diE/LPB2Wju3X0vO769OayIS/cyMh4b1oFPjFE/8gc/n8fOSjT5lFVlUwH3w3fx1PDl2oSfWp1Uadw/t5lNGIiKhl5wQx/8u7EtKUtzvsaJix7WjprE+J9/HzCKDCniIrdycW2Zt74Z1EnhandZEpAbKaFibkWf38sTW5+Rz/VvTNMnLXqiAh1B+YRFXvzGVLbkFv8diDJ48tw+N66rTmojUTEd3acw1f/AuPzph8Uae+HZhBUcIhLCAm9ntZjbdzLab2Uoze8LM6uz9yOhx76dzmLEy2xP763GdGNCugU8ZiYiEhxuP6Uj/Nt4OvE+MXcj4hesrOEJCeQc+EHgY6AOcB/wReDKEn++rD6at5PWfl3tix3RpzFWHt6vgCBGRmiMuNoYnz+3tmeQlsHLZdLK25vmYWfgKWQF3zp3onHvTOTffOTcOuBM4JVSf76f5a3O4dfRMT6xV/WT+fdZBxMRopjUREYD0ukk8cU5vSk5AuXH7Tq59cxqFRcX+JRam/HwG3hDYUt4LZpZmZhklN6BFKJOrKjl5Bfz59SnkFez+yxeYTrCPlgcVESllYPuG3HB0R09sUuYm/v21Fj0pzZcCbmapwF+BFyvY5QZgaaltfGiyqzrOOW4dPZMlpeb4vfeUbnRvnupTViIi4e3/jmrP4A7eVRif/n4xY+dp0ZOSQl7AzSwReB9YAjxYwW4jgTaltsEhSbAKjZq0gk9nrPHEzjq4BWcf0sqnjEREwl9sjPHY2b1oXDfRE//LO7+xNlvPw3cJaQE3szjgLSAFOM05V+7M9c65Lc65zJIbsDKEqR6wuWu2MuKT2Z5Y5yYp3HNKd58yEhGJHA3rJPLkud5FTzbnFnDj29M1PjwolMPIYoBXgfbACc65baH67FDbnl/I/705lfzC3c+9kxNieer8PiTFa7IWEZHK6NemPjcd630ePnHJRp7+fpFPGYWXUN6BPwscCVwIJJhZk+AWdRXtro9ms3i997n3/ad1p12jGjXsXUTkgP35iHYMau+dK+OxbxYyZdkmnzIKH6Es4JcBTYFpwJoSW8sQ5lDt3p+ykvenelv7z+zbgtN6R2QnehERX8XEGI+e1Yv6tXePDy8qdlw3ajrZOwr2cGT0C+U4cKtgywxVDtVt0bpt3PnRLE+sfXodRpyiRUpERPZX47pJ/PvMgzyxVVt2cOvomr1+uOZCryJ5BUX835tTyd1Z9HssMS6Gp87rQ3JC3B6OFBGRvflD53QuHdTGExszcy1v/brCp4z8pwJeRe79dA7z1uZ4YiOGdqNTk5QKjhARkX1x8wmd6Nasric24pPZLMzKqeCI6KYCXgW+mLWWN37xznM+9KBmnH1IVD3eFxHxVWJcLE+e25vkhN19n/MKirl21DTyCor2cGR0UgE/QGuz87hl9AxPLKNBMvef1h0zzXMuIlKV2jaqU2Y+jXlrc3jky/k+ZeQfFfADUFzsuOmd6Z71veNjjSfP7UNKkuY5FxGpDsP6NOeUXs08sRd+XMqPCzf4lJE/VMAPwLPjlzBh8UZP7C9/7ESPFprnXESkupgZ957aneZptTzxv7w7nS25O33KKvRUwPfTjJVb+FepJpuB7RowfHBbnzISEak56ibF89jZvTxLj2Ztzef2D2bVmKFlKuD7YXt+Ide/NZ3CEvPxpiXH8+hZvbS+t4hIiPRrU5+rjmjniX02cw0fTFvlU0ahpQK+H+75ZA5LSy0R+tCwnjRJTfIpIxGRmunGYzqWGVp210ezWbEp16eMQkcFfB99PnMNb0/2Thxwbr9WHNetiU8ZiYjUXAlxMTx+Ti8S43aXs235hdz0TvSvWqYCvg/WZO/gltEzPbF2jWpz50ldfMpIRETap6dw+4ne/4d/zdzMMz8s9imj0FABr6TiYsdf3/3NM3l+fKzx+Dm9NVWqiIjPLjy0NUd0bOSJPfb1AmauzPYpo+qnAl5Jr0zM5KdF3iFjfz+uM92ba8iYiIjfzIxHzuzpWbWssNjxl3enk18YnbO0qYBXwqJ1OTz4+TxPbGC7Blx2WJsKjhARkVBLT0nigdN7eGILsrbx2NcLfcqoeqmA70VBUTE3vv0b+YXFv8dSkuL415kHaciYiEiYOa5bE4b1aeGJPTtuMVOWbfIpo+qjAr4X/xm7iJmrvM9QRgztRrNSMwCJiEh4uOvkrjQtMay32MFf353Bjp3R1ZSuAr4H01ds4T/fLfLETujehNN6N/cpIxER2ZvUWvE8NKynJ7Z0w3Ye+mJeBUdEJhXwCuzYWcRNb3vHETask8j9p/XQKmMiImHu8I6NOL9/K0/s5QmZTFgcPQueqIBX4MHP57Kk1GxrD5/Rw9PDUUREwtdtQ7rQsr73ceff3p1BTl5BBUdEFhXwcoxfuJ5XJi7zxM7t14qjOjf2KSMREdlXtRPjeOSMgzwLnqzasoN/jpnrX1JVSAW8lOwdBfzt3RmeWOsGydxxomZbExGJNIe2bcClg7xDfkdNWsF389f5lFHVUQEv5b5P57B2a97vP8cYPHrWQdRO1GxrIiKR6G/HdaJto9qe2G2jZ7I1wpvSVcBL+G7eOt6dstITG354O/q2ru9TRiIicqCS4mP595kHUXLqjjXZeTwwJrJ7pauAB2XvKOCW0d6m8w7pdbjx2A4+ZSQiIlWld6t6DD/cu3b4qEnL+WlR5PZKVwEPuvfTOWRtzf/959gY419nHkRiXKyPWYmISFW54ZgOtG3obUq/ZfQMcncW+pTRgQlpATez083sWzPLNrOwWah17Lws3ivVdH7l4W05qGWaTxmJiEhVS4qP5eEzenp6pa/YtINHvpzvX1IHINR34MnAWODBEH9uhbJzC7i11BrfHRvX4fpj1HQuIhJtDs6oz0UDMjyxlydkMjkz8uZKD2kBd8697py7H5gYys/dk3vUdC4iUqP8/fhOnglenIO/vzeDvILImis9LJ+Bm1mamWWU3IAWezlsn42dl8X7U71N51cd0ZaeLdR0LiISrZIT4njwdO9c6Us2bGfkN5G17GhYFnDgBmBpqW18VX5AeU3nnRqncN3RajoXEYl2g9o35Nx+LT2xZ8ctZsbKLT5ltO/CtYCPBNqU2gZX5QfcP0ZN5yIiNdmtQ7rQpK532dG/vzeDgqJiH7OqvLAs4M65Lc65zJIbsHJvx1XWT4s28M5k79v9+Yh29GiRWlUfISIiYa5uUjz/PL27JzZvbQ7PjlviU0b7JiwLeHXK3VlYZsKW9ul1uPbo9j5lJCIifjmqc2NO693cE3v824UsLbUaZTgK9Tjw+mbWC2gf/LlXcAvZGp3//moBKzbtKJETPDSsp5rORURqqDtO7EK95Pjff95ZWMztH8zEubCZrqRcob4DHwpMA54L/jwtuDULxYdPW76Zl35a6oldNCCDvq3rheLjRUQkDDWok8gdJ3b1xCYs3lhmgq9wE+px4C8756ycLbO6P3tnYTG3vD+T4hJfqJqn1eJvx3Wq7o8WEZEwd3qf5hzWvqEndv+YuWzYll/BEf6rMc/An/5+MfOzcjyxB07voWVCRUQEM+P+07qTFL+7LG7JLeDeT+f4mNWe1YgCviArh/985x2gP6xPCw7v2MinjEREJNy0blCbG47p6Il9NH01389f51NGexb1Bbyo2HHz+zMoKNrddt6wTgJ3ntTFx6xERCQcXXZYG7o0reuJ3fHhrLBcsSzqC/grEzKZttw7s86Iod1JSw5Zx3cREYkQ8bExPHh6D2JKrFi2cvMOHvt6gX9JVSCqC/jqLTv411feZeKO7dqYIT2a+JSRiIiEu4NapnHxwDae2As/LmXWqmyfMipf1BZw5xx3fTSb3J27V5dJSYzj3lO6YyUXgxURESnlL3/sSPO03SuWFTu4/cNZFBWHz9jwqC3gX87O4pu5WZ7YzSd0pklqUgVHiIiIBNROjOPeU7t5Yr+t2MKoSct9yqisqCzgOXkF3P3xbE+sT6s0zuvXyqeMREQk0hzVuTHHd/M+cn34i3mszwmPseFRWcD//dUC1m7N+/3nuBjjn6f3ICZGTeciIlJ5d53cleSE3VNtb80r5J9j5vqY0W5RV8B/W7GFVyZmemJXHN6Wzk3qlru/iIhIRZql1eLGUmPDP5i2igmLN/iU0W5RVcALi4q5dfRMSs4/36p+Mtcd1cG/pEREJKJdPCiDzk1SPLE7PpxFfmFRBUeERlQV8Jd+ymTOmq2e2L2ndqdWglYaExGR/RMfG8P9p3nXDV+yfjvP+bxueNQU8JWbc3m01ED7oQc14whNlyoiIgeob+v6nHNIS0/sybGLWL4x16eMoqSA7xrzvaNgd3NG3aQ47jyp6x6OEhERqbybj+9M/dq7Z/HMLyzmro9n+bZueFQU8C9nr2XsPO9k87cO6UKjlESfMhIRkWhTr3YCt57Q2RP7fv56vpy91pd8Ir6A5+4s5J5PvMu9HZJRj7MPblnBESIiIvvnjL4t6JdR3xO799O57NgZ+g5tEV/Anxy7iNXZu8d8x8YY952qMd8iIlL1zIz7TutOXIkas2rLDp7+flHIc4noAr5o3TaeH+/tBXjpoAw6leruLyIiUlU6Nk7h4oEZntgz45aEvENbxBbwQMe1WZ51vhvXTeT6UgPuRUREqtr1x3SgYZ3d/ax2FhZzz6dz9nBE1YvYAv7pjDVMWLzRE7vzpK7USYzzKSMREakpUpLiy3Ro+2ZuFt+V6lBdnSKygG/LL+S+z7zfdA5r35ATezT1KSMREalpTuvdnL6t63liIz6ZHbIZ2iKygI/8egFZW3evBhMfa4w4pZvW+RYRkZCJiTFGDO1GydKTuTGX58cvDc3nh+RTqtDi9Tm8NCHTE7ticFvaNarjT0IiIlJjdW+eyvn9vUtV/2fsItZk76j2z464Av7oVwspKt7dca15Wi3+76j2PmYkIiI12V+O7URacvzvP+8oKOL+z6p/ydGIK+C/rdzi+TmwVqs6romIiD/q1U7gb8d18sQCHa2rd8nRkBdwM7vVzFabWa6ZfWhm6fv7Xkd2asQfuzauyvRERET22TmHtKJ787qe2N0fz6agqLjaPjOkBdzMLgFuA64BBgJpwKj9ea+EuJhg5wF1XBMREX/FxhgjhnqXHF2QtY3XJi6rts8M9R34tcCjzrkPnHPTgUuBo8ys+16OK+PPR7SjdYPaVZ6giIjI/ujbuh5n9G3hiT329QLW5+RXcMSBCdnDYzNLBA4CbtwVc84tMbNMoD8wq8S+aQTuzktqDVC4dQPNUmsxJCOWzMzM6k5bRESk0s7tksQnP25ge3Bxk83Aba9+y+0n7vvy1itXrtz129jyXrdQrWNqZs2AVUBX59zcEvFJwAfOuQdKxO4G/hGSxERERMLbYOfcj6WDoey+vS8Pq0cCL5eKtQW+BY4AlldRTpGgBTAeGAys3Mu+0UTnrfOuCXTeOu89iQWaAr+W92IoC/gGoBhIB0oOkGsEeCaPdc5tATzjxUp0VlvunMustizDTInzXqnzjn46b513TaDz3qfzXlzRCyHrxOacywd+A/6wK2ZmbYAM4JdQ5SEiIhINQj0Dyn+AkWY2DcgEHgO+c87N2uNRIiIi4hHSAu6ce9HMGgPPAKnA18DwUOYgIiISDUI+B2mwt/kDe92xrC3ACEo9G68BdN41i867ZtF51yxVet4hG0YmIiIiVSfiFjMRERERFXAREZGIpAIuIiISgSKmgFflMqSRwMxeNjNXarvB77yqmpmdbmbfmlm2mZXpkGFm/c1sspnlmdkcMzvBjzyr2p7O28yOLOfaT/cr16pkZreb2XQz225mK83sCTOrU2qfqLvmezvvKL/mI8xsgZntMLMsM3vDzJqUeD3qrjfs+byr6npHRAGvymVII8w7BKbR27U962861SIZGAs8WPoFM2sAfA78BPQBXgM+MLMOIc2welR43iWUvPZHhyKpEBgIPEzgep4H/BF4cteLUXzN93jeJUTjNZ8HXAl0AU4GWgGvQFRfb9jDeZdwYNfbORf2GzAVGFHi57aAA7r7nVs1nvPLwMt+5xHC8z0y8NfRE7uOwIQ/ViI2DviX3/lW83mXiUXrBpwJbKpJ17yC865J1/xkYFtNut7lnHeVXO+wvwMvsQzp2F0x59wSAhe9v09phcpQM1tvZjOCjxBCPm7fZ/0IzNRXson5W6L/ugNgZplmttzM3jKzVn7nU00a4h0TW1OueenzBqL/mptZKoEWiJ+CoRpxvcs5713xA7reYV/AgQYE8lxXKr6ewMIo0WoMgQt+FPAocBNwj68ZhV46Ne+6A6wBLgNOAS4icL7fmVktX7OqYsH/1P4KvFgiHPXXvILzjuprbmbnm9k2Al9a2gDnBl+K6uu9h/OukusdCXd0+7IMadRwzr1T4seZZlYEPG1mt5f6thrNauq1nw/M3/WzmU0msITuScC7fuVVlYIta+8DS/D2A4jqa17RedeAa/4xMAloSWAmsv8ReIwQ1debCs67qq53JBTwSi9DGuWmArUJNL2t9zmXUMmi7Dfxmnbdcc7lmNkiAiv3Rbzgo6C3gBTgaOdcYYmXo/aa7+W8PaLtmjvncoAcYKGZzQdWmllXovh6Q8Xn7ZybU3q//bneYd+E7rQM6S7dgVwCX2hqikkEOnuUdBQ167oTbFZrCyzzO5cDZWYxwKtAe+AE59y2UrtE5TWvxHmX3j9qrnk5dt11FxGl17sCJc/b+8L+Xm+/e+ZVsvfepcBWAs8LdnVoG+t3XtV8zo8S6MiRAZwOrAIe9TuvajjP+kAv4HICIwt6BbcEAv0fNgEjCQzFuBnIAzr4nXc1n/dwYCjQDuhLoBluGVDH77yr4LyfB1YHz7VJiS02+HpUXvNKnHdUXnMgHrgfOARoDRwGfAdMIXADGa3Xe2/nXSXX2/cT3Yc/kFsJPPjPBT4CGvudUzWf7xcEmsrzgYXAXUCi33lVw3leHCxgpbeM4OuHBv/S5xN4hDLE75yr+7wJzHewKPgfWRYwGmjvd85VdN7lnfPv1ztar/nezjtarzmBx7Sjg/935weL1HNA0yi/3ns876q63lqNTEREJAKF/TNwERERKUsFXEREJAKpgIuIiEQgFXAREZEIpAIuIiISgVTARUREIpAKuIiISARSARcREYlAKuAiIiIRSAVcRMowswZmtsLMHi8RSzezNWb2kJ+5iUiAplIVkXKZ2eHAtwQW0/mUwPz8qcBg51yBn7mJSGSsBy4iPnDOjTOz+4CXgFcIrI7XW8VbJDzoDlxEKhRcx/pHYABwjnPubZ9TEpEgPQMXkT1pCnQEioK/ikiY0B24iJQrePf9LYEv+o8B7wJ/cM796GtiIgLoGbiIVOx2oAfQyzm30syeAd4ws17Ouc0+5yZS4+kOXETKMLOBwA/AMOfcx8FYIjAJWOScG+ZnfiKiAi4iIhKR1IlNREQkAqmAi4iIRCAVcBERkQikAi4iIhKBVMBFREQikAq4iIhIBFIBFxERiUAq4CIiIhFIBVxERCQC/T9QxKTcUsHwKQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 576x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "b = 0.006  # [Ns/m]\n",
    "t = np.arange(0, 2.05, 0.01)\n",
    "x2a = vx0*m/b*(1 - np.exp(-b/m*t))\n",
    "y2a = (vy0*m/b + g*m**2/b**2) * (1 - np.exp(-b/m*t)) - g*m/b*t\n",
    "plt.figure(figsize=(8, 4))\n",
    "plt.plot(x2a, y2a, lw=4)\n",
    "plt.xlim(0, 36)\n",
    "plt.ylim(0, 6)\n",
    "plt.title('Ball trajectory')\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('y')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "fjWZclG7M8jK",
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "#### Numerical solution\n",
    "\n",
    "Now, we will solve the same situation using a numerical method (Euler method).  \n",
    "We start from the equations previously found for each coordinate:\n",
    "<span class=\"notranslate\">\n",
    "$$ \\frac{d^2x}{dt^2} = -\\frac{b}{m} \\frac{dx}{dt} $$\n",
    "</span>\n",
    "<span class=\"notranslate\">\n",
    "$$ \\frac{d^2y}{dt^2} = -\\frac{b}{m}\\frac{dy}{dt} - g $$\n",
    "</span>\n",
    "We can separate each equation into two first order equations and apply the Euler method:\n",
    "<span class=\"notranslate\">\n",
    "$$ \\frac{dv_x}{dt} = -\\frac{b}{m} v_x $$\n",
    "</span>\n",
    "<span class=\"notranslate\">\n",
    "$$ \\frac{dx}{dt} = v_x $$\n",
    "</span>\n",
    "<span class=\"notranslate\">\n",
    "$$ \\frac{dv_y}{dt} = -\\frac{b}{m}v_y - g $$\n",
    "</span>\n",
    "<span class=\"notranslate\">\n",
    "$$ \\frac{dy}{dt} = v_y $$\n",
    "</span>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "colab": {},
    "colab_type": "code",
    "id": "YkEz1B5VM8jL",
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [],
   "source": [
    "x  = x0\n",
    "y  = y0\n",
    "vx = vx0\n",
    "vy = vy0\n",
    "r  = np.array([x, y])\n",
    "while y >= 0:\n",
    "    dxdt = vx\n",
    "    x = x + dt * dxdt\n",
    "    dydt = vy\n",
    "    y = y + dt * dydt\n",
    "    dvxdt = -b/m*vx\n",
    "    vx = vx + dt * dvxdt\n",
    "    dvydt = -g - b/m*vy\n",
    "    vy = vy + dt * dvydt\n",
    "    r = np.vstack((r, np.array([x, y])))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "colab": {},
    "colab_type": "code",
    "id": "-UgCUusuM8jO",
    "outputId": "27e3ee61-eb8a-4c9b-9ff1-69def6ce0b6a",
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 576x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(8, 4))\n",
    "x2n = r[:, 0]\n",
    "y2n = r[:, 1]\n",
    "plt.plot(x2n, y2n, lw=4)\n",
    "plt.xlim(0, 36)\n",
    "plt.ylim(0, 6)\n",
    "plt.title('Ball trajectory')\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('y')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "M_NMu46AM8jU",
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Example 3: Ball kicked into the air considering the air drag proportional to square of speed\n",
    "\n",
    "Now, we will consider the drag force due to the air resistance proportional to the square of speed and still in the opposite direction of the velocity vector. \n",
    "           \n",
    "<figure><center><img src=\"../images/ballGravSquareRes.png\" alt=\"free-body diagram of the ball\" width=\"500\"/><figcaption><i>Figure. Free-body diagram of a ball under the influence of gravity and drag proportional to the square of speed.</i></figcaption></center></figure>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "36ViR3o1M8jU"
   },
   "source": [
    "So the forces being applied on the ball are (for a revision of Time-varying frames and the meaning of the $\\hat{\\bf{e_t}}$, see [Time-varying frames notebook](https://nbviewer.jupyter.org/github/BMClab/bmc/blob/master/notebooks/Time-varying%20frames.ipynb)):\n",
    "<span class=\"notranslate\">\n",
    "$$ \\vec{\\bf{F}} = -mg \\; \\hat{\\bf{j}} - bv^2\\hat{\\bf{e_t}} \\\\\n",
    "\\vec{\\bf{F}} = -mg \\; \\hat{\\bf{j}} - b (v_x^2+v_y^2) \\frac{v_x \\; \\hat{\\bf{i}} + v_y \\; \\hat{\\bf{j}}}{\\sqrt{v_x^2+v_y^2}} \\\\\n",
    "\\vec{\\bf{F}} = -mg \\; \\hat{\\bf{j}} - b \\sqrt{v_x^2+v_y^2} \\,(v_x \\; \\hat{\\bf{i}}+v_y \\; \\hat{\\bf{j}}) \\\\\n",
    "\\vec{\\bf{F}} = -mg \\; \\hat{\\bf{j}} - b \\sqrt{\\left(\\frac{dx}{dt} \\right)^2+\\left(\\frac{dy}{dt} \\right)^2} \\,\\left(\\frac{dx}{dt} \\hat{\\bf{i}} + \\frac{dy}{dt} \\; \\hat{\\bf{j}}\\right) $$\n",
    "</span>\n",
    "Writing down the Newton's second law:\n",
    "<span class=\"notranslate\">\n",
    "$$ \\vec{\\bf{F}} = m \\frac{d^2\\vec{\\bf{r}}}{dt^2} \\\\\n",
    "-mg \\; \\hat{\\bf{j}} - b \\sqrt{\\left(\\frac{dx}{dt} \\right)^2+\\left(\\frac{dy}{dt} \\right)^2} \\,\\left(\\frac{dx}{dt} \\hat{\\bf{i}}+\\frac{dy}{dt}\\hat{\\bf{j}}\\right) = m\\left(\\frac{d^2x}{dt^2}\\hat{\\bf{i}}+\\frac{d^2y}{dt^2}\\hat{\\bf{j}}\\right) $$\n",
    "</span>\n",
    "Now, we can separate into one equation for each coordinate:\n",
    "<span class=\"notranslate\">\n",
    "$$ - b \\sqrt{\\left(\\frac{dx}{dt} \\right)^2+\\left(\\frac{dy}{dt} \\right)^2} \\,\\frac{dx}{dt} = m\\frac{d^2x}{dt^2} \\quad \\rightarrow \\\\\n",
    "\\frac{d^2x}{dt^2} = - \\frac{b}{m} \\sqrt{\\left(\\frac{dx}{dt} \\right)^2+\\left(\\frac{dy}{dt} \\right)^2} \\,\\frac{dx}{dt} $$\n",
    "</span>\n",
    "<span class=\"notranslate\">\n",
    "$$ -mg - b \\sqrt{\\left(\\frac{dx}{dt} \\right)^2+\\left(\\frac{dy}{dt} \\right)^2} \\,\\frac{dy}{dt} = m\\frac{d^2y}{dt^2} \\quad \\rightarrow \\\\\n",
    "\\frac{d^2y}{dt^2} = - \\frac{b}{m} \\sqrt{\\left(\\frac{dx}{dt} \\right)^2 + \\left(\\frac{dy}{dt} \\right)^2} \\,\\frac{dy}{dt} -g $$\n",
    "</span>\n",
    "These equations are very difficult to solve analytically, but they can be easily  solved by using numerical methods. Below we  will use the same numerical method (Euler method) to solve these equations.  \n",
    "For that, again we must break each equation into two first-order differential equations:\n",
    "<span class=\"notranslate\">\n",
    "$$ \\frac{dv_x}{dt} = - \\frac{b}{m} \\sqrt{v_x^2+v_y^2} \\,v_x $$\n",
    "</span>\n",
    "<span class=\"notranslate\">\n",
    "$$ \\frac{dx}{dt} = v_x $$\n",
    "</span>\n",
    "<span class=\"notranslate\">\n",
    "$$ \\frac{dv_y}{dt} = - \\frac{b}{m} \\sqrt{v_x^2+v_y^2} \\,v_y -g $$\n",
    "</span>\n",
    "<span class=\"notranslate\">\n",
    "$$ \\frac{dy}{dt} = v_y $$\n",
    "</span>\n",
    "Now, we can apply the Euler method to find a solution. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "colab": {},
    "colab_type": "code",
    "id": "QyHkIliGM8jV",
    "outputId": "09a23149-de85-41e7-9282-169a7c0f158c",
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [],
   "source": [
    "x  = x0\n",
    "y  = y0\n",
    "vx = vx0\n",
    "vy = vy0\n",
    "r  = np.array([x, y])\n",
    "while y >= 0:\n",
    "    dxdt = vx\n",
    "    x = x + dt * dxdt\n",
    "    dydt = vy\n",
    "    y = y + dt * dydt\n",
    "    dvxdt = -b/m * np.sqrt(vx**2 + vy**2)*vx\n",
    "    vx = vx + dt * dvxdt\n",
    "    dvydt = -b/m * np.sqrt(vx**2 + vy**2)*vy - g\n",
    "    vy = vy + dt * dvydt\n",
    "    r = np.vstack((r, np.array([x, y])))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 576x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(8, 4))\n",
    "x3n = r[:, 0]\n",
    "y3n = r[:, 1]\n",
    "plt.plot(x3n, y3n, lw=4)\n",
    "plt.xlim(0, 36)\n",
    "plt.ylim(0, 6)\n",
    "plt.title('Ball trajectory')\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('y')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### All numerical solutions plotted together"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 576x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(8, 4))\n",
    "plt.plot(x1n, y1n, lw=4, c='g', label='$g$')\n",
    "plt.plot(x2n, y2n, lw=4, c='b', label='$g+bv$')\n",
    "plt.plot(x3n, y3n, lw=4, c='r', label='$g+bv^2$')\n",
    "plt.xlim(0, 36)\n",
    "plt.ylim(0, 6)\n",
    "plt.title('Ball trajectory')\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('y')\n",
    "plt.legend(loc='best', frameon=False)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Further reading\n",
    "\n",
    "- Read the chapter 0 of the [Ruina and Rudra's book](http://ruina.tam.cornell.edu/Book/index.html) about mechanics;\n",
    "- Read pages 478-494 of the chapter 10 of the [Ruina and Rudra's book](http://ruina.tam.cornell.edu/Book/index.html) about 1D dynamics;\n",
    "- Read the chapter 13 of the [Hibbeler's book](https://drive.google.com/file/d/1sDLluWCiBCog2C11_Iu1fjv-BtfVUxBU/view) (available in the Classroom)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Video lectures on the internet\n",
    "\n",
    "- Khan Academy: [Forces and Newton's laws of motion](https://www.khanacademy.org/science/ap-physics-1/ap-forces-newtons-laws) (Em português: [Forças e as Leis do Movimento de Newton](https://pt.khanacademy.org/science/physics/forces-newtons-laws)).  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "R8JH_WGNM8ja",
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Problems\n",
    "\n",
    "1. Solve exercises 12.1.16, 12.1.19, 12.1.24, 12.1.29, 12.1.30, 12.1.31(a, b, d) and 12.1.32 from Ruina and Pratap's book (2019).\n",
    "2. Check examples 13.1, 13.4 and 13.5 from Hibbeler's book.\n",
    "3. Solve exercises 13-5, 13-8, 13-11 and 13-16 from Hibbeler's book.\n",
    "4. Consider a block with mass of 1 kg attached to a spring hanging from a ceiling (the spring constant k = 100 N/m). At t = 0 s, the spring is stretched by 0.1 m from the equilibrium position of the block + spring system and then it's released (the initial velocity is not specified). Find the motion of the block."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "3l-W0R08M8jb"
   },
   "source": [
    "## References\n",
    "\n",
    "- R. C. Hibbeler (2010) [Engineering Mechanics Dynamics](https://drive.google.com/file/d/1sDLluWCiBCog2C11_Iu1fjv-BtfVUxBU/view). 12th Edition. Pearson Prentice Hall.\n",
    "- Ruina A, Rudra P (2019) [Introduction to Statics and Dynamics](http://ruina.tam.cornell.edu/Book/index.html). Oxford University Press.  \n",
    "- Taylor JR (2005) [Classical Mechanics](https://books.google.com.br/books?id=P1kCtNr-pJsC). University Science Books."
   ]
  }
 ],
 "metadata": {
  "colab": {
   "collapsed_sections": [],
   "name": "Copy of newtonLawForParticles.ipynb",
   "provenance": [],
   "version": "0.3.2"
  },
  "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": "3.7.3"
  },
  "latex_envs": {
   "LaTeX_envs_menu_present": true,
   "autoclose": false,
   "autocomplete": true,
   "bibliofile": "biblio.bib",
   "cite_by": "apalike",
   "current_citInitial": 1,
   "eqLabelWithNumbers": true,
   "eqNumInitial": 1,
   "hotkeys": {
    "equation": "Ctrl-E",
    "itemize": "Ctrl-I"
   },
   "labels_anchors": false,
   "latex_user_defs": false,
   "report_style_numbering": false,
   "user_envs_cfg": false
  },
  "nbTranslate": {
   "displayLangs": [
    "*"
   ],
   "hotkey": "alt-t",
   "langInMainMenu": true,
   "sourceLang": "en",
   "targetLang": "fr",
   "useGoogleTranslate": true
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}