{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy.integrate import odeint"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The sky diving problem\n",
    "\n",
    "$$\\frac{dv_x}{dt} = -\\frac{b}{m}\\sqrt{v_x^2 + v_y^2} \\cdot v_x $$\n",
    "$$\\frac{dv_y}{dt} = - g -\\frac{b}{m}\\sqrt{v_x^2 + v_y^2} \\cdot v_y $$\n",
    "\n",
    "$\\vec{S} = (v_x, v_y)$. To solve ODEs in python, \n",
    "\n",
    "1. define function that takes in $t$ and $\\vec{S}$ and returns $d\\vec{S}/dt$ "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def dSdt(S, t, g, m, b):\n",
    "    vx = S[0]\n",
    "    vy = S[1]\n",
    "    return [\n",
    "        -b/m * np.sqrt(vx**2+vy**2) * vx, #dvx/dt\n",
    "        -g - b/m * np.sqrt(vx**2+vy**2) * vy #dvy/dt\n",
    "    ]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2. Supply the function and initial conditions to the ODE solver. Provide the times at which you want the solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "t= np.linspace(0, 20 ,100)\n",
    "m = 80\n",
    "g = 9.81\n",
    "vt = -55\n",
    "b = m*g/vt**2\n",
    "v0x, v0y = 50, 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "sol = odeint(dSdt, y0=[v0x, v0y], t=t, args=(g, m, b))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "vx = sol.T[0]\n",
    "vy = sol.T[1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x2ba45477130>]"
      ]
     },
     "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": [
    "plt.plot(t, vx)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.lines.Line2D at 0x2ba4589c7c0>"
      ]
     },
     "execution_count": 14,
     "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": [
    "plt.plot(t, vy)\n",
    "plt.axhline(vt, color='r')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "How long until one reaches terminal velocity?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "ind = np.abs(vy - vt)/np.abs(vt) < 0.01"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([17.17171717, 17.37373737, 17.57575758, 17.77777778, 17.97979798,\n",
       "       18.18181818, 18.38383838, 18.58585859, 18.78787879, 18.98989899,\n",
       "       19.19191919, 19.39393939, 19.5959596 , 19.7979798 , 20.        ])"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t[ind]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}