{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from modsim import *"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "pint.unit.build_unit_class.<locals>.Unit"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 165,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "class Vector(np.ndarray):\n",
    "    \"\"\"Represents a vector, usually in 2D or 3D.\n",
    "    \n",
    "    Based on https://gist.github.com/eigencoder/c029d7557e1f0828aec5\n",
    "    \"\"\"\n",
    "    def __new__(cls, *args):\n",
    "        \"\"\"\n",
    "        cls:  class object\n",
    "        args: tuple of coordinates\n",
    "        \"\"\"\n",
    "        if len(args) < 2:\n",
    "            raise ValueError('Vector must contain at least two elements.')\n",
    "        \n",
    "        # if any of the elements have units, store the first one\n",
    "        for elt in args:\n",
    "            units = getattr(elt, 'units', None)\n",
    "            if units:\n",
    "                break\n",
    "            \n",
    "        if units:\n",
    "            # if there are units, remove them\n",
    "            args = [getattr(elt, 'magnitude', elt) for elt in args]\n",
    "            \n",
    "        obj = np.asarray(args).view(cls)    \n",
    "        return obj\n",
    "    \n",
    "    def __mul__(self, other):\n",
    "        if isinstance(other, UNITS.Unit):\n",
    "            return VectorQuantity(self, other)\n",
    "        else:\n",
    "            return super().__mul__(other)\n",
    "            \n",
    "    def __div__(self, other):\n",
    "        if isinstance(other, UNITS.Unit):\n",
    "            return VectorQuantity(self, 1/other)\n",
    "        else:\n",
    "            return super().__truediv__(other)\n",
    "        \n",
    "    __truediv__ = __div__\n",
    "            \n",
    "    @property\n",
    "    def x(self):\n",
    "        return self[0]\n",
    "\n",
    "    @property\n",
    "    def y(self):\n",
    "        return self[1]\n",
    "\n",
    "    @property\n",
    "    def z(self):\n",
    "        return self[2]\n",
    "    \n",
    "    def __repr__(self):\n",
    "        t = map(str, self)\n",
    "        s = ', '.join(t)\n",
    "        return '%s(%s)' % (self.__class__.__name__, s)\n",
    "        \n",
    "    def __eq__(self, other):\n",
    "        return np.array_equal(self, other)\n",
    "\n",
    "    def __ne__(self, other):\n",
    "        return not np.array_equal(self, other)\n",
    "\n",
    "    def __iter__(self):\n",
    "        for x in np.nditer(self):\n",
    "            yield x.item()\n",
    "\n",
    "    @property\n",
    "    def mag(self):\n",
    "        \"\"\"\n",
    "        \"\"\"\n",
    "        return np.sqrt(np.dot(self, self))\n",
    "\n",
    "    @property\n",
    "    def mag2(self):\n",
    "        \"\"\"\n",
    "        \"\"\"\n",
    "        return np.dot(self, self)\n",
    "\n",
    "    def hat(self):\n",
    "        \"\"\"\n",
    "        \"\"\"\n",
    "        return self / self.mag\n",
    "\n",
    "    @property\n",
    "    def angle(self):\n",
    "        \"\"\"\n",
    "        \"\"\"\n",
    "        return np.arctan2(self.y, self.x)\n",
    "\n",
    "    def polar(self):\n",
    "        return self.mag, self.angle\n",
    "\n",
    "    def dot(self, other):\n",
    "        \"\"\"\n",
    "        \"\"\"\n",
    "        return np.dot(self, other)\n",
    "\n",
    "    def cross(self, other):\n",
    "        \"\"\"\n",
    "        \"\"\"\n",
    "        return np.cross(self, other)\n",
    "\n",
    "    def proj(self, other):\n",
    "        \"\"\"\n",
    "        \"\"\"\n",
    "        print('hello1')\n",
    "        b_hat = other.hat()\n",
    "        return self.dot(b_hat) * b_hat\n",
    "\n",
    "    def comp(self, other):\n",
    "        \"\"\"\n",
    "        \"\"\"\n",
    "        return self.dot(other.hat())\n",
    "\n",
    "    def dist(self, other):\n",
    "        \"\"\"Euclidean distance from self to other.\n",
    "        \"\"\"\n",
    "        return np.linalg.norm(self - other)\n",
    "\n",
    "    def diff_angle(self, other):\n",
    "        \"\"\"\n",
    "        \"\"\"\n",
    "        #TODO: see http://www.euclideanspace.com/maths/algebra/vectors/angleBetween/\n",
    "        raise NotImplementedError()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 192,
   "metadata": {},
   "outputs": [
    {
     "ename": "IndentationError",
     "evalue": "unindent does not match any outer indentation level (<ipython-input-192-40c5da61710c>, line 13)",
     "output_type": "error",
     "traceback": [
      "\u001b[0;36m  File \u001b[0;32m\"<ipython-input-192-40c5da61710c>\"\u001b[0;36m, line \u001b[0;32m13\u001b[0m\n\u001b[0;31m    def __div__(self, other):\u001b[0m\n\u001b[0m                             ^\u001b[0m\n\u001b[0;31mIndentationError\u001b[0m\u001b[0;31m:\u001b[0m unindent does not match any outer indentation level\n"
     ]
    }
   ],
   "source": [
    "def __mul__(self, other):\n",
    "        if isinstance(other, UNITS.Unit):\n",
    "            return VectorQuantity(self, self.units * other)\n",
    "        \n",
    "        if isinstance(other, UNITS.Quantity):\n",
    "            prod = super().__mul__(other.magnitude)\n",
    "            prod._units = self.units * other.units\n",
    "            return prod\n",
    "        else:\n",
    "            prod = super().__mul__(other)\n",
    "            return prod\n",
    "        \n",
    "    def __div__(self, other):\n",
    "        if isinstance(other, UNITS.Unit):\n",
    "            return VectorQuantity(self, self.units / other)\n",
    "        \n",
    "        if isinstance(other, UNITS.Quantity):\n",
    "            quot = super().__div__(other.magnitude)\n",
    "            quot._units = self.units / other.units\n",
    "            return quot\n",
    "        else:\n",
    "            quot = super().__div__(other)\n",
    "            return quot\n",
    "        \n",
    "    __truediv__ = __div__"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 193,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "class VectorQuantity(Vector, UNITS.Quantity):\n",
    "    \n",
    "    def __new__(cls, vector, units):\n",
    "        \"\"\"\n",
    "        cls:  class object\n",
    "        args: tuple of coordinates\n",
    "        \"\"\"\n",
    "        obj = np.asarray(vector).view(cls)\n",
    "        obj._units = units    \n",
    "        return obj\n",
    "    \n",
    "    def __repr__(self):\n",
    "        return super().__repr__() + ' ' + str(self.units)\n",
    "    \n",
    "    def __mul__(self, other):\n",
    "        if isinstance(other, UNITS.Unit):\n",
    "            return VectorQuantity(self, self.units * other)\n",
    "        \n",
    "        if isinstance(other, UNITS.Quantity):\n",
    "            prod = super().__mul__(other.magnitude)\n",
    "            prod._units = self.units * other.units\n",
    "            return prod\n",
    "        else:\n",
    "            prod = super().__mul__(other)\n",
    "            return prod\n",
    "        \n",
    "    def __div__(self, other):\n",
    "        if isinstance(other, UNITS.Unit):\n",
    "            return VectorQuantity(self, self.units / other)\n",
    "        \n",
    "        if isinstance(other, UNITS.Quantity):\n",
    "            quot = super().__div__(other.magnitude)\n",
    "            quot._units = self.units / other.units\n",
    "            return quot\n",
    "        else:\n",
    "            quot = super().__div__(other)\n",
    "            return quot\n",
    "        \n",
    "    __truediv__ = __div__\n",
    "            \n",
    "    #def __getitem__(self, key):\n",
    "    #    return super().__getitem__(key) * self.units\n",
    "\n",
    "    @property\n",
    "    def x(self):\n",
    "        return self[0] * self.units\n",
    "\n",
    "    @property\n",
    "    def y(self):\n",
    "        return self[1] * self.units\n",
    "\n",
    "    @property\n",
    "    def z(self):\n",
    "        return self[2] * self.units\n",
    "    \n",
    "    @property\n",
    "    def mag(self):\n",
    "        return super().mag * self.units\n",
    "    \n",
    "    @property\n",
    "    def mag2(self):\n",
    "        return super().mag2 * self.units\n",
    "\n",
    "    def hat(self):\n",
    "        return super().hat() * self.units\n",
    "\n",
    "    def dot(self, other):\n",
    "        prod = super().dot(other) * self.units\n",
    "        try:\n",
    "            return prod * other.units\n",
    "        except AttributeError:\n",
    "            return prod\n",
    "        \n",
    "    def cross(self, other):\n",
    "        prod = super().cross(other) * self.units\n",
    "        try:\n",
    "            return prod * other.units\n",
    "        except AttributeError:\n",
    "            return prod\n",
    "        \n",
    "    def proj(self, other):\n",
    "        \"\"\"\n",
    "        \"\"\"\n",
    "        print('hello1')\n",
    "        return super().proj(other) / self.units\n",
    "    \n",
    "    def comp(self, other):\n",
    "        \"\"\"\n",
    "        \"\"\"\n",
    "        return super().comp(other) / self.units"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 194,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "m = UNITS.m\n",
    "N = UNITS.newton"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 195,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array(2)"
      ]
     },
     "execution_count": 195,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = Vector(3, 4)\n",
    "b = Vector(1, 2)\n",
    "a.cross(b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 196,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "__main__.VectorQuantity"
      ]
     },
     "execution_count": 196,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = Vector(3, 4) * m\n",
    "type(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 197,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "VectorQuantity(3, 4) meter"
      ]
     },
     "execution_count": 197,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 198,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "VectorQuantity(3, 4) meter"
      ]
     },
     "execution_count": 198,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a * m"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 199,
   "metadata": {},
   "outputs": [
    {
     "ename": "TypeError",
     "evalue": "units must be of type str, Unit or UnitsContainer; not <class 'pint.unit.build_quantity_class.<locals>.Quantity'>.",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mTypeError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[0;32m/home/downey/anaconda3/lib/python3.6/site-packages/IPython/core/formatters.py\u001b[0m in \u001b[0;36m__call__\u001b[0;34m(self, obj)\u001b[0m\n\u001b[1;32m    670\u001b[0m                 \u001b[0mtype_pprinters\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mtype_printers\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    671\u001b[0m                 deferred_pprinters=self.deferred_printers)\n\u001b[0;32m--> 672\u001b[0;31m             \u001b[0mprinter\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mpretty\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mobj\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    673\u001b[0m             \u001b[0mprinter\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mflush\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    674\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0mstream\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mgetvalue\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/downey/anaconda3/lib/python3.6/site-packages/IPython/lib/pretty.py\u001b[0m in \u001b[0;36mpretty\u001b[0;34m(self, obj)\u001b[0m\n\u001b[1;32m    381\u001b[0m                             \u001b[0;32mif\u001b[0m \u001b[0mcallable\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mmeth\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    382\u001b[0m                                 \u001b[0;32mreturn\u001b[0m \u001b[0mmeth\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mobj\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcycle\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 383\u001b[0;31m             \u001b[0;32mreturn\u001b[0m \u001b[0m_default_pprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mobj\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcycle\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    384\u001b[0m         \u001b[0;32mfinally\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    385\u001b[0m             \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mend_group\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/downey/anaconda3/lib/python3.6/site-packages/IPython/lib/pretty.py\u001b[0m in \u001b[0;36m_default_pprint\u001b[0;34m(obj, p, cycle)\u001b[0m\n\u001b[1;32m    501\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0m_safe_getattr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mklass\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'__repr__'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0;32min\u001b[0m \u001b[0m_baseclass_reprs\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    502\u001b[0m         \u001b[0;31m# A user-provided repr. Find newlines and replace them with p.break_()\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 503\u001b[0;31m         \u001b[0m_repr_pprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mobj\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mp\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcycle\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    504\u001b[0m         \u001b[0;32mreturn\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    505\u001b[0m     \u001b[0mp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mbegin_group\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'<'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/downey/anaconda3/lib/python3.6/site-packages/IPython/lib/pretty.py\u001b[0m in \u001b[0;36m_repr_pprint\u001b[0;34m(obj, p, cycle)\u001b[0m\n\u001b[1;32m    699\u001b[0m     \u001b[0;34m\"\"\"A pprint that just redirects to the normal repr function.\"\"\"\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    700\u001b[0m     \u001b[0;31m# Find newlines and replace them with p.break_()\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 701\u001b[0;31m     \u001b[0moutput\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mrepr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mobj\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    702\u001b[0m     \u001b[0;32mfor\u001b[0m \u001b[0midx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0moutput_line\u001b[0m \u001b[0;32min\u001b[0m \u001b[0menumerate\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0moutput\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msplitlines\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    703\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0midx\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m<ipython-input-193-daab7023f8bb>\u001b[0m in \u001b[0;36m__repr__\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m     11\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     12\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0m__repr__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 13\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0msuper\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m__repr__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0;34m' '\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0mstr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0munits\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     14\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     15\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/downey/anaconda3/lib/python3.6/site-packages/pint/quantity.py\u001b[0m in \u001b[0;36munits\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m    175\u001b[0m         \u001b[0;34m:\u001b[0m\u001b[0mrtype\u001b[0m\u001b[0;34m:\u001b[0m \u001b[0mUnitContainer\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    176\u001b[0m         \"\"\"\n\u001b[0;32m--> 177\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_REGISTRY\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mUnit\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_units\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    178\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    179\u001b[0m     \u001b[0;34m@\u001b[0m\u001b[0mproperty\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/downey/anaconda3/lib/python3.6/site-packages/pint/unit.py\u001b[0m in \u001b[0;36m__new__\u001b[0;34m(cls, units)\u001b[0m\n\u001b[1;32m     69\u001b[0m         \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     70\u001b[0m             raise TypeError('units must be of type str, Unit or '\n\u001b[0;32m---> 71\u001b[0;31m                             'UnitsContainer; not {0}.'.format(type(units)))\n\u001b[0m\u001b[1;32m     72\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     73\u001b[0m         \u001b[0minst\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m__used\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;32mFalse\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mTypeError\u001b[0m: units must be of type str, Unit or UnitsContainer; not <class 'pint.unit.build_quantity_class.<locals>.Quantity'>."
     ]
    }
   ],
   "source": [
    "a / m"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 173,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3"
      ]
     },
     "execution_count": 173,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 174,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "3 meter"
      ],
      "text/latex": [
       "$3 meter$"
      ],
      "text/plain": [
       "<Quantity(3, 'meter')>"
      ]
     },
     "execution_count": 174,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 175,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "4 meter"
      ],
      "text/latex": [
       "$4 meter$"
      ],
      "text/plain": [
       "<Quantity(4, 'meter')>"
      ]
     },
     "execution_count": 175,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 176,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "5.0 meter"
      ],
      "text/latex": [
       "$5.0 meter$"
      ],
      "text/plain": [
       "<Quantity(5.0, 'meter')>"
      ]
     },
     "execution_count": 176,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.mag"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 177,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "25 meter"
      ],
      "text/latex": [
       "$25 meter$"
      ],
      "text/plain": [
       "<Quantity(25, 'meter')>"
      ]
     },
     "execution_count": 177,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.mag2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 178,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "__main__.VectorQuantity"
      ]
     },
     "execution_count": 178,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 179,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "pint.unit.build_quantity_class.<locals>.Quantity"
      ]
     },
     "execution_count": 179,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(a.mag)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 180,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "VectorQuantity(15.0, 20.0) meter ** 2"
      ]
     },
     "execution_count": 180,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c = a * a.mag\n",
    "c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 181,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "VectorQuantity(0.6, 0.8) dimensionless"
      ]
     },
     "execution_count": 181,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c = a / a.mag\n",
    "c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 182,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "0.6 dimensionless"
      ],
      "text/latex": [
       "$0.6 dimensionless$"
      ],
      "text/plain": [
       "<Quantity(0.6, 'dimensionless')>"
      ]
     },
     "execution_count": 182,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c.x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 183,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "VectorQuantity(0.6, 0.8) meter"
      ]
     },
     "execution_count": 183,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ah = a.hat()\n",
    "ah"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 184,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.6  0.8]\n"
     ]
    }
   ],
   "source": [
    "print(ah)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 185,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "meter"
      ],
      "text/latex": [
       "$meter$"
      ],
      "text/plain": [
       "<Unit('meter')>"
      ]
     },
     "execution_count": 185,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ah.units"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 186,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "b = Vector(1, 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 187,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "11 meter"
      ],
      "text/latex": [
       "$11 meter$"
      ],
      "text/plain": [
       "<Quantity(11, 'meter')>"
      ]
     },
     "execution_count": 187,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.dot(b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 188,
   "metadata": {},
   "outputs": [],
   "source": [
    "b = Vector(1, 2) * N"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 189,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "11.0 meter newton"
      ],
      "text/latex": [
       "$11.0 meter \\cdot newton$"
      ],
      "text/plain": [
       "<Quantity(11.0, 'meter * newton')>"
      ]
     },
     "execution_count": 189,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.dot(b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 190,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "2.0 meter newton"
      ],
      "text/latex": [
       "$2.0 meter \\cdot newton$"
      ],
      "text/plain": [
       "<Quantity(2.0, 'meter * newton')>"
      ]
     },
     "execution_count": 190,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c = a.cross(b)\n",
    "c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 191,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "hello1\n",
      "hello1\n",
      "maximum recursion depth exceeded while calling a Python object\n"
     ]
    }
   ],
   "source": [
    "c = a.proj(b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 133,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "[ 2.2  4.4] 1/meter"
      ],
      "text/latex": [
       "$[ 2.2  4.4] \\frac{1}{meter}$"
      ],
      "text/plain": [
       "<Quantity([ 2.2  4.4], '1 / meter')>"
      ]
     },
     "execution_count": 133,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 134,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "4.919349550499537 newton"
      ],
      "text/latex": [
       "$4.919349550499537 newton$"
      ],
      "text/plain": [
       "<Quantity(4.919349550499537, 'newton')>"
      ]
     },
     "execution_count": 134,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.comp(b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 135,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "x_hat = Vector(1, 0)\n",
    "y_hat = Vector(0, 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 136,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "3.0 dimensionless"
      ],
      "text/latex": [
       "$3.0 dimensionless$"
      ],
      "text/plain": [
       "<Quantity(3.0, 'dimensionless')>"
      ]
     },
     "execution_count": 136,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.comp(x_hat)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 137,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "4.0 dimensionless"
      ],
      "text/latex": [
       "$4.0 dimensionless$"
      ],
      "text/plain": [
       "<Quantity(4.0, 'dimensionless')>"
      ]
     },
     "execution_count": 137,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.comp(y_hat)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 138,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "0.9272952180016122 radian"
      ],
      "text/latex": [
       "$0.9272952180016122 radian$"
      ],
      "text/plain": [
       "<Quantity(0.9272952180016122, 'radian')>"
      ]
     },
     "execution_count": 138,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.angle"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 139,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "1.1071487177940904 radian"
      ],
      "text/latex": [
       "$1.1071487177940904 radian$"
      ],
      "text/plain": [
       "<Quantity(1.1071487177940904, 'radian')>"
      ]
     },
     "execution_count": 139,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b.angle"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 140,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.25"
      ]
     },
     "execution_count": 140,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Vector(1, 1).angle / pi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 141,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.75"
      ]
     },
     "execution_count": 141,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Vector(-1, 1).angle / pi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 142,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.75"
      ]
     },
     "execution_count": 142,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Vector(-1, -1).angle / pi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 143,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.25"
      ]
     },
     "execution_count": 143,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Vector(1, -1).angle / pi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 144,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1.4142135623730951, -0.78539816339744828)"
      ]
     },
     "execution_count": 144,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Vector(1, -1).polar()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 145,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[5, 3.5355, 0, -10]"
      ]
     },
     "execution_count": 145,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = [5, 3.5355, 0, -10]\n",
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 146,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0, 3.5355, 10, 0]"
      ]
     },
     "execution_count": 146,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y = [0, 3.5355, 10, 0]\n",
    "y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 147,
   "metadata": {},
   "outputs": [],
   "source": [
    "theta, rho = cart2pol(x,y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 148,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.        ,  0.78539816,  1.57079633,  3.14159265])"
      ]
     },
     "execution_count": 148,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "theta"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 149,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([  5.        ,   4.99995205,  10.        ,  10.        ])"
      ]
     },
     "execution_count": 149,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rho"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 150,
   "metadata": {},
   "outputs": [],
   "source": [
    "theta = [0, pi/4, pi/2, pi]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 151,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "rho = [5, 5, 10, 10]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 152,
   "metadata": {},
   "outputs": [],
   "source": [
    "x2, y2 = pol2cart(theta,rho)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 153,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[  5.00000000e+00   3.53553391e+00   6.12323400e-16  -1.00000000e+01]\n"
     ]
    }
   ],
   "source": [
    "print(x2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 154,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[  0.00000000e+00   3.53553391e+00   1.00000000e+01   1.22464680e-15]\n"
     ]
    }
   ],
   "source": [
    "print(y2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 155,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "degree = UNITS.degree\n",
    "kg = UNITS.kg\n",
    "m = UNITS.m\n",
    "s = UNITS.s"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 156,
   "metadata": {},
   "outputs": [],
   "source": [
    "condition = Condition(x = 0 * m, \n",
    "                      y = 0 * m,\n",
    "                      g = 9.8 * m/s**2,\n",
    "                      mass = 145e-3 * kg,\n",
    "                      diameter = 73e-3 * m,\n",
    "                      rho = 1.2 * kg/m**3,\n",
    "                      C_d = 0.3,\n",
    "                      angle = 45 * degree,\n",
    "                      velocity = 40 * m / s,\n",
    "                      duration = 5 * s)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 229,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def make_system(condition):\n",
    "    unpack(condition)\n",
    "    \n",
    "    theta = np.deg2rad(angle)\n",
    "    vx, vy = pol2cart(theta, velocity)\n",
    "    init = State(x=x, y=y, vx=vx, vy=vy)\n",
    "    v = State(vx=vx, vy=vy)\n",
    "    \n",
    "    area = np.pi * (diameter/2)**2\n",
    "    ts = linspace(0, duration, 101)\n",
    "    \n",
    "    return System(init=init, g=g, mass=mass, v=v,\n",
    "                  area=area, rho=rho, C_d=C_d, ts=ts)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 230,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style>\n",
       "    .dataframe thead tr:only-child th {\n",
       "        text-align: right;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: left;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>value</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>x</th>\n",
       "      <td>0 meter</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>y</th>\n",
       "      <td>0 meter</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>vx</th>\n",
       "      <td>28.284271247461902 meter / second</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>vy</th>\n",
       "      <td>28.2842712474619 meter / second</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "x                               0 meter\n",
       "y                               0 meter\n",
       "vx    28.284271247461902 meter / second\n",
       "vy      28.2842712474619 meter / second\n",
       "dtype: object"
      ]
     },
     "execution_count": 230,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "system = make_system(condition)\n",
    "system.init"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 235,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([<Quantity(28.284271247461902, 'meter / second')>,\n",
       "       <Quantity(28.2842712474619, 'meter / second')>], dtype=object)"
      ]
     },
     "execution_count": 235,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v = system.v.values\n",
    "v"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 233,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "1600.0 meter<sup>2</sup>/second<sup>2</sup>"
      ],
      "text/latex": [
       "$1600.0 \\frac{meter^{2}}{second^{2}}$"
      ],
      "text/plain": [
       "<Quantity(1600.0, 'meter ** 2 / second ** 2')>"
      ]
     },
     "execution_count": 233,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.dot(v, v)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 222,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def slope_func(state, t, system):\n",
    "    x, y, vx, vy = state\n",
    "    unpack(system)\n",
    "    \n",
    "    a_grav = np.array([0, -9.8]) * m / s**2\n",
    "\n",
    "    v = np.array([vx.magnitude, vy.magnitude]) * m / s\n",
    "    \n",
    "    f_drag = -rho * np.linalg.norm(v) * v.units * v * C_d * area / 2\n",
    "    a_drag = f_drag / mass\n",
    "\n",
    "    a = a_grav + a_drag\n",
    "    \n",
    "    return v, a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 223,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(<Quantity([ 28.28427125  28.28427125], 'meter / second')>,\n",
       " <Quantity([ -5.87820989 -15.67820989], 'meter / second ** 2')>)"
      ]
     },
     "execution_count": 223,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "slope_func(system.init, 0, system)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 161,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "a = 5 * m"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 162,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "b = Vector(1, 2) * s"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 163,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "VectorQuantity(5, 10) meter * second"
      ]
     },
     "execution_count": 163,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b * a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 164,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "maximum recursion depth exceeded in comparison\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([ 5, 10])"
      ]
     },
     "execution_count": 164,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a * b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
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