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    "<div style=\"display:inline-block;background-color:#f6f6f6;border:3px solid #ff9900;padding:0.6em 1em;box-sizing:border-box;\">\n",
    "    <span style=\"color:#ff9900;font-weight:bold;\">WORK IN PROGRESS</span>&nbsp;&nbsp;This notebook is under development, please bear with us...\n",
    "</div>"
   ]
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Falling objects - computational exercise\n",
    "\n",
    "## 1. Get into the \"Falling objects\" virtual lab\n",
    "\n",
    "A. First, do the [exercise with the \"Falling objects\" virtual lab](FallingObjects-exercise.ipynb)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "from lib.fallingobjects import *\n",
    "FallingObjectsLab();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "B. Draw a diagram representing the different elements of the code [available in this python file](lib/suspendedobjects.py).  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2. Extend the virtual lab\n",
    "\n",
    "You can extend the virtual lab by redefining the following elements:\n",
    "* The list of the objects available in the lab so that you can add properties to these objects\n",
    "* The functions representing the equations of motion\n",
    "    \n",
    "**Goal:** redefine the necessary elements to implement **a quadratic model of air friction** and **Archimed's principle** in the virtual lab.\n",
    "\n",
    "### 2.1. Rewrite the functions below"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "###--- Functions representing the equations of the movement as functions of time and the problem parameters\n",
    "def accel_time_withair(obj, g, h_0, v_0, t):\n",
    "    # TODO to replace!! -->\n",
    "    lamb = obj.k / obj.mass\n",
    "    return -g * np.exp(- lamb * t)\n",
    "    # <--\n",
    "\n",
    "def veloc_time_withair(obj, g, h_0, v_0, t):\n",
    "    # TODO to replace!! -->\n",
    "    lamb = obj.k / obj.mass\n",
    "    return (v_0 + (g / lamb)) * np.exp(- lamb * t) - (g / lamb)\n",
    "    # <--\n",
    "\n",
    "def height_time_withair(obj, g, h_0, v_0, t):\n",
    "    # TODO to replace!! -->\n",
    "    lamb = obj.k / obj.mass\n",
    "    return (1 / lamb) * (v_0 + (g / lamb)) * (1 - np.exp(- lamb * t)) - (g / lamb) * t + h_0\n",
    "    # <--"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2.2. Extend the list of objects with the characteristics that you need in the equations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "###--- Static list of objects with which we can experiment.\n",
    "# Objects come with a number of attributes, of which:\n",
    "# - a name and a color : mandatory so that the virtual lab executes\n",
    "# - a mass and a friction coefficient : used by the functions implementing the motion equations, can be replaced or renamed \n",
    "#       as long as the three functions accel_time_withair, veloc_time_withair and height_time_withair are redefined\n",
    "objects_with_k = [{\n",
    "    'name':'Bowling ball',\n",
    "    'mass':5.0,\n",
    "    'k': (6*np.pi*0.11) * 1.8*10**-5,\n",
    "    'color':'#DC143C'\n",
    "    },{\n",
    "    'name':'Tennis ball',\n",
    "    'mass':0.0567,\n",
    "    'k': (6*np.pi*0.032) * 1.8*10**-5,\n",
    "    'color':'#2E8B57'\n",
    "    },{\n",
    "    'name':'Ping-pong ball',\n",
    "    'mass':0.0027,\n",
    "    'k': (6*np.pi*0.02) * 1.8*10**-5,\n",
    "    'color':'#FF4500'\n",
    "    },{\n",
    "    'name':'Balloon',\n",
    "    'mass':0.013,\n",
    "    'k': 0.02,#(6*np.pi*0.28) * 1.8*10**-5,\n",
    "    'color':'#000080'\n",
    "}]\n",
    "\n",
    "## Utility function to print objects nicely\n",
    "def object_string(obj):\n",
    "    return '{!s}:\\n  mass = {} kg \\n  friction coeff. = {:.2e}'.format(obj.name, obj.mass, obj.k)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2.3. Test your code \n",
    "\n",
    "Execute the following cell:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "FallingObjectsLab(objects = objects_with_k, show_withair = True,\n",
    "                  accel_time_withair = accel_time_withair, veloc_time_withair = veloc_time_withair, height_time_withair = height_time_withair);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## References : \n",
    "\n",
    "frottements linéaires vs. quadratiques : \n",
    "\n",
    "http://www.physagreg.fr/mecanique/m12/M12-chute-libre-frottements.pdf\n",
    "\n",
    "https://femto-physique.fr/mecanique/problemes-de-chute.php\n",
    "\n",
    "equations avec frottements linéaires :\n",
    "\n",
    "https://fr.wikipedia.org/wiki/Frottement_fluide\n",
    "\n",
    "equations avec frottements quadratiques :\n",
    "\n",
    "http://hyperphysics.phy-astr.gsu.edu/hbase/Mechanics/fallq.html\n",
    "\n",
    "https://studylibfr.com/doc/4875224/chute-libre-avec-frottement---gilles-auriol\n",
    "\n",
    "https://fr.wikipedia.org/wiki/Chute_avec_r%C3%A9sistance_de_l%27air"
   ]
  }
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