Motion_along_trajectories-CORRECT.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "nELnU4JpmpoH"
   },
   "source": [
    "# Companion Notebook for the lecture in frame calculations\n",
    "\n",
    "This Notebook serves as a guide to do an physical *analysis numericaly* for the exercices of the lecture. \n",
    "\n",
    "Each cell has to be executed one after the other."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "id": "FC7VMRfnmpoR"
   },
   "outputs": [],
   "source": [
    "# Please enter the your Surname in capital letters\n",
    "SURNAME = \" \"\n",
    "# If you did associate with other students please enter their Surname also\n",
    "COLLABORATORS = \" \" \n",
    "# Date : Paris January the 17, 2021"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "id": "tTKxsUOkmpoS"
   },
   "outputs": [],
   "source": [
    "# Important libraries to import\n",
    "#\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "97EMVPsampoT"
   },
   "source": [
    "## A) Trajectory \n",
    "\n",
    "We need to know the velocity at any specific instant of time or specific point along the path. This is called **instantaneous velocity** .\n",
    "\n",
    "\n",
    "\n",
    "- CAUTION : How long is an instant? Note that the word \"instant\" has a somewhat different definition in physics than in everyday language. You might use the phrase \"It lasted just an instant\" to refer to something that lasted for a very shan time interval. But in physics an instant has no duration at all; it refers to a single value of time\n",
    "\n",
    "We call $t$ any time in seconds to the initial position of the plane"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "id": "c_A9ZeKFmpoU"
   },
   "outputs": [],
   "source": [
    "t=[0.0, 20.0, 40.0, 60.0, 80.0, 100.0, 120.0, 140.0, 160.0, 180.0, 200.0, 220.0\n",
    "   , 240.0, 260.0, 280.0, 300.0, 320.0, 340.0]\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "86xQHyIVmpoU"
   },
   "source": [
    "The formal definition of instant velocity is: \n",
    "\n",
    "$$ v_{(\\vec {r})}=\\frac{d\\,\\vec {r}}{dt} $$\n",
    "\n",
    "Where $\\vec {r}$ is the position vector in a frame."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "tJzd4P0lmpoV",
    "outputId": "f0597f1e-e0c9-40c8-efb3-7e0fec37c3a7"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "18"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# First check that t contains 18 values\n",
    "len(t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "id": "075HyFQLmpoW"
   },
   "outputs": [],
   "source": [
    "# Declare the coordinates for the first plane\n",
    "x1=[3.52E4, 3.79E4, 4.04E4, 4.27E4, 4.52E4, 4.62E4, 4.60E4, 4.60E4, 4.60E4, \n",
    "    4.37E4, 4.13E4, 3.89E4, 3.65E4, 3.41E4, 3.21E4, 3.02E4, 2.81E4, 2.61E4]\n",
    "y1=[-9.67E3, -9.09E3, -8.38E3, -7.68E3, -6.92E3, -4.69E3, -2.52E3, -1.76E2, \n",
    "     1.93E3,  3.46E3,  3.99E3,  3.93E3,  3.75E3,  3.63E3,  3.46E3,  3.34E3, \n",
    "     3.17E3,  2.99E3]\n",
    "\n",
    "# Declare the coordinates for the second plane\n",
    "x2=[4.89E4, 4.69E4, 4.49E4, 4.28E4, 4.11E4, 3.92E4, 3.73E4, 3.53E4, 3.34E4,\n",
    "    3.14E4, 2.95E4, 2.76E4, 2.59E4, 2.41E4, 2.24E4, 2.08E4, 1.92E4, 1.75E4]\n",
    "y2=[1.15E4, 1.01E4, 8.79E3, 7.50E3, 6.27E3, 4.98E3, 3.87E3, 3.40E3, 3.52E3,\n",
    "    3.34E3, 3.22E3, 3.11E3, 2.99E3, 2.87E3, 2.70E3, 2.64E3, 2.46E3, 2.40E3]\n",
    " "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 504
    },
    "id": "I-CqnISempoX",
    "outputId": "d4c43121-664e-49f4-b394-27354e0b2e44"
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 720x560 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#----- Plotting of points and trajectories y1=f(x1) and y2=f(x2)-----  \n",
    "\n",
    "# initialization of the figure, name and size of the viewing window\n",
    "\n",
    "plt.figure('Planes fly in the sky',figsize=(9,7),dpi=80)\n",
    "\n",
    "plt.title('Trajectory of each plane') # title\n",
    "plt.xlabel('x(in m)')              # legend of the abscissa  \"ex\" axis\n",
    "plt.ylabel('y(in m)')              # legend of the ordinate \"ey\" axis\n",
    "plt.axis('equal')               # same sclale for both axis\n",
    "\n",
    "\n",
    "# Representation of coordinate points (x1(t),y1(t)),red 'r',shape 'o',\n",
    "# size 2, connected by fine dotted lines 0.2 min thick \n",
    "\n",
    "plt.plot(x1,y1,'ro--',ms=2,lw=0.2)\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "Wodi1AcimpoX"
   },
   "source": [
    "#### Question 1\n",
    "Trace add the corresponding lines to trace the trajectory of the second plane : blue 'b',shape '-', size 3, connected by fine dotted lines 0.3 dots thick \n",
    "\n",
    "Answer below"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "id": "OWiWJgW6mpoX"
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 720x560 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "### BEGIN SOLUTION\n",
    "#----- Plotting of points and trajectories y1=f(x1) and y2=f(x2)-----  \n",
    "\n",
    "# initialization of the figure, name and size of the viewing window\n",
    "\n",
    "plt.figure('Planes fly in the sky',figsize=(9,7),dpi=80)\n",
    "\n",
    "plt.title('Trajectory of each plane') # title\n",
    "plt.xlabel('x(in m)')              # legend of the abscissa  \"ex\" axis\n",
    "plt.ylabel('y(in m)')              # legend of the ordinate \"ey\" axis\n",
    "plt.axis('equal')               # same sclale for both axis\n",
    "\n",
    "\n",
    "# Representation of coordinate points (x1(t),y1(t)),red 'r',shape 'o',\n",
    "# size 2, connected by fine dotted lines 0.2 min thick \n",
    "\n",
    "plt.plot(x1,y1,'ro--',ms=2,lw=0.2)\n",
    "plt.plot(x2,y2,'bo--',ms=2,lw=0.3)\n",
    "#plt.plot(x3,y3)\n",
    "\n",
    "plt.show()\n",
    "### END SOLUTION"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 295
    },
    "id": "euCvBOC6mpoY",
    "outputId": "b6ca1de2-1fa3-4039-9def-9a9d044fd455"
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Continuation of the program\n",
    "\n",
    "# DESCRIPTION OF THE SPEED\n",
    "\n",
    "#-----Tracing speed vectors every 2 positions-----\n",
    "# e corresponds to the scale for the representation of the velocity vector\n",
    "\n",
    "e=15\n",
    "\n",
    "# for loop to plot the different speed vectors\n",
    "\n",
    "for j in range(0,9):\n",
    "    i=2*j\n",
    "    \n",
    "    dx=(x1[i+1]-x1[i])/(t[i+1]-t[i])\n",
    "    dy=(y1[i+1]-y1[i])/(t[i+1]-t[i])\n",
    "    \n",
    "    plt.arrow(x1[i], y1[i], e*dx, e*dy,\n",
    "              length_includes_head=\"true\",width=80,fc='r',ec='r')\n",
    "\n",
    "plt.plot(x1,y1,'ro--',ms=2,lw=0.2)  \n",
    "\n",
    "\n",
    "# Display the same attributes to the graphic\n",
    "plt.title('Speed vectors of each plane') # title\n",
    "plt.xlabel('x (in m)')              # legend of the abscissa  \"ex\" axis\n",
    "plt.ylabel('y (in m)')              # legend of the ordinate \"ey\" axis\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'y (in m)')"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Representation of coordinate points (x1(t),y1(t)),red 'r',shape 'o',\n",
    "# size 2, connected by fine dotted lines 0.2 min thick \n",
    "\n",
    "plt.plot(x1,y1,'ro--',ms=2,lw=0.2)\n",
    "plt.plot(x2,y2,'bo--',ms=2,lw=0.2)\n",
    "\n",
    "\n",
    "# Display the same attributes to the graphic\n",
    "plt.title('Speed vectors of each plane') # title\n",
    "plt.xlabel('x (in m)')              # legend of the abscissa  \"ex\" axis\n",
    "plt.ylabel('y (in m)')              # legend of the ordinate \"ey\" axis\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "hECzpMwnmpoY"
   },
   "source": [
    "#### Question 2\n",
    "Trace add the corresponding lines to trace the speed vector of the second plane : blue 'b',with length including head, width of 80 points, . \n",
    "\n",
    "Answer below"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "id": "vsDX9rTqm_6Z"
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "### BEGIN SOLUTION\n",
    "for j in range(0,9):\n",
    "    i=2*j\n",
    "    \n",
    "    dx1=(x1[i+1]-x1[i])/(t[i+1]-t[i])\n",
    "    dy1=(y1[i+1]-y1[i])/(t[i+1]-t[i])\n",
    "    \n",
    "    dx2=(x2[i+1]-x2[i])/(t[i+1]-t[i])\n",
    "    dy2=(y2[i+1]-y2[i])/(t[i+1]-t[i])\n",
    "    \n",
    "    plt.arrow(x1[i], y1[i], e*dx1, e*dy1,\n",
    "              length_includes_head=\"true\",width=80,fc='r',ec='r')\n",
    "    \n",
    "    plt.arrow(x2[i], y2[i], e*dx2, e*dy2,\n",
    "              length_includes_head=\"true\",width=80,fc='g',ec='g')\n",
    "    \n",
    "\n",
    "plt.plot(x1,y1,'ro--',ms=1,lw=0.2) \n",
    "plt.plot(x2,y2,'go--',ms=1,lw=0.2) \n",
    "\n",
    "\n",
    "# Display the same attributes to the graphic\n",
    "plt.title('Speed vectors of each plane') # title\n",
    "plt.xlabel('x (in m)')              # legend of the abscissa  \"ex\" axis\n",
    "plt.ylabel('y (in m)')              # legend of the ordinate \"ey\" axis\n",
    "plt.show()\n",
    "### END SOLUTION"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "2viZVo4enBcw"
   },
   "source": [
    "## B) Operators on vectors\n",
    "\n",
    "The sclalar product of two vectors returns a scalar.\n",
    "\n",
    "Program function `scal_prod` inside of a cell in order to get a scalar product done:\n",
    "$$ \\overrightarrow{OM}_1 \\bullet \\overrightarrow{OM}_2 $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "id": "yHcgzmvinBcx"
   },
   "outputs": [],
   "source": [
    "from operator import mul\n",
    "def scal_prod(OM1,OM2):\n",
    "    return sum(map(mul,OM1,OM2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "yvKeD1WYnBcx",
    "outputId": "33c8a0ee-b5be-4d6d-9183-57948b26f4db",
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "12"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Check you function here with tho orthogonal vectors :\n",
    "def scal_prodV2(OM1,OM2):\n",
    "    x2=OM1[0]*OM2[0] # in m2\n",
    "    y2=OM1[1]*OM2[1] # in m2\n",
    "    z2=OM1[2]*OM2[2] # in m2\n",
    "    return (x2+y2+z2) # in m\n",
    "\n",
    "OM1=np.array([1,1,1])\n",
    "OM2=np.array([1,5,6])\n",
    "                                                        \n",
    "scal_prodV2(OM1,OM2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "12"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "OM1.dot(OM2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "MKnOZ0NCndyV"
   },
   "source": [
    "### The norm of a Vector\n",
    "To acces the magnitude of a vector $ \\biggl\\lVert  \\overrightarrow{M_1M_2}\\biggr\\rVert$ we can make the scalar product of this vector by itself.\n",
    "Programm a function `norm` that gives the norm of a vector in a cartesian frame."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "id": "fra-rKbcomb5"
   },
   "outputs": [],
   "source": [
    "def norm(vector):\n",
    "    return (vector[0]**2 + vector[1]**2 + vector[2]**2)**0.5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.7320508075688772"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "norm(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "def NORM(Vector_a):\n",
    "    \"\"\"\n",
    "    This is a \n",
    "    scalar product\n",
    "    of the same vector\n",
    "    Vector_a = np.array([])\n",
    "    \"\"\"\n",
    "    return (Vector_a.dot(Vector_a))**0.5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.7320508075688772"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x=np.array([1,1,1])\n",
    "NORM(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'Vector_a' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-16-f440606bbfd2>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mNORM\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mVector_a\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mVector_b\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;31mNameError\u001b[0m: name 'Vector_a' is not defined"
     ]
    }
   ],
   "source": [
    "\n",
    "NORM(Vector_a,Vector_b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "JcaGiR8od32E",
    "outputId": "be5da948-45ad-4792-c182-34f337758a5d"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5.196152422706632"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M1M2=np.array([3,3,3])\n",
    "norm(M1M2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5.196152422706632"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "drnorm = np.sqrt(M1M2.dot(M1M2))\n",
    "drnorm"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "oyieECOqmpoZ"
   },
   "source": [
    "# C) The lenght of a path\n",
    "\n",
    "We walk along a the following path. What is the distance covered ?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "g63Qvn1epcA9"
   },
   "source": [
    "The first exercice is a solved problem."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 113,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 279
    },
    "id": "Aiyv6Y2RmpoZ",
    "outputId": "2f387060-fb4a-4b08-aea1-78c1157c01a5"
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Variables \n",
    "c = 1  # meter . second -3\n",
    "a = 1  # meter . second -2\n",
    "\n",
    "\n",
    "t = np.linspace(0,2,30)\n",
    "x = c*t**3\n",
    "z = a*t**3\n",
    "\n",
    "plt.plot(x,z,'ro--',ms=2,lw=0.2)\n",
    "plt.xlabel('x (in m)')\n",
    "plt.ylabel('z (in m)')\n",
    "plt.axis('equal')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 114,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "11.313708498984761"
      ]
     },
     "execution_count": 114,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(x[-1]**2+z[-1]**2)**0.5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 110,
   "metadata": {},
   "outputs": [],
   "source": [
    "D=0\n",
    "t = np.linspace(0,2,500)\n",
    "x = c*t**3\n",
    "z = a*t**3\n",
    "\n",
    "\n",
    "\n",
    "for i in np.arange(0,len(t),1):\n",
    "    dx = (x[i]-x[i-1])\n",
    "    dz = (z[i]-z[i-1])\n",
    "    dl  = (dx**2+dz**2)**0.5\n",
    "    D += dl"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "22.62741699796953"
      ]
     },
     "execution_count": 111,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "fUsY9Iskmpoa"
   },
   "source": [
    "#### Question 3 (solved)\n",
    "Propose a piece of code to estimate the lenght corresponding to the distance covered between $t_O=0\\,s$ and $t_F=2\\,s$.\n",
    "\n",
    "Compare the obtained result with the distance corresponding to the straight line (shorter path).\n",
    "\n",
    "Answers below\n",
    "\n",
    "For a piecewise constant average speed $$dl=v(M)|_{\\mathcal{R}}.dt$$  \n",
    "\n",
    "where \n",
    "\n",
    "- $v(M)$ is the norm of the speed at a point (M)\n",
    "- $v(M)|_{\\mathcal{R}}$ denotes that this speed is to be taken in the same referential $\\mathcal{R}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 115,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "8oB8T4Iqmpoa",
    "outputId": "dfb0ca98-7c2b-45b8-b1cf-56bfadb3ab6a"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The lenght of the full path is L = 10.23 m\n"
     ]
    }
   ],
   "source": [
    "### BEGIN SOLUTION\n",
    "\n",
    "# The covered distance is calculated as follows :\n",
    "\n",
    "# Initialize variables\n",
    "Integral=[]\n",
    "Var_int=0\n",
    "\n",
    "npoints = 300\n",
    "t0 = 0 # time in s\n",
    "tF = 2 # time in s\n",
    "dt = tF/(npoints-1) # time in s\n",
    "\n",
    "\n",
    "# list of times in seconds\n",
    "t = np.arange(t0, tF+dt , dt)\n",
    "\n",
    "for ti in t:\n",
    "    Var_int += ti*(9*c**2+9*a**3)**0.5*dt\n",
    "    \n",
    "\n",
    "print(\"The lenght of the full path is L = {:0.2f} m\".format(Var_int)) \n",
    "\n",
    "### END SOLUTION"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "DQH7V36YmpoZ"
   },
   "source": [
    "The straight line is obtained by integrating the vectorial elementary displacement ; it is the norm of the vector that connects the starting point $O$ to the ending point $P$.\n",
    "\n",
    "$$ D = \\biggl\\lVert \\int_{t_O}^{t_F} d\\vec{l} \\biggr\\rVert $$\n",
    "\n",
    "$$ D = \\biggl\\lVert \\vec{OF} \\biggr\\rVert = \\sqrt{{(x_F-x_O)}^2 + {(x_F-x_O)}^2} $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 116,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "IbzUGErQnBcy",
    "outputId": "82b5051d-e15c-413f-8913-045e3a77c016"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The lenght of the shortest path is L = 11.31 m\n"
     ]
    }
   ],
   "source": [
    "# The shorter distance is calculated as follows :\n",
    "tF = 2 # in s\n",
    "D=((c*tF**3)**2 + (a*tF**3)**2)**0.5\n",
    "print(\"The lenght of the shortest path is L = {:0.2f} m\".format(D))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "17.88854381999832"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "7lnQXW7mrYtb"
   },
   "source": [
    "#### Question 3 (to be solved)\n",
    "From the following curve of CORNU calculate the lenght of the path covered during the lapse of time.\n",
    "\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 363
    },
    "id": "09S-Rlz-qOvn",
    "outputId": "feb42c8a-e74b-4777-bcb3-945afaf701cf"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-0.8578827404145053,\n",
       " 0.8578827404145053,\n",
       " -0.7851231243437429,\n",
       " 0.7851231243437429)"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAY0AAAEWCAYAAACaBstRAAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADl0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uIDMuMC4zLCBodHRwOi8vbWF0cGxvdGxpYi5vcmcvnQurowAAIABJREFUeJzt3XmYVNW57/Hv292KcWAQUCaRgIA0aDQi0ZNoNA4BtTGanGPURI1THBL1GI16Y0yiSZyiRxNJrkMcYganTKAocSJe4xDhhqsy06UMgogIKIpAd7/3j1WV2l1d3V3dNeyu6t/neeqpXVWrdq0ucb+1pneZuyMiIpKLqrgrICIi5UNBQ0REcqagISIiOVPQEBGRnCloiIhIzhQ0REQkZwoaInkys2oz22hmQ+Oui0ixKWhIt5O8wKduTWa2KfL45I6ez90b3X1Hd1+WZ72eN7PT8jmHSLHVxF0BkVJz9x1Tx2b2JnCmuz/VWnkzq3H3hlLUrbPMrArA3ZvirotUNrU0RDKY2Y/N7EEz+4OZfQB8zcwONLOXzGy9ma0ys5+b2TbJ8jVm5mY2LPl4OzO72cyWm9lqM/ulmW0XOf/xZjbHzN43syVmdqSZXQ8cCPzvZIvnlmTZz5nZLDPbYGb/NLPPRM7zvJldY2YvAh8Cl5nZyxl/y2Vm9kiRvzLpRhQ0RLI7Dvg90At4EGgALgT6AZ8FJgLfbOW9PwM+CewNjASGAd8DMLP/AO4GvgP0Bg4Flrr7ZcCLwDnJrq6LzKwf8BhwE9AX+Dkw3cz6RD7r68DpQE/gNmC0mY2MvP414P5OfwsiGRQ0RLJ73t2nuXuTu29y91fc/WV3b3D3BHAH8PnMNyW7ic4ELnL3de7+PnAt8NVkkTOAO9396eS5l7v7wlbqUAfMdfc/JD/3t0ACODpS5m53n+/uW939A+BhQqDAzPYBBgLT8/42RJI0piGS3fLoAzPbk/CLfz9ge8L/Oy9ned8AoAfw/8zs32+PvL4b8EqOdRgELM14bikwuLV6AvcB9wI/IASPB919a46fJ9IutTREsstM/3w78Dqwh7v3BK6ieTBIWQ1sAUa7e+/krZe790q+vhwYkeNnrgR2z3huKPBWa+9x9+cBzOyzwImoa0oKTEFDJDc7ARuAD81sDK2MZ7h7I3AXcIuZ9bdgiJkdmSzya+BMMzvUzKqSr41OvrYaGB453aPAWDM7ITnYfhKwB+13N90P/Ar40N1f6swfK9IaBQ2R3HwHOBX4gNDqeLCdskuBfxICzd8IA+K4+wvAWYRB7Q3As4QuK4BbgBOTM7Rudvc1wGTgMmAt8N/AMe7+Xjt1/Q0wDrUypAhMmzCJ5MfMtgU2A4PdfWUXqM8OwDvAOHd/I+76SGVRS0Mkf+OAjwgX6q7gfOAfChhSDJo9JZIHMzsBmAJ8tyusGjezFcBW4Ni46yKVSd1TIiKSM3VPiYhIzique6pfv34+bNiwuKshIlJWZs+e/a6792+vXMUFjWHDhjFr1qy4qyEiUlbMLDP7QFbqnhIRkZwpaIiISM4UNEREJGcKGiIikjMFDRERyZmChoiI5ExBQ0Til0jA2LFQUxPuE4m4ayStUNAQkfh94Qswbx40Nob7z3wGfvELeOgh+PvfYf58BZUuouJyT40fP961uE+kC0kk4JhjYOFC6N8fDjwwPHfmmfDtb4cy1dXQ1NT6OUaPhsWLQxkz+OQnob6+NPXvJsxstruPb69cxa0IF5EupKEBDjoIVia3GVm9Gv7yl3Ac/XE3ciQsWgTuISj06wdf+Uoo/+678I9/pIOKewg6e+8NX/86nHQSDB6MlIZaGiJSPF/+MvzpT82fq6oKQWCvvWCHHcJziQTU1YXWyOjRMG0aDI/sfDt2LCxYkG5pmKWDiFlovbz5Zggy2d4v7cq1paExDREpnIYG2LAh/fiUU2CbbcKFHULA2HNPOOCAdMCAcIGfOze8f+7clhf8adPC+6qrYcyYUObPfw5BaZtt4IUXQmumsTEEl7q64v+t3ZRaGiKSv0QCPv95WLECeveG2bPDhd89jEUcd1zrrYh8rVsXurOiYyLV1bB1azpYSbvU0hCR0jnwwBAwANavDwPfEC7ao0a13YrIV58+oRVSlbycVVWFgfJPfQpmzizsZ4mChojkwR2uugreydgefdGi0tYj2n21555hyu5rr8Ghh8JFF8GmTaWtTwVT0BCRznGHiy+Ga64Jj6PjFqNHl7YumWMi99wDP/hBCCK33hq6zKqrtcajABQ0RKRzfvELuOWWMBA9ZUoYoE790p82Ld66bbMN/PCH8PLL0KMHbNkSxjw0SJ43DYSLSO6iU2NHjoS+feHKK2HixLhr1rqamjCrKqW6OrRIpBkNhItI4dXVhV/rjY1h3GLdus4HjFLlmxo9Oj1Iblb6rrMKo6AhIrlLLbCDcL9wYcfPsWVLmNU0YUI631Sq2+if/wzTdTdvLlydM9d4xN11VuYUNEQkNx9/HC68Ke0NeEdbEqNGwY9/HNZr9OsXZjWtXZsumwpAV14J48fDjjtCbW2YTpu62He2JZJt4eDs2fDVrxY2OHUTsQYNM5toZgvNbImZXd5Kmf8ys3lmNtfMfl/qOopI0m23hQVzPXrkNuAd7cpavBi+//2Qd+qDD0Iw6du35Yyr0aNDgGhsDJlt169PD2AfeWRh/o6GhpDX6sEH4ROfCMFJM6pyFlvCQjOrBqYARwArgFfMbKq7z4uUGQlcAXzW3deZ2S7x1Fakm1u7NrQUIFz4cxnHWLiw+SptM7jjDvjiF2G33drON7VxI/Tq1fz9b7xRmL+lpiY9xuGe7hqbO7cw569wcWa5nQAscfcEgJk9ABwLzIuUOQuY4u7rANz9nRZnEZHiu+aakFPqiCPCRb8t7nDTTTBsWLjQNzWlc06deWa6XKrbKJsddwzlo0kK99yz+WfkkyJk6dLm5+rM2Ew3FWf31GBgeeTxiuRzUaOAUWb2DzN7ycyy/rwxs7PNbJaZzVqzZk2RqivSTS1ZEtZhmMGNN7Z9sXaHSy6BSy8Nq7BHj+782o3WBrAXLYKDD85v1Xl0RhVAz56dP1c3E2fQyPYvL3PRSA0wEjgEOBG4y8x6t3iT+x3uPt7dx/fv37/gFRXpthIJ+NznwjhAr16w006tl21shLPOgptvDovrbr01zI7qbM6p1jLfXnEFPP98ejFhZ6brRgMShLETdU/lJM6gsQLYLfJ4CLAyS5m/uvtWd38DWEgIIiJSCnV1kGq9v/9+9tXUiUQYTN5mG/j1r8NA+dSpYbC5GO65J7QMmpo6v8o7GpDOOy/M6Ip2WUmrYlsRbmY1wCLgMOAt4BXgJHefGykzETjR3U81s37Av4B93H1ttnOCVoSLFFQuq6nHjg0tipTUWEbc9crVunWhq6pXr8LUrUx1+RXh7t4AfAuYAcwHHnL3uWZ2tZlNThabAaw1s3nAs8ClbQUMESmwUaOapxzPti4jcxB5+fKWZQotc0xi6NDOn6tPnxAwSrVCvczFuk7D3ae7+yh3H+HuP0k+d5W7T00eu7tf7O617r6Xuz8QZ31Fup1Bg8Kv+NTsp2yD2aNGlT7DbWpMIvW5Q4bkf86jjw5rQ7T7X5u0IlxEWrdwYVjQN39+64PZjz5a+gy3qTGJNWvCQP0ll+R/zkWLwuwv6HyKlG4gznUaItKVvftu2I1vhx1gjz1aL9fWeoti69sX/s//Kcy5Ro5MB4o49gQpE2ppiEh2c+aE+099qvn4QdQjj8Bpp8FLL5WsWlkVYjxi+vQQIAEGDFBiw1YoaIhIS4kEnHxyOJ4/v/WL8L33wn33wf/9vyWrWlZHHdUyY25HDR8O110Xjg84oPB7mVcIBQ0RaamuLr3v9/r12S/C//oXPPZYOL711nhnGy1Zkj7OZzxicnLi5owZIauvtKCgISItRS+6reVmigaSJUvinW00alT6OJ/xiKFDYd994cMP4ZlnClO3CqOgISItjR7d/jTaVavSx6WebZQ5hvHLX6brO2pUfuMRRx4Z0qXU1Wm9RhYKGiLS0rRp7U+jjfb5l3q2UXSvjgUL4PzzQyoQCLsC5jMeMW1aaGl0NkVJhdOUWxFpKZdptNOnh0DhHqarlnK2UXSvjlQrZ+jQsG7jww8Lf275N7U0RKR1998PV12VPa/TyJFw7rlw0UXw97+XdrZRNI1IqpWTmi67cWPhzy3/FlvCwmJRwkKRAtptt7DAr76+a01Bzdz1b8qUkFX3vfdgxIgw+6mz9U0kYO+9Q4tl6FB49tmu9bcXSZdPWCgiXVwiES7CAIce2nUGhOfNC/tpzJyZ3mvj/PNDtlr3dEDprOHD4UtfCsdXX90tAkZHKGiISHZ1dfDRR+F4+fLsF+JFi8Iv+85uhtQZ998Pp54KP/lJ+rlCj0MMHBjuozPEBFDQEJHW5LpWI5Eo7Uyj6dPD/dFHp5/bfffCZtpV0GiVgoaIZJe5Z8Wuu7YsU1+fPi72TKNEIqzBePXVECCi6dD32ScENrPCZNpN/d2/+IXWamRQ0BCR7FJ7VkRnEm3Z0rxMZmAp5u53dXXpdCHu6e1k16+Hxx8Px4sXd24/8ky33JL+HK3VaEZBQ0SyS63V2LIFJk2CH/84jF1EZQaW994L+4QXw4IF6f0uIN2q+f3vYdMm+MIXwvhKIUR3H9RajWYUNESkbdXVYRzh1FNbBo1UYGlsTAeLc88Ns5sKaePG5i2a1LiFO9x5Z3jurLMK93laq9EqBQ0RyU17e1acfnpY6LfddvC1rxV2r+0dd4Q//SmMq6RSm0yZEloWc+aEC/unPpX/56Q8+mj4jFLuRlgmYl3cZ2YTgVuBauAud7+ulXJfAR4G9nf3NlfuaXGfSJHU1qa7iFJ7hmemGmloCOXq60O3TmvlcvHxx/DEE+k1E5nGjg1rNiAMgI8ZE98OghWgyy/uM7NqYAowCagFTjSz2izldgIuAF4ubQ1FpJlc9tCuqUlPwU2VmzcvrKlYtiy3z0kkQnfQJz4Bxx0HN9+cvVwuU4I765e/DAPtTz5ZuHNWiDi7pyYAS9w94e5bgAeAY7OUuwa4AdCOKCJxypwpVVUFb7zRfjmAK6+EYcPg8MPhpptCqyDafTV9Olx+eRhwHz06BKiUKVPar0+hxx1eeAH++EdYubJw56wQcQaNwUBkigIrks/9m5ntC+zm7o+2dSIzO9vMZpnZrDVr1hS+piKSnilVXQ09esDWrfAf/xHWTbRWbswYuPtu+OpXYdtt4emn4ZJLmqc1r6sL28Zef33ojspMjrh0afrYHS68MOxNHv2cQo87vP12uB8woHDnrBTuHssN+E/COEbq8deBX0QeVwEzgWHJxzOB8e2dd7/99nMRKbL1690POcQd3Hv1cn/ppfbfs26d++23h/dEb9XV7g8/7H7VVe5//KP7Hnu4V1WF16qq3Gtrw/ubmtzPOy88v+OO7mvWFOdvq69379EjfM7w4eFxNwDM8hyu3XG2NFYAu0UeDwGibcGdgHHATDN7EzgAmGpm7Q7UiEiR9eoVFtQdf3zY/Mis7ZlVAL17w9lnh4HyzG6lr3wFfvSjcL4ZM5q3IKZMCeesrg5jDdtuG1oaqU2XCq2uDjZvDsdvvqmFfRlimz1lZjXAIuAw4C3gFeAkd886/cHMZgKXuGZPiXQdjY1h46PDDgtdTU1N7c9kykxrPm1a6yu4N28O02rfeiv93NChzbusCq26Oj2Qn3qcbT+RCtPlZ0+5ewPwLWAGMB94yN3nmtnVZjY5rnqJSAdUV4d+/2iWWfcwY+qWW2Dt2pbvSS0ITKU1byvlx/XXNw8Y0PJxoUVbMFrY10Ksi/vcfbq7j3L3Ee7+k+RzV7n71CxlD2mvlSEiMck2Y+q//xsGDYLJk8PMqba6rhKJ0G1VXR1aEqkyp50WFgsWMoNte669Fnr2TK8x0cK+ZrRzn4jkL9rlNGoUfOtbYVX1E080zxeVuhCfey787Gfp51euDLOxUmpr091bHenOkk7LtXuqphSVEZEKl+pyijrvvLCgb9iwlosCN2xoe1wiulAv27klNso9JSLFM3RoGBTPnC11/vlhYWDqNnJk10gQ+NBDoYWU2rFQWlDQEJHiyrYIr3fv0AJJ3Z54Iv4EgfX1IdFiXR2MG6eNl1qhMQ0REYA99kjvRJhPosUy1eWn3IqIdCnRloU2XmqVgoaIyPLlLWd5aX1GVgoaIiKp9Os9e2rjpXZoyq2IdG/vvgt33BGOn3uusDsAViAFDRHp3rbbDn74Q5g/XwEjBwoaItI9ZVtpLu3SmIaIdE91dS03g5J2KWiISPcUzcyrKbY5U9AQke7HPYxlpGiKbc40piEi3c+vfw0ffhiChZnGNDpALQ0R6V6WLoWLLw7H99+f22ZQ8m8KGiLSfTQ1wemnwwcfhP3ITzwx7hqVHXVPiUj3kEjAZz8Lb78dVn1fdll6R0DJWawtDTObaGYLzWyJmV2e5fWLzWyemb1qZk+b2e5x1FNEKkBdHbzzTjhuaoJvfCPe+pSp2IKGmVUDU4BJQC1wopnVZhT7FzDe3fcGHgFuKG0tRaRiRKfYumuKbSfF2dKYACxx94S7bwEeAI6NFnD3Z909tYXWS8CQEtdRRMrdhg1w221hhlRX2B2wzMUZNAYDyyOPVySfa80ZwOPZXjCzs81slpnNWrNmTQGrKCJl7eOPYfJk+Pa34fDD498dsALEGTSyjUBl3UbQzL4GjAduzPa6u9/h7uPdfXz//v0LWEURKVsNDWF21HPPwaBBYZrt3LmaYpunOGdPrQB2izweAqzMLGRmhwPfAz7v7ptLVDcRKWfucO658Je/hP3IZ8yA3TWPphDiDBqvACPN7JPAW8BXgZOiBcxsX+B2YKK7v1P6KopI2amvhwMOCPtkmMHtt8O4cXHXqmLE1j3l7g3At4AZwHzgIXefa2ZXm9nkZLEbgR2Bh81sjplNjam6IlIuPve5EDBSfvSj+OpSgcw96zBC2Ro/frzPmjUr7mqISFxqakK685Tq6jCOIW0ys9nuPr69ckojIiLlL5GArVvDsabWFpWChoiUt6eegk9/OkyrdQ9TaTW1tmiUe0pEyk9qq9YFC9KrvFevDq2N4cPDlFopCgUNESk/Rx3VPA1I377wxz+mu6WkaPQNi0h5efjhlnmj1q9XwCgRfcsiUl4ee6z5Yw12l5SChoh0be7plOYAN98M114LtbUa7I5Bm+s0zGwIYaX2QcAgYBPwOvAY8Li7N5Wikh2hdRoiFSKRgCOPDPfbbANz5sCYMXHXqmLlvU7DzO4B7ga2ANcDJwLnAU8BE4HnzezgwlRXRCRi8eIwjba+PrQ0tmyBY46Ju1ZC27OnbnL317M8/zrwJzPbFhhanGqJSLdUXw/XXAO//W3zVd0AS5fGUydpptWg0UrAiL6+BVhS8BqJSPeRWm+xcGEYzN64EZYtC2MVvXvD+++HdRga7O4y2h0IN7NjzOxfZvaemb1vZh+Y2fulqJyIVDB3OOywsECvsTHcNzSEvbsXLYLZs7WyuwvKZXHfLcDxwGteadkNRaT01qyB+++Hu+6CN99MP9/UFFZ13313+jmt7O5ycgkay4HXFTBEpMOi3U+DB4fZT888k04uWF0dgoW7uqDKRC5B47vAdDP7O/DvnfPc/eai1UpEKkM0P9SyZeFWVRVmQp1xRggixx+fHtNQF1SXl0vQ+AmwEdgO2La41RGRsrZ6NTzxRFi1/aUvhWDQFFnOZRYCx+DB6efUBVVWcgkaO7v7kUWviYiUn7lzQ6th6VLo0QM+/jj9WkNDaD2kWhpVVWFAOxowpOzkkkbkKTNT0BDpjhIJGDs27IZXWwvTp8OmTenXDzwwDGa7h4BhBpMmwW23wU03aW+LCtTudq9m9gGwA2E8YytggLt7z+JXr+OURkQkT6nxh8WL4ZRTQpdT9Drx7LNwyCHhuKqq+WvaWrVsFWy7V3ffyd2r3P0T7t4z+bggAcPMJprZQjNbYmaXZ3m9h5k9mHz9ZTMbVojPFekWoq2EsWPDYwgpOZYvh1mzwrTXQYPCxT5VZsMG+OQnQ96nt99uHhQgLLhLGTNGW6t2M7FluTWzamAKMAmoBU40s9qMYmcA69x9D+B/CDmwRIqrtYttrq8X4hyZry9YEFZLv/de2DsiVSaV6XXYMPjVr+C++0LX0E9/ChMmNF84d8ABYbOiHj1g6FDYf3846yxYtSq0LhYsCLOd+vQJn3nQQWFVtln4vKqq8HmTJ6frqe6nbqfd7qmifbDZgcAP3f2LycdXALj7tZEyM5JlXjSzGuBtoH9ba0bUPSVteu21MMUzkYAhQ+A734Httw+/no87LvzCHjsW5s9P/8LeYQfYb7/wC33LljAjaNOm9ODuttuGi7F7es3Bu++mcyelBoBvvhlOPjk8t25d81lFVVWwYgUMHBge9+wJH3yQ/W849lj4y1/SwSR6nraYhbpVV8Muu8CAASFzbFvdS5lpPqZNC9upSsXJtXsqzu1eBxMWDqasAD7TWhl3bzCzDUBf4N1oITM7GzgbYOhQ5VDsllIX6pkz4aKLwq/n3r3hnHPCr24IM3z23jv9nmXL4MIL049HjAhBY+HC5hfSDz+E557L/rlNTWEA+K23Wq9bU1M459atsHZt62WiF/+NG1uW2WGHEKC23z48zjad9eSTQ8DZaSe4557wnaSC28iR4e/o2zcEB2geeLJ1L2m/bcnk7u3egGrCfhpDU7dc3tfOOf8TuCvy+OvALzLKzAWGRB7XA33bOu9+++3nUsEWLHAfPty9utq9tta9vt79oovce/d2D5f65rftt0+/d9Omlq+buX/jG+4XXug+Z04oV1vrXlWVfn333d1nznR/4QX3WbPcR4xIv15V5T5ypPuyZe4rVri/9Zb7qlXuo0Y1L1Nb6755s/uaNeE2enTz10ePdm9sTNd1zJiW788UrWe2MvX14bnod5UplzLSLQCzPJdrd7sF4NuEX/ZzgdeSt1dzOXk75z0QmBF5fAVwRUaZGcCByeOaZD2srfMqaJS5zIvYM8+4/+534aJ+4IHhIp664KculBdeGB7vtFPLoFBV1fz87V1os9Uh80JaiItxKT5DpANyDRq5TLldAnzG3VtpV3dOcoxiEXAY8BbwCnCSu8+NlDkf2MvdzzGzrwLHu/t/tXVejWmUuY7201dXh7EAs9BPP25cy8Vk0e4V9dGLZFXIMY3lwIb8q9SchzGKbxFaE9XA3e4+18yuJkS8qcCvgfuTges9wtazUknWroUZM0Laic2bW/bTQ7jIT5gQbhdcENYPRPvgBwxIl502rWVQiFIfvUhecmlp/BoYTdgXvMsnLFRLo4urrw/z/994A7bbLgwip/4N1tTAHnuEvRTUUhApqUK2NJYlb9uihIWSr4MPhpUrw3EqHcVhh8HRR8NRR4XAMXmyWgoiXVS7QcPdf1SKikgFamqCp54KaxJOOCE8t3p18zLV1aFMlIKCSJfVatAws1vc/SIzmwa06MNy98lZ3ibdXSIBEyfCkiXphWKDBsGXvxxaEZlZT5V2QqSstNXSuD95/7NSVEQqQH19WDm9ITlvoqEBttkGzj03rKSuqWl/oFpEurTY0ogUiwbCYzJ7NnzmM+nUGSnKeipSFvLOcmtm08yszsy2yfLacDO72sxOz7eiUsZSifMA9t0X9tknpO5Q1lORitVWltuzgIOABWb2iplNN7NnzCwB3A7Mdve7S1JL6ToSCRg1KgSEnXdO52SqqoIXXwwtDmU9FalYOXVPJfexGAhsAha5+0fFrVbnqXuqiJqaQmbYVavSzw0cmJ5CKyJlq2CbMCVXba939xfdfU5XDhhSRPX1YT1FNGAAvPNOPPURkVjksgnTAGCWmT2U3GnPil0p6WIeegj22iukHa+ubr4pj8YsRLqVXLZ7vRIYScgDdRqw2Mx+amYjilw3iUvmrnG9eoVZUSefDC+9FLb41JiFSLeU0yZM7u5m9jZh57wGoA/wiJk96e7fLWYFJQZ1demd6xYsgIsvDo9TOZ60Yluk22o3aJjZBcCphL0s7gIudfetZlYFLAYUNCrJ5s3NtzpN7TqnpIAiQm4tjX6EfSyWRp909yYzO6Y41ZJYrFkT9s+OzqjTuIWIROSSsPCqNl6bX9jqSGzmzYNjjgkpy3fdNexHvXSpUn2ISDM5jWlIhXviiZCF9v33Yfx4+OtfQ5JBEZEMChrdWSIRWhfzkw3GSZPgkUdg++3jrZeIdFm5rNOQSpXKNgth7cXSpQoYItImBY3uavHi5vtxu6cDiIhIK2IJGma2s5k9aWaLk/d9spTZx8xeNLO5ZvaqmZ0QR10r0hNPQG0t9OmjjLQi0iFxtTQuB55295HA08nHmT4CTnH3scBE4BYz613COlaml18Ou+g1NMBxxykjrYh0SFwD4ccChySP7wNmApdFC7j7osjxSjN7B+gPrEc6Z/58OOoo+OgjOO00uP32dB4pEZEcxNXS2NXdVwEk73dpq7CZTQC2Bepbef1sM5tlZrPWrFlT8MqWvdQeGLW18N57cOihcOedChgi0mFFa2mY2VOEDLmZvtfB8wwk7Fd+qrs3ZSvj7ncAd0DYT6ODVa18xxwTBr5TVq0KyQhFRDqoaFcOdz+8tdfMbLWZDXT3VcmgkHVTBjPrCTwGXOnuLxWpqpVv0aLmj6MBRESkA+LqnppKSIJI8v6vmQXMbFvgz8Bv3P3hEtat8owerVlSIlIQcQWN64AjzGwxcETyMWY23szuSpb5L+Bg4DQzm5O87RNPdctUKvHgtGmaJSUiBZHTHuHlRHuER1x6KWzaBNdfHxIQioi0omB7hEuZev55uOkm+NWv0rmlRETypKBRaRKJsB3rQQeF7qlvfjNkrhURKQAFjUoTTUIIMHNmbFURkcqjoFFpFi5svvNe5nRbEZE8KGhUmr5908eaXisiBaagUWkOPjjcV1Vpeq2IFJxySVSahx8OXVSjRim3lIgUnIJGJVKXlIgUibqnKkEiAcOGhRXfY8eGxyIiRaCgUQnq6sL+3k1NYSFfXV3cNRKRCqWgUQkWLEgfa69vESkiBY1K0K9f+ljTbEWkiBQ5RTgUAAASaElEQVQ0KsHuu4f7OKfZJhJhPKWmJvu4Snuvi0hZ0Oypcvf++/Cvf4VB8Hffhd69S1+HDz6Az30u7AgIMG9eGFeZOxeWLQvTf7dsSa9UnzcPDjssBA5NCxYpK2pplLtnnoGGBjjggNIHjDlz4JxzYNCgdMBIiY6rbN7cPLUJwJtvwrhx8PrrRa+miBSOgka527gRhgyBL36xuJ+T2b102WWw775w++2hDttvn241RMdVdtst7OkxZkx690Cz0DJatixMFU6df+RIdV+JdHEKGuUskYBrrw2/8v/wh+JeaOvqwiytxsZw/8gj0KsXXHBB6IZ67bUQGDJ3BzSD7baDRx9N7x44Zkx4z3PPwY47hnLHHANLlqTPr2nDIl2Sdu4rZ2PHhgtsU1N6EHzu3OJ8VnV1+Jzo440bQ0AohJqaEDBSqqqaPxaRourSO/eZ2c5m9qSZLU7e92mjbE8ze8vMbitlHcvCwoXpC3lTU/HWZ9x5Z/OAkep+ai9g1NeHcrl0OY0ene6+gjAG8uST+dVbRAouru6py4Gn3X0k8HTycWuuAf5eklqVm1KkQb/zTjj77HC8667p7qcpU7JPof3gAzj//DCbatSosJ9HY2NYqd7WuMu0aenuq169QtA4+mh44IHC/00i0mlxBY1jgfuSx/cBX8pWyMz2A3YF/laiepWXSZPCfbHWZ0QDxk03wdtvh5lac+eGwJAa40hNsYUwIH7PPfCPfzRvnbiHMYu//CX7Zw0fHs7b0ADvvQcXXwxbt8LJJ2sjKZEuJK6gsau7rwJI3u+SWcDMqoCbgEvbO5mZnW1ms8xs1po1awpe2S5r06Zw/9vfhgvu8OH5nzM1S6q6unnAuPji5uVSYykpqa6x6uowo+pvfwstjWiXU3U1HHpo+nOGDMmeZLGqKnzmDTfA5ZfDccdpVpVIF1G0oGFmT5nZ61lux+Z4ivOA6e6+vL2C7n6Hu4939/H9+/fPr+Ll5O23w/2AAYU7Z2qWVCog7Lpry4Dx+99nH+NI+frX4Ygj4PHH011OtbUhsPXqFcoccwy89VbbSRYvvTS0TKKztjSrSiRWRVsR7u6Ht/aama02s4HuvsrMBgLvZCl2IHCQmZ0H7Ahsa2Yb3b2t8Y/upRhBIzq4DmGVeYp7aAFcmmz87bwzbNgQAka2rrFUl1M20S4n9+ZJF1urTzEH+0UkJ3F1T00FTk0enwr8NbOAu5/s7kPdfRhwCfAbBYwMq1eH+0IGjegspswWxL33pgPGTTfB2rXpMY6Odo1lmy312mvtl+tOLUmRLiiuoHEdcISZLQaOSD7GzMab2V0x1am8zJsXfuVDmKlUqL7+c89Nb+iUOUvqhhtgwoSwkDCzy6qjorOlevYMQWPyZMgck0qVSwWOhgb46KP8PltEOk2L+8pVbW3o0nEv3MK+Dz8MeaTefz90H40cWZoFhJs2wec/D6+8EhIZPvlky0SG7rD//jB7Nlx/PXz3u4Wtg0g316UX90kBLFqUTgJYqL7+hx8OAeOAA0LAgNKMKXziE/DnP8Nee8E3vhESGWbOljKDn/wkHF93XbqVJSIlpaBRrtoae+isO+8M92edlX5ut93Sx8Xc4Gnw4JA196c/bX221JFHhtbGunXQp4+m4IrEQEGjXE2bFqbDmsEee+S3sC+RCOd44YUQGCZMSL82cWK4Nyv+Bk9VVW23bMzSYx6pGVeagitSUgoa5Wr48LA4zj3MaspnYV9dXfoXuzuccEL6tRdeCPczZhRuAWFbUt1ikL1lszyybEdTcEVKTkGjnA0cGO4zN0DqqIUL0+Mj7ukL8fLl8OqrsMMOcPDB+X1Grh59NN3tNmJEy5ZNMbrlRCRnChrlrFBBY/To9Gwls/SFePr0cH/44dCjR36fkasRI8IUYoCf/7xlyyY6VTeu/dBFujEFjXKWupB/+9v5DQpPm5beQGnMmPA4kYBLLgmv//OfpR1w3nvvcJ9tsd/w4fDQQ3DssWF6brG7y0SkmaKlEZESePjhcB8dFO7MGops6T7Gjg2bLEFYed7Zc3fGXnuF+2xBA8JYxp/+pK4pkRiopVHOUmlEoDCDwk1NsGVLOI6eq9QDzjvvHO7vvz97CyrVukgktLufSIkpaJSz6C/tfAeFH3ggdE+ddlr63HENOH//++njbNNqV68Oi/+2bg3daVqrIVIyChrl7NFH02MR+Q4Kr18f7h94IPy6nzIlvgHnxYvTx9laOXV1IQcVhI2dtFZDpGSUe0qCYcNg6dJwXKwcU7lqL99VTU3zbqnq6nQQEZFOUe6p7mTLlpDk7733On+OFSvSx3EvmmtvWm0hu+VEpEMUNCrBCSeEvExTp3b+HNELb3StRqlt3BhyX51zThizyLYKfepU6NcvXU+t1RApGQWNSrDvvuH+9NM7v15j2rSwsA7CFN4pUwpXv4549FF45pmwZ0dmevSUESNCDqqmprCviNZqiJSMgkYl+O1vw30+SfyGDw+DyqeeGtKG5NPVlY+HHgr30fxXItJlKGhUgmjLIt/xiGuvhTfegOOPz79eHZFIhJlgf/5zeBzNtJvp2WfhpZfg449LUzcR+TcFjUoQzR0F+XXXDBwYzz7cdXXNg92ZZ7Ze9uKL4cADQbPkREpOQaMSpHJHpQJHKndTPhYsCFu/VleXZrOj1Na1Ka21lhYsCJs1QRgw18I+kZKKJWiY2c5m9qSZLU7e92ml3FAz+5uZzTezeWY2rLQ1LROp3FGLF4dB4kJ0LR18cMie29QE8+cXfwFdn8g/gbam0R55ZPp40SIt7BMpsbhaGpcDT7v7SODp5ONsfgPc6O5jgAnAOyWqX3kaMSJcSE86Kf9zRQfC3UPgKOYCupdfDlvLtrUCvaFBmzCJxCyuoHEscF/y+D7gS5kFzKwWqHH3JwHcfaO7f1S6KpapqqrQZTN2bFg53dmupWjuKQiB44gj8t+7I3q+W29NJ10cMQKWLQuBobUdAu+9t/ljLewTKbm4gsau7r4KIHm/S5Yyo4D1ZvYnM/uXmd1oZtXZTmZmZ5vZLDObtSa1h3R3dvTRoWXQ2Nj5KbjRVdm77w59+8LMmWFNSD6/7lMBrboaLroobPCUS6bajRvTiQwHD9YmTCIxKVrQMLOnzOz1LLdjczxFDXAQcAmwPzAcOC1bQXe/w93Hu/v4/nHM/OlqFi9ODyp3tgsnNU7S0ABvvgmvvw6HHBK6kI47LrRiamtDcMrFhg1hD4z99gsL8lL127AhBID21NTAhReG1s7y5W23SESkaGJJWGhmC4FD3H2VmQ0EZrr76IwyBwDXufshycdfBw5w9/PbOrcSFtI84R+EQea1a1tfYZ2rxsZw7sWLw7nNwsV/773h058OrZABA+AHPwhlUik+1q8P6y6ytSg6mmzQPf+/Q0Ra6OoJC6cCpyaPTwX+mqXMK0AfM0s1Hb4AzCtB3cpfqmupqipcYNetg2uuyf+81dVh1XgqGKV+cLz6ahhvuPDCsJJ7wYLmXWPjxkHPnnDQQbDLLumLfi5jEolEaNGkxmfeeCP/v0NEOi2uoHEdcISZLQaOSD7GzMab2V0A7t5I6Jp62sxeAwy4M6b6lpdU11JjI/zxj+Hi/IMfwO2353/uzM2Z9twTXngBbrst5L6KSnWNbbstrFwJzz0HL77YsT1ADjkk//EZESkY7afRHdxxB3zzm+EiPWcO9OjR+XMlEunV26nup+i4Qnt7YXTEH/7Qcvqw9s4QKYpcu6dqSlEZidnZZ4funXHjwthDaxf8XKRaMa2ZNq1lUOmMu+4K9Yb02Imm2IrETi2N7iTaCjALrYB5XXCY6H/+J+SXAvjOd+Dxx/MLdCLSrq4+EC5xWLiw+SD2/PnxbenamhtuSAeMW2+Fn/0sPfVXU2xFYqeg0Z1krvKG0F114425LbArptSivyuuCHW87jq44IJ46yQiLShodCeZe2+fcELYX/y73w3rKEaOzC/1SGc0NIQZXnV1zdeW/OY3pfl8EekQjWl0d48/Dmec0TynVL6znnLR0BBmR11zTVgIWFWVDhigWVIiJaYxDcnNpEkhRUh0lXVqfcUDDxRu29doEsUhQ0Kr5pRT0uncBw9uvv5Ds6REuiQFDYGddw4L7qIX7WHD4MQTQ1qQ44+HX/2q+crsjnZf1dWlF+m99VbIZzViRFhJvmBBSIYY7TpTIkKRLkndUxJkLtq79lqYMgWeeqp5t1HKmDHp6brPPRdSnP+v/xXOM2BASCz45pthJtTkySHYRAfbq6pg8+bwvIjELtfuKQUNadvKlWHs4ZJLmj8fHXMYNSp0M2VzxRXw058WdqW4iBScxjSkMAYNCgvsamvT3Vdmzccc9t+/5fvMYPr0sGcGtJy5pe4nkbKkoCG5iV70x4xpftH/3e+aB5WqqlBm0qSQ1Raa78+hRXoiZUsdypKbUuWcEpEuTUFDCqO9oCIiFUHdUyIikjMFDRERyZmChoiI5ExBQ0REcqagISIiOVPQEBGRnFVcGhEzWwMsLfHH9gPeLfFnlgN9Ly3pO2lJ30lLcXwnu7t7//YKVVzQiIOZzcolZ0t3o++lJX0nLek7aakrfyfqnhIRkZwpaIiISM4UNArjjrgr0EXpe2lJ30lL+k5a6rLficY0REQkZ2ppiIhIzhQ0REQkZwoanWBm/2lmc82sycxanRZnZhPNbKGZLTGzy0tZxziY2c5m9qSZLU7e92mlXKOZzUneppa6nqXQ3n97M+thZg8mX3/ZzIaVvpallcN3cpqZrYn82zgzjnqWkpndbWbvmNnrrbxuZvbz5Hf2qpl9utR1zKSg0TmvA8cDz7VWwMyqgSnAJKAWONHMaktTvdhcDjzt7iOBp5OPs9nk7vskb5NLV73SyPG//RnAOnffA/gf4PrS1rK0OvD/w4ORfxt3lbSS8bgXmNjG65OAkcnb2cCvSlCnNilodIK7z3f3he0UmwAscfeEu28BHgCOLX7tYnUscF/y+D7gSzHWJU65/LePflePAIeZmZWwjqXWHf9/aJe7Pwe810aRY4HfePAS0NvMBpamdtkpaBTPYGB55PGK5HOVbFd3XwWQvN+llXLbmdksM3vJzCoxsOTy3/7fZdy9AdgA9C1J7eKR6/8PX052wzxiZruVpmpdWpe7jmi711aY2VPAgCwvfc/d/5rLKbI8V/bzm9v6XjpwmqHuvtLMhgPPmNlr7l5fmBp2Cbn8t6/Ifx9tyOXvnQb8wd03m9k5hJbYF4pes66ty/07UdBohbsfnucpVgDRX0pDgJV5njN2bX0vZrbazAa6+6pkE/qdVs6xMnmfMLOZwL5AJQWNXP7bp8qsMLMaoBdtd1OUu3a/E3dfG3l4JxU+zpOjLncdUfdU8bwCjDSzT5rZtsBXgYqcKRQxFTg1eXwq0KJFZmZ9zKxH8rgf8FlgXslqWBq5/LePfldfAZ7xyl5p2+53ktFXPxmYX8L6dVVTgVOSs6gOADakuoBj4+66dfAGHEf4BbAZWA3MSD4/CJgeKXcUsIjwK/p7cde7BN9LX8KsqcXJ+52Tz48H7koe/wfwGvD/kvdnxF3vIn0XLf7bA1cDk5PH2wEPA0uAfwLD465zF/hOrgXmJv9tPAvsGXedS/Cd/AFYBWxNXlPOAM4Bzkm+boRZZ/XJ/1/Gx11npREREZGcqXtKRERypqAhIiI5U9AQEZGcKWiIiEjOFDRERCRnChoieTKzgWb2aPJ4vJn9vAifsZeZ3Vvo84p0lKbciuTJzG4Envfc0svk8zlPAae7+7Jifo5IW9TSEMmBme2fTKS3nZntkNxPZVzy5S8DTyTLHRJpdfwwuV/CTDNLmNkFrZx7o5ldb2azzewpM5sQeU80dfw0wkpqkdgoaIjkwN1fIaR0+DFwA/Bbd3/dzD5J2Bdjcytv3RP4IiE1+A/MbJssZXYAZrr7fsAHyc84gpB54OpIuVnAQYX4e0Q6SwkLRXJ3NSGH0sdAqtUwEFjTxnseSwaUzWb2DrArIV1E1BaSLRVCqojN7r7VzF4DhkXKvUNIVSMSG7U0RHK3M7AjsBMhdxTApshxNtEWSCPZf6ht9fTgYlPqPe7elFF+u+TnicRGQUMkd3cA3wd+Rzpt9yKatwaKaRRhq2GR2Kh7SiQHZnYK0ODuv0/ud/2CmX3B3Z8xs3oz28PdlxS5GocCjxX5M0TapCm3Inkys+OA/dz9yiJ+Rg/g78DnPGwPKxILtTRE8uTufzazYu/vPRS4XAFD4qaWhoiI5EwD4SIikjMFDRERyZmChoiI5ExBQ0REcqagISIiOfv/oDdvjB1uZKMAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.special import fresnel\n",
    "  \n",
    "tO=-3\n",
    "tF=3\n",
    "\n",
    "t = np.linspace(tO, tF, 100)\n",
    "y, x = fresnel(t)\n",
    "\n",
    "\n",
    "plt.plot(x, y,'ro--',ms=4,lw=2)\n",
    "plt.title('Trajectory') # title\n",
    "plt.xlabel('x(in m)')              # legend of the abscissa  \"ex\" axis\n",
    "plt.ylabel('y(in m)')              # legend of the ordinate \"ey\" axis\n",
    "plt.axis('equal')                  # same sclale for both axis\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "id": "jo-hRwSHsc90"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The lenght of the path is L = 5.973 m\n"
     ]
    }
   ],
   "source": [
    "### BEGIN SOLUTION\n",
    "from scipy.special import fresnel\n",
    "  \n",
    "tO=-3\n",
    "tF=3\n",
    "npoints=100\n",
    "D=0\n",
    "\n",
    "t = np.linspace(tO, tF, npoints) # The time is s\n",
    "y, x = fresnel(t) # The lenght is m\n",
    "\n",
    "for ti in np.arange(0,npoints-1,1):\n",
    "    dx2 = (x[ti+1]-x[ti])**2\n",
    "    dy2 = (y[ti+1]-y[ti])**2\n",
    "    dl  = (dx2+dy2)**0.5\n",
    "    D += dl\n",
    "\n",
    "print(\"The lenght of the path is L = {:0.3f} m\".format(D))\n",
    "\n",
    "### END SOLUTION"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "colab": {
   "collapsed_sections": [],
   "name": "Motion_along_trajectories.ipynb",
   "provenance": []
  },
  "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": "3.7.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}