{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(motion_in_1d)=\n",
    "# Motion in 1D"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Position, velocity, and acceleration\n",
    "\n",
    "The position of a body as a function of time, denoted by \\\\(x(t)\\\\), is relative to some arbitrary chosen origin where \\\\(x=0\\\\).\n",
    "\n",
    "**Velocity $v(t)$** is a measure of **how rapidly and the direction which the position of the body is changing**:\n",
    "\n",
    "\\\\[v=\\frac{\\Delta x}{\\Delta t}=\\frac{x(t+\\Delta t)-x(t)}{\\Delta t}\\\\]\n",
    "\n",
    "As \\\\(\\Delta t\\to 0\\\\):\n",
    "\n",
    "\\\\[v=\\frac{dx}{dt}\\\\]\n",
    "\n",
    "**Acceleration \\\\(a(t)\\\\)** is a measure of **how fast and in which direction the velocity is changing**:\n",
    "\n",
    "\\\\[a=\\frac{\\Delta v}{\\Delta t}=\\frac{v(t+\\Delta t)-v(t)}{\\Delta t}\\\\]\n",
    "\n",
    "As \\\\(\\Delta t\\to 0\\\\):\n",
    "\n",
    "\\\\[a=\\frac{dv}{dt}=\\frac{d^2x}{dt^2}\\\\]\n",
    "\n",
    "## Force\n",
    "\n",
    "A force is **any interaction that changes the motion of an object**. This is encapsulated by Newton's first and second law of motion.\n",
    "\n",
    "**Newton's first law of motion** states that a body continues to move at the same velocity if not acted upon by external forces. This concept is referred to as intertia, the tendency of a body to maintain its motion when no net force acts on that body.\n",
    "\n",
    "**Newton's second law of motion** states that the acceleration of a body is proportional to the net force acting on that body, and inversely proportional to the mass of that body. In other words:\n",
    "\n",
    "\\\\[F=ma\\\\]\n",
    "\n",
    "From this, the unit of force in SI units is \\\\(kgms^{-2}\\\\), which is defined as Newton (\\\\(N\\\\)).\n",
    "\n",
    "Forces are broadly classified into **contact forces**, which involves physical contact between bodies, and **non-contact forces**, which can act at a distance. Therefore, pulling of a rope is a contact force, whereas magnetism is a non-contact force.\n",
    "\n",
    "### Gravity\n",
    "\n",
    "Gravitational force is a non-contact, attractive force that acts between any pair of bodies with mass. It is proportional to the mass of each body, inversely proportional to the square of the distance between the centre of those bodies, and acts in a direction that aligns with a straight line connecting the centre of the bodies. Mathematically:\n",
    "\n",
    "\\\\[F=\\frac{Gm_am_b}{R_{ab}^2}\\\\]\n",
    "\n",
    "where \\\\(m_a\\\\) is the mass of body A, \\\\(m_b\\\\) is the mass of body B, \\\\(R_{ab}\\\\) is the distance between the two bodies, and \\\\(G\\\\) is the gravitational constant equal to \\\\(6.674*10^{-11}m^3kg^{-1}s^{-2}\\\\).\n",
    "\n",
    "The force acting upon a body of unit mass m by the Earth with mass \\\\(5.972\\times10^{24}kg\\\\) and radius \\\\(6.371\\times10^3m\\\\) is:\n",
    "\n",
    "\\\\[F=\\frac{m(5.972\\times10^{24})(6.674\\times10^{-11})}{(6.371\\times10^3)^2}=9.8m=mg\\\\]\n",
    "\n",
    "where \\\\(g=9.8ms^{-2}\\\\), or acceleration due to gravity.\n",
    "\n",
    "### Falling objects\n",
    "\n",
    "Considering objects free-falling solely under gravity:\n",
    "\n",
    "\\\\[\\frac{d^2x}{dt^2}=-g,\\\\\\\\\\\\\n",
    "\\frac{dx}{dt}=v=v_0-gt,\\\\\\\\\\\\\n",
    "x=x_0+v_0t-\\frac{1}{2}gt^2\\\\]\n",
    "\n",
    "where \\\\(x_0\\\\) and \\\\(v_0\\\\) are initial position and initial velocity respectively."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First we will include libraries needed in the solutions:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "scrolled": false,
    "tags": [
     "remove-output"
    ]
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "from sympy import Function, Symbol, diff, dsolve, pprint, integrate\n",
    "from matplotlib import animation, rc\n",
    "from IPython.display import HTML"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Tutorial Problem 1.2\n",
    "\n",
    "If a rock were ejected from a volcano at an initial speed of \\\\(200m/s\\\\), what would be the maximum height that it reaches above the top of the volcano? And how long will it take to reach that height?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Max height reached = 2039 m\n",
      "Time needed for the body to reach that height = 20.40s\n"
     ]
    },
    {
     "data": {
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nn25HMyHIioXJMs6ePcvzzz9Pjx49OHXqFMuWLePtt98mIiLC7WghS0Ro2rQpmzdvpmPHjgwbNoyKFSvyzTffuB3NhBgrFiZL2Lx5M5UrV+aVV16hTp06bNq0iTp16rgdK9MoUKAAU6ZM4eOPPyYhIYGqVasyZMgQLlxI7X65JruxYmEyNVVl2rRpVK5cmQMHDrBgwQKee+456xMig+677z42bNhAixYt6NevH7Vr12bPnj1uxzIhwIqFybROnjxJu3bt6NSpE1WqVGHDhg00btzY7ViZXoECBZgzZw7Tp0/n22+/pUKFCixZssTtWMZlVixMprRlyxYqVarEnDlzePnll/n444+59tpr3Y6VZYgIHTp04Pvvv+e6666jUaNGvPjii3bLkGzMioXJdBYuXMidd97J8ePHWb58Of3797crnQKkdOnSfPXVV7Rt25bo6GiaNGnCsWPWlUx2ZMXCZBqqyiuvvELTpk0pU6YMa9eu5d5773U7VpZ3xRVX8OabbzJhwgQ+/vhjKleuzE8//eR2LBNkVixMpnDq1Ckeeughnn/+eVq3bs3KlSspXrx4+jMavxARnnjiCVasWMGJEyeoUqUKy5cvdzuWCSIrFibkHTx4kBo1ajBv3jyGDx/OzJkzyZMnj9uxsqW7776bb775hiJFilC3bt2/+gg3WZ8VCxPStm/fzt13382mTZv44IMP+N///ofn7vfGLSVLluTLL7+kRo0aPProo/Tr189OfGcDVixMyFqzZg133303x48fJy4uzi6LDSEFChRg8eLFREVFMWTIEB555BG7GWEWF7BblBtzKRYvXkzLli2JiIhg6dKllClTxu1IJpnLLruMiRMncv311/P888/zxx9/8O6779ohwiwqkD3lTRORgyKyyavtHa+OkH4TkfVOewkRSfAaN9FrnkgR+UFEtovIOLFjEFlebGwsTZs2pWzZsnz99ddWKEKYiNCvXz9iYmJYvHgx9evX58SJE27HMgEQyMNQM4B63g2q+pCqVlTVinj65p7vNXpH0jhV7eLVHgNEAaWd4R/LNFnLjBkzaNOmDffccw9xcXF2E8BMokuXLsyePfuvcxmHDh1yO5Lxs4AVC1X9Akix+1Nn76AlMCetZYhIYeBKVf3a6UnvLaCZv7Oa0DBx4kQ6duxIrVq1+Oijj8iXL5/bkcxFaNWqFR9++CE//vgj1apVY//+/W5HMn4kgey8XURKAItUtXyy9mrAKFWt5DXdZuBn4ATQX1VXikglYKiq1nam+z/gOVVtlMr6ovDshRAREREZGxubodzx8fGEh4dnaN5ACcVM4L9cc+fOZcKECVSpUoXo6Ghy5crleiZ/CsVMEJhcGzZsoE+fPlxzzTWMHj2aQoUKuZ7JH0Ixl78z1ahRY13S5/K/qGrABqAEsCmF9higp9fr3MBVzvNIYDdwJVAZ+NRruv8DFvqy7sjISM2ouLi4DM8bKKGYSdU/uYYPH66AtmjRQs+cORMSmfwtFDOpBi7X559/rnnz5tWyZcvqgQMHQiLTpQrFXP7OBKzVVD5Tg37prIjkBO4H3klqU9UzqnrEeb4O2AGUAfYAxbxmLwbsC15aE2gTJkygV69ePPTQQ8TGxl7SHoUJHdWqVWPx4sXs3LmTmjVr8vvvv7sdyVwiN35nURv4SVX/ukm+iPxHRMKc5zfgOZH9i6ruB06KyF3OeY5HgA9dyGwCYNq0aXTv3p2mTZvy9ttvkzOnXcmdldx7770sXryY3377jZo1a3Lw4EG3I5lLEMhLZ+cAXwP/FZE9ItLJGdWKf5/YrgZsFJENwFygi6omnRzvCkwBtuPZ4/goUJlN8MyZM4fOnTtTt25d3nnnHS677DK3I5kAqF69OosXL+bXX3+lXr16dlltJhawr3Kq2jqV9g4ptM3DcyltStOvBcqnNM5kTu+//z7t2rWjWrVqzJ8/n9y5c7sdyQRQ9erVmTdvHk2aNKFJkyZ89NFH9sO9TMhu92GCKi4ujlatWlG5cmUWLlzIFVdc4XYkEwT169fnrbfe4osvvqBVq1acP3/e7UjmIlmxMEGzceNGmjVrRqlSpViyZIn9jiKbad26NePGjWPBggV07tzZbj6YydgZRRMUu3bton79+uTLl4+lS5dSsGBBtyMZF3Tv3p0jR44QHR3NVVddxciRI92OZHxkxcIE3NGjR6lXrx6nTp1i1apV1mlRNvfCCy9w+PBhRo0aRYkSJejRo4fbkYwPrFiYgEpISKBx48b88ssvLFu2jPLl7VqF7E5EGDNmDLt37+bpp5+mRIkSdvv5TMDOWZiASUxMpH379nz99dfMmjXL+ss2fwkLC2PWrFncfvvttGrVinXr1rkdyaTDioUJmOjoaN577z1effVVWrRo4XYcE2Ly5s3LwoULufrqq2ncuDG7d+92O5JJgxULExCzZ8/m5Zdf5tFHH6Vnz55uxzEh6tprr2XJkiWcOnWKhg0b2o/2QpgVC+N3q1ev5tFHH6VatWrExMRYn9kmTTfffDPz589ny5YttGnTxi6pDVFWLIxf7dq1i2bNmlG0aFHmzZtnNwY0PqlVqxZjx45l0aJFTJ8+3e04JgVWLIzfnD59miZNmpCQkMCiRYu4+uqr3Y5kMpGuXbvSuXNnZs6cyXvvved2HJOMFQvjF6rKY489xsaNG4mNjaVs2bJuRzKZjIgwfvx4br75Zjp06MCGDRvcjmS8WLEwfvHaa68xe/ZsXnrpJerXr+92HJNJ5c6dmxdffJECBQrQrFkzDh8+7HYk47BiYS7ZypUr6dmzJ02aNKFfv35uxzGZ3FVXXcX777/P/v37eeihh+ymgyHCioW5JIcPH+bBBx+kZMmSvPXWW+TIYf+lzKW74447iImJYfny5URHR7sdxxDYzo+michBEdnk1RYtIntFZL0zNPAa11dEtovIVhGp69Vez2nbLiJ9ApXXXLyzZ88SHR1NfHw88+fPJ3/+/G5HMllIx44defTRRxk8eDAffWR9nrktkF8DZwD1UmgfraoVnWEJgIiUw9OD3s3OPK+LSJjT1eoEoD5QDmjtTGtCwHPPPcfmzZuZNm2a3fPJBMT48eOpUKECbdu2ZdeuXW7HydYCVixU9QvgaLoTejQFYlX1jKr+iqcL1TucYbuq/qKqZ4FYZ1rjsg8//JAxY8bQvHlzWrZs6XYck0XlyZOHuXPncu7cOVq2bMnZs2fdjpRtiaoGbuEiJYBFqlreeR0NdABOAGuBnqp6TETGA6tVdaYz3VT+7mu7nqp2dtrbAXeqavdU1hcFRAFERERExsbGZih3fHw84eHhGZo3UEIp04EDB3jssccoXLgww4YNC7m+KUJpWyUJxUwQmrlSyvT5558THR1NixYt6N49xT9/V3K5zd+ZatSosU5VK6U4UlUDNgAlgE1eryOAMDx7NIOBaU77BKCt13RTgRbAg8AUr/Z2wGu+rDsyMlIzKi4uLsPzBkqoZDp79qxWqVJF8+XLp9u2bQuZXN4sk+9CMVdqmZ566ikFdN68ecEN5MhM2yqjgLWaymdqUC9dUdXfVfWCqiYCb+A5zASwB/DuEacYsC+NduOSAQMG8PXXXzN58mRKlSrldhyTjbz66qtUrlyZzp072x1qXRDUYiEihb1eNgeSrpRaALQSkdwiUhIoDawBvgVKi0hJEcmF5yT4gmBmNn9bunQpw4YNIyoqilatWrkdx2QzuXLlYvbs2Zw7d4527dpx4cIFtyNlK4G8dHYO8DXwXxHZIyKdgFdF5AcR2QjUAJ4BUNXNwLvAj8BSoJuzB3Ie6A4sA7YA7zrTmiD7/fffeeSRRyhfvjxjxoxxO47JpkqVKsWECRP4/PPPGTp0qNtxspWAdauqqq1TaJ6axvSD8ZzHSN6+BFjix2jmIqkqjz76KCdPniQuLo48efK4HclkY+3atWPp0qUMHDiQWrVqcdddd7kdKVuwn9uadE2aNIklS5YwbNgwbr75ZrfjmGxORIiJiaF48eI8/PDD/PHHH25HyhasWJg0bd26lWeffZY6deq4dsmiMcnlz5+f2bNns2vXLrp16+Z2nGzBioVJ1blz52jTpg158uRh+vTpdt8nE1KqVKnCCy+8wKxZs5g7d67bcbI8++s3qXrxxRdZt24db7zxBkWKFHE7jjH/0q9fPypXrkyXLl04cOCA23GyNCsWJkVfffUVQ4YMoWPHjtx///1uxzEmRTlz5uStt97i1KlTREVFJf141wSAFQvzLwkJCXTo0IHixYszduxYt+MYk6abbrqJIUOGsHDhQmbMmOF2nCzLioX5lwEDBrBt2zamTZtGvnz53I5jTLqefPJJ7r33Xp566il27tzpdpwsyYqF+YevvvqKUaNG0aVLF2rWrOl2HGN8kiNHDqZPn/7Xb4ISExPdjpTlWLEwf0lISODRRx+lePHivPrqq27HMeailCxZktGjR7N8+XImTpzodpwsx4qF+cvAgQPZunUrU6dOtcNPJlPq1KkTderU4bnnnrPOkvzMioUBYPXq1YwcOZKoqChq167tdhxjMkREmDRpEqpK165d7eooP7JiYfjzzz/p2LEjRYsWZfjw4W7HMeaSlChRgldeeYUlS5YwZ84ct+NkGVYsDC+++CI//fQTU6ZM4corr3Q7jjGXrFu3btx11108+eSTHDp0yO04WYIVi2xu48aNDB8+nI4dO1KnTh234xjjF2FhYUyZMoUTJ07w9NNPux0nS7BikY1duHCBqKgoChUqxIgRI9yOY4xf3XzzzfTv35/Zs2ezaNEit+NkelYssrFJkybxzTffMGrUKAoVKuR2HGP8rk+fPpQvX56uXbty4sQJt+NkaoHsKW+aiBwUkU1ebcNF5CcR2Sgi74tIAae9hIgkiMh6Z5joNU+k07vedhEZJyISqMzZyb59++jbty+1a9emTZs2bscxJiBy5crF1KlT2bdvH/369XM7TqaWZrEQkctF5AERGSsi74nIWyLSW0R86QFnBlAvWdsnQHlVrQD8DPT1GrdDVSs6Qxev9hggCk+/3KVTWKbJgKeeeoqzZ88SExOD1V+Tld1xxx10796d119/nbVr17odJ9NKtViISDTwJVAF+AaYhKef7PPAUBH5REQqpDa/qn4BHE3W9rHTrzbAaqBYWuFEpDBwpap+rZ4Lpt8CmqX3pkzaFi1axNy5cxkwYAClSpVyO44xAffSSy8RERFB165duXDhgttxMiVJ7UcrItJQVRenOqPINcB1qppqqRaREsAiVS2fwriFwDuqOtOZbjOevY0TQH9VXSkilYChqlrbmef/gOdUtVEq64vCsxdCREREZGxsbGrR0hQfH094eHiG5g0Uf2VKSEigY8eO5MmTh8mTJ3PZZZeFRC5/sky+C8Vcgcq0fPlyXn75ZZ566imaNbv475zZYVvVqFFjnapWSnGkqgZsAEoAm1Jofx54n7+LVW7gKud5JLAbuBKoDHzqNd//AQt9WXdkZKRmVFxcXIbnDRR/ZXr22WcV0FWrVvlleVl5W/lTKGZSDc1cgcqUmJiotWrV0vz58+v+/fsvev7ssK2AtZrKZ2q6J7hFpJJzMvo758T0DyKyMaOVS0TaA42ANk44VPWMqh5xnq8DdgBlgD3881BVMWBfRted3f3www+MHTuWqKgo7rnnHrfjGBNUIsLrr79OQkICvXr1cjtOpuPL1VCzgOlAC6Axng/6xhlZmYjUA54Dmqjqaa/2/4hImPP8Bjwnsn9R1f3ASRG5y7kK6hHgw4ysO7tTVXr06EH+/Pl55ZVX3I5jjCvKlClD7969mTlzJnFxcW7HyVR8KRaHVHWBqv6qqjuThvRmEpE5wNfAf0Vkj4h0AsYD+YBPkl0iWw3YKCIbgLlAF1VNOjneFZgCbMezx/HRRb1DA8A777zD559/ziuvvMJVV13ldhxjXNOvXz9KlizJE088wdmzZ92Ok2nk9GGagSIyBfgMOJPUqKrz05pJVVun0Dw1lWnnAfNSGbcW+NcJcuO7+Ph4evbsye23307nzp3djmOMq/LkycP48eNp2LAhI0eOpG/fvunPZHwqFh2Bm4DLgKTupxRIs1iY0DFo0CD27dvH3LlzCQsLczuOMa5r0KABzTWXFy4AACAASURBVJs3Z9CgQbRr145ixdK8it/g22GoW1W1kqq2V9WOzvBowJMZv9i6dSujRo2iY8eOVKlSxe04xoSMkSNHcuHCBZ577jm3o2QKvhSL1SJSLuBJjN+pKk8++SRXXHEFQ4YMcTuOMSGlZMmS9OrVi9mzZ/Pll1+6HSfk+VIsqgLrRWSrPy6dNcHz4Ycf8vHHH//161VjzD/16dOHokWL8tRTT5GYmJj+DNmYL8WiHp5LWetwiZfOmuBJSEjgmWeeoXz58jzxxBNuxzEmJOXNm5dhw4axbt06ZsyY4XackJbWvaHCAbwvl01+6WzSNCb0jB49mt9++43XXnuNnDl9uY7BmOzp4Ycf5u6776Zv3752G/M0pLVn8aGIjBSRaiKSN6lRRG4QkU4isgy7A2xIOnDgAEOGDKF58+ZUr17d7TjGhDQRYezYsRw6dIhBgwa5HSdkpVosVLUWnt9WPA5sFpE/ROQIMBO4FmivqnODE9NcjAEDBnDmzBmGDRvmdhRjMoVKlSrRsWNHxowZw88//+x2nJCU5jkLVV2iqm1UtYSq5lfVq1T1blUdrKoHghXS+G7Dhg1MnTqV7t27U7p0abfjGJNpDB48mMsvv5xnn33W7SghybpVzUJUlZ49e1KwYEEGDBjgdhxjMpVrr72WAQMGsHjxYj799FO344QcKxZZyOLFi/nss8+Ijo6mYMGCbscxJtPp0aMH119/Pb169bJLaZOxYpFFnDt3jv/973/897//pUuXLunPYIz5l8svv5whQ4awfv16Zs6c6XackOJLfxZv+9Jm3DVx4kS2bt3KiBEjLrn3O2Oys4ceeohKlSrRv39/EhIS3I4TMnzZs7jZ+4XT70RkYOKYjDh27BjR0dHUqlWLhg0buh3HmEwtR44cjBgxgt27dzN27Fi344SMtH6U11dETgIVROSEM5wEDmIdEIWUIUOGcOzYMUaOHImnjyhjzKW49957ady4MUOGDOHQoUNuxwkJaf3OYoiq5gOGq+qVzpDPuXzWpxvAi8g0ETkoIpu82gqJyCciss15LOi0i4iME5Htzj2obveap70z/TanW1bj2L17N+PGjaNdu3bceuutbscxJssYNmwYp06d4uWXX3Y7SkhI9zCUqvYVkaIicrfza+5qIlLNx+XP4N+/8u4DfKaqpfH86K+P014fzz2oSgNRQAx4igswELgTuANPZ0x2qY8jOjoaVeWll15yO4oxWUrZsmXp3LkzMTExbNu2ze04rvPlBPdQ4EugP9DLGf7ny8JV9QvgaLLmpsCbzvM3gWZe7W+px2qggIgUBuoCn6jqUVU9BnyC3WYEgB9//JEZM2bQrVs3rr/+erfjGJPlREdHkzt3bvr06ZP+xFmcL3eYaw78V1XPpDulbyJUdT+Aqu4XkWuc9qLAbq/p9jhtqbX/i4hE4dkrISIighUrVmQoYHx8fIbnDZSUMvXv3588efJw7733upY3s2wrt4ViJgjNXKGWqWXLlkyfPp277rrL7Sj/EtRtpappDsBHQHh606Uxfwlgk9fr48nGH3MeFwNVvdo/w3PVVS+gv1f7AKBneuuNjIzUjIqLi8vwvIGSPNOqVasU0EGDBrkTyJEZtlUoCMVMqqGZK9QyxcfHa0REhN56662amJjodpx/8Pe2AtZqKp+paV0N9ZqIjANO4+n8aJJzAnqc055RvzuHl3AeDzrte4DiXtMVA/al0Z5tqSp9+vTh2muv5emnn3Y7jjFZWt68eenfvz8bNmzgk08+cTuOa9I6Z7EWWAcsAF4GvnJeJw0ZtQBIuqKpPX9fhrsAeMS5Kuou4A/1HK5aBtQRkYLOie06Tlu2tWjRIlatWsXAgQPJmzdv+jMYYy5JVFQU1157Lf369Us6wpHtpHrOQlXfTG2cr0RkDlAduFpE9uC5qmko8K6IdAJ2AQ86ky8BGgDb8ezNdHRyHBWRl4FvneleUtXkJ82zjQsXLtC3b19Kly5Np06d3I5jTLaQK1cu2rdvz7Bhw5g/fz4tWrRwO1LQpXuCW0R+AJKX0j/w7HkMUtUjqc2rqq1TGVUrhWkV6JbKcqYB09LLmh3MnDmTzZs38+6779ptPYwJovvuu48FCxbQv39/mjZtmu16oPTldh8f4Tn53MYZFgIrgQN4fkdhguTs2bNER0cTGRnJAw884HYcY7KVsLAwBg0axE8//ZQtbzLoS2m8R1Xv8Xr9g4h8qar3iEjbQAUz/zZt2jR+++03Xn/9dbuthzEuaN68OZUqVWLgwIG0bt2a3Llzux0paHzZswgXkTuTXojIHUC48/J8QFKZfzl79iyDBg2iSpUq1Ktnv0k0xg0iwiuvvMKuXbuYPHmy23GCypdi0RmYIiK/ishvwBTgMRHJCwwJZDjzt4ULF7J3714GDRpkexXGuKh27dpUr16dQYMGER8f73acoPHl3lDfquotQEWgoqpWUNU1qnpKVd8NfERz+vRpZs2aRfXq1alZs6bbcYzJ1pL2Lg4ePMi4cZfyk7PMJdVzFiLSVlVnisizydoBUNVRAc5mHBMmTODYsWN290tjQkSVKlVo2LAhI0eOpHv37lx55ZVuRwq4tPYskn7tlS+VwQTByZMnGTZsGJUrV6Zq1apuxzHGOAYOHMjRo0d57bXX3I4SFGn9KG+S8/hi8OKY5MaNG8eRI0dsr8KYEFO5cmUaNWrEyJEj6dGjR5bfu/DlFuVlROSzpA6MRKSCiPQPfDRz/PhxRowYQePGjSlbtqzbcYwxyURHR3Ps2LFssXfhy9VQbwB9gXMAqroRaBXIUMZj9OjRHD9+3Do2MiZERUZG0rhxY0aOHMkff/zhdpyA8qVYXKGqa5K12e8rAuzYsWOMGTOGFi1aULFiRbfjGGNSkV32LnwpFodF5Eac+0OJyAPA/oCmMowbN44TJ07wwgsvuB3FGJOG22+/nSZNmmT5vQtfikU3YBJwk4jsBZ4GugY0VTZ34sQJxowZQ9OmTalQoYLbcYwx6YiOjub48eNZ+ncXvvwo7xdVrQ38B7hJVauq6m8BT5aNjR8/nuPHjzNgwAC3oxhjfHDbbbfRtGlTRo0axfHjx92OExC+XA2VW0QeBp4CnhGRF0TEjo0ESHx8PKNGjaJBgwZERka6HccY46OBAwdy/Phxxo4d63aUgPDlMNSHQFM8J7VPeQ0ZIiL/FZH1XsMJEXlaRKJFZK9XewOvefqKyHYR2SoidTO67swgJiaGI0eO2F6FMZnMbbfdRrNmzRg7diwnT550O47f+XKL8mKq6rfbnKrqVjz3mUJEwoC9wPt4esYbraojvKcXkXJ4LtW9GSgCfCoiZVT1gr8yhYrTp08zYsQI7rvvPu666y634xhjLtLzzz/PBx98QExMDL1793Y7jl/5smfxlYjcEqD11wJ2qOrONKZpCsSq6hlV/RVPt6t3BCiPqyZPnszBgwftCihjMqlKlSpRp04dRo0aRUJCgttx/EpS63zcqzvVnEBp4BfgDCB4ekG95Mt0RGQa8J2qjheRaKADcAJPl609VfWYiIwHVqvqTGeeqcBHqjo3heVFAVEAERERkbGxsRnKFR8fT3h4ePoT+tHZs2d5+OGHKV68OKNHjw6JTL4IxVyWyXehmCsUM4HvuTZs2MDTTz/Nk08+SfPmzUMik69q1KixTlUrpThSVVMcgOvTGlKbz9cByAUcBiKc1xFAGJ69ncHANKd9AtDWa76pQIv0lh8ZGakZFRcXl+F5M2r8+PEK6PLly1Mc70YmX4RiLsvku1DMFYqZVC8uV9WqVbV48eJ65syZwAVS/28rYK2m8pma6mEoVd2Z1nBp9QuA+nj2Kn531ve7ql5Q1UQ8txhJOtS0ByjuNV8xYJ8f1h8yzp49y7Bhw7jnnnuoXr2623GMMZfo+eefZ/fu3bz99ttuR/EbX85ZBEprYE7SCxEp7DWuObDJeb4AaOVcwlsSzyGx5LcfydRmzZrF7t276d+/v/WCZ0wWULduXSIjIxk6dCjnz2eNuyO5UixE5ArgPmC+V/OrIvKDiGwEagDPAKjqZuBd4EdgKdBNs9CVUImJibz66qtUrFiRunWz9FXBxmQbIkK/fv3Yvn077733nttx/MKXS2f9TlVPA1cla2uXxvSD8ZzHyHIWLFjATz/9xJw5c2yvwpgspFmzZpQrV45XXnmFhx56iBw53DyQc+kyd/pMTlUZMmQIN9xwAw888IDbcYwxfpQjRw769u3Lpk2bWLhwodtxLpkVCxd9/vnnrFmzhl69epEzpys7ecaYAGrVqhU33HADgwcPTrqaM9OyYuGioUOHEhERQYcOHdyOYowJgJw5c9K7d2++/fZbVqxY4XacS2LFwiXff/89y5Yt4+mnn+byyy93O44xJkAeeeQRrrnmGl599VW3o1wSKxYuGTZsGFdeeSVdu1rXIMZkZXny5OHJJ59k6dKlbNy40e04GWbFwgU7duzgvffeo0uXLuTPn9/tOMaYAOvatSt58+ZlxIgR6U8coqxYuGDEiBHkzJmTp59+2u0oxpggKFSoEI899hhz5sxh165dbsfJECsWQXbgwAGmT59Ohw4dKFy4cPozGGOyhGeeeQZVZcyYMW5HyRArFkE2duxYzp07R69evdyOYowJouuuu47WrVszefJkjh075naci2bFIohOnDjB66+/zgMPPECpUqXcjmOMCbJevXpx6tQpYmJi3I5y0axYBNHUqVM5ceKE7VUYk01VqFCBevXqMXbsWP7880+341wUKxZBcv78ecaOHUu1atWoVCnlvkWMMVlf7969OXjwIG+99ZbbUS6KFYsgmT9/Pjt37uTZZ591O4oxxkXVq1enUqVKjBgxggsXMs8NtK1YBIGqMnLkSEqVKkWjRo3cjmOMcZGI0Lt3b7Zt28aCBQvcjuMzKxZB8NVXX7FmzRqeeeYZwsLC3I5jjHFZ8+bNuf766xk9erTbUXzmWrEQkd+czo7Wi8hap62QiHwiItucx4JOu4jIOBHZLiIbReR2t3JnxKhRoyhYsCDt27d3O4oxJgTkzJmTJ598kpUrV7Ju3Tq34/jE7T2LGqpaUVWTzvj2AT5T1dLAZ85r8PTXXdoZooBMc93Zjh07eP/99//6ub8xxgB06tSJ8PDwTLN34XaxSK4p8Kbz/E2gmVf7W+qxGiiQrM/ukDV27Fhy5sxJt27d3I5ijAkh+fPnp1OnTrzzzjvs3bvX7TjpErc65BCRX4FjgAKTVHWyiBxX1QJe0xxT1YIisggYqqqrnPbPgOdUdW2yZUbh2fMgIiIiMjY2NkPZ4uPjCQ8Pz9C83k6ePEnLli2599576dOnT/ozBCGTv4ViLsvku1DMFYqZIDC59u3bR9u2bWndujWPPfaY65lq1KixzutIzz+pqisDUMR5vAbYAFQDjieb5pjzuBio6tX+GRCZ1vIjIyM1o+Li4jI8r7ehQ4cqoOvXr7/kZfkrk7+FYi7L5LtQzBWKmVQDl6t58+ZaqFAhPXXq1EXP6+9MwFpN5TPVtcNQqrrPeTwIvA/cAfyedHjJeTzoTL4HKO41ezFgX/DSXryzZ88ybtw4atWqxa233up2HGNMiHrmmWc4evRoyP9Iz5ViISJ5RSRf0nOgDrAJWAAkXTLUHvjQeb4AeMS5Kuou4A9V3R/k2Bfl3XffZd++ffTs2dPtKMaYEFa1alUiIyMZM2YMiYmJbsdJlVt7FhHAKhHZAKwBFqvqUmAocJ+IbAPuc14DLAF+AbYDbwBPBD+y71SVUaNGUbZsWerWret2HGNMCBMRnnnmGbZu3crSpUvdjpOqnG6sVFV/Af51bEZVjwC1UmhXINNcTrRy5Uq+//57Jk2aRI4coXbBmTEm1Dz44IP07t2b0aNH06BBA7fjpMg+yQLgtddeo2DBgrRt29btKMaYTCBXrlx0796dTz/9lB9++MHtOCmyYuFnu3fv5v3336dz585cccUVbscxxmQSjz/+OHny5AnZnvSsWPhZTEwMqsoTT4T0aRVjTIgpVKgQ7du3Z9asWRw8eDD9GYLMioUfJSQkMHnyZJo0aUKJEiXcjmOMyWSeeuopzpw5wxtvvOF2lH+xYuFHsbGxHDlyhCeffNLtKMaYTOimm27ivvvuIyYmhvPnz7sd5x+sWPiJqjJu3DjKly9P9erV3Y5jjMmkunfvzt69e/nwww/TnziIrFj4yZdffsn69evp0aMHIuJ2HGNMJtWwYUOuv/56xo8f73aUf7Bi4Sfjxo2jQIECtGnTxu0oxphMLCwsjCeeeIIVK1awadMmt+P8xYqFH+zZs4f58+fTuXNn67PCGHPJOnXqxOWXX86ECRPcjvIXKxZ+EBMTQ2Jiol0ua4zxi6uuuorWrVvz1ltvcfz4cbfjAFYsLtmff/751+WyJUuWdDuOMSaL6N69O6dPn2bGjBluRwGsWFyy2NhYDh8+TI8ePdyOYozJQm6//XbuvvtuJkyYEBJ3o7VicQlUlddee41y5cpRs2ZNt+MYY7KY7t27s337dj7++GO3o1ixuBTffPMN3333Hd27d7fLZY0xfteiRQsiIiJC4jJaKxaXICYmhvDwcLu7rDEmIHLlysXjjz/OkiVL2LFjh6tZrFhk0NGjR3nnnXdo27Yt+fLlczuOMSaLevzxxwkLCyMmJsbVHEEvFiJSXETiRGSLiGwWkaec9mgR2Ssi652hgdc8fUVku4hsFZGQ6HpuxowZnDlzhq5du7odxRiThRUpUoT777+fqVOncurUKddyuLFncR7oqaplgbuAbiJSzhk3WlUrOsMSAGdcK+BmoB7wuoiEuZD7L4mJiUycOJG7776bChUquBnFGJMNdOvWjePHj/POO++4liHoxUJV96vqd87zk8AWoGgaszQFYlX1jKr+iqcf7jsCnzR1y5cvZ9u2bbZXYYwJiv/7v/+jXLlyTJw40bUM4une2qWVi5QAvgDKA88CHYATwFo8ex/HRGQ8sFpVZzrzTAU+UtW5KSwvCogCiIiIiIyNjc1Qrvj4eMLDw1Md/8ILL7Bhwwbee+89cuXKlaF1+DuTW0Ixl2XyXSjmCsVM4H6uefPmMX78eCZNmkSZMmUCkqlGjRrrVLVSiiNV1ZUBCAfWAfc7ryOAMDx7O4OBaU77BKCt13xTgRbpLT8yMlIzKi4uLtVxe/bs0bCwMO3Vq1eGl58RaWVyUyjmsky+C8VcoZhJ1f1cR48e1Tx58mhUVNRfbf7OBKzVVD5TXbkaSkQuA+YBs1R1PoCq/q6qF1Q1EXiDvw817QGKe81eDNgXzLzepkyZwoULF3j88cfdimCMyYYKFixIq1atmD17NidPngz6+t24Gkrw7B1sUdVRXu2FvSZrDiTdm3cB0EpEcotISaA0sCZYeb2dP3+eN954g7p163LjjTe6EcEYk4116dKF+Ph4Zs2aFfR1u7FncQ/QDqiZ7DLZV0XkBxHZCNQAngFQ1c3Au8CPwFKgm6pecCE3CxcuZO/evXTp0sWN1RtjsrnKlStTsWJFJk6cmHRYPmhyBnVtgKquAlK6N8aSNOYZjOc8hqtiYmIoVqwYjRo1cjuKMSYbEhG6dOlCly5dWLMmuAdY7BfcPtq2bRuffPIJjz32GDlzBr3GGmMMAA8//DDh4eFBv4zWioWPJk+eTFhYGJ07d3Y7ijEmG8uXLx9t2rQhNjY2qCe6rVj44M8//2T69Ok0a9aMIkWKuB3HGJPNPf744/z5559BvXW5FQsfvP/++xw5csQulzXGhITbbruNO++8k4ULFwbtRLcVCx9MmTKFEiVKUKtWLbejGGMM4LmMdufOnaxcuTIo67NikY4dO3awfPlyOnXqRI4ctrmMMaGhZcuW5M2bl0mTJgVlffbpl44pU6aQI0cOOnbs6HYUY4z5yxVXXEHdunWZO3cuhw4dCvj6rFik4dy5c8yYMYOGDRtStGhaN8Y1xpjga9SoEWfPnuXtt98O+LqsWKRh8eLFHDhwwC6XNcaEpJIlS1KlShWmTJkS8BPdVizS8MYbb1CkSBEaNGiQ/sTGGOOCzp07s2XLFr766quArseKRSp2797N0qVL6dixo/1i2xgTslq2bEl4eDhTpkwJ6HqsWKRi+vTpJCYm0qlTJ7ejGGNMqsLDw3n44Yd59913+eOPPwK2HisWKbhw4QJTp06ldu3alCxZ0u04xhiTps6dO3P69GnmzJkTsHVYsUjBunXr2LVrF4899pjbUYwxJl2VKlWiQoUKAT0UZcUiBYsXL+bqq6+madOmbkcxxph0iQidO3dm3bp1fP/99wFZR6YpFiJST0S2ish2EekTqPX8/vvvfPnll7Rv357cuXMHajXGGONXbdu2JXfu3AHbu8gUxUJEwoAJQH2gHNBaRMoFYl1vvvkmFy5csBPbxphMpWDBgjzwwAPMnj2bM2fO+H35maJYAHcA21X1F1U9C8QCfj9GpKpMmTKFW265hbJly/p78cYYE1ADBw5kzZo1ATkqIsHuxzUjROQBoJ6qdnZetwPuVNXuyaaLAqIAIiIiImNjYy9qPQkJCYwfP57y5ctTv359/4T3k/j4eMLDw92O8S+hmMsy+S4Uc4ViJgjNXP7OVKNGjXWqWinFkaoa8gPwIDDF63U74LW05omMjNSMiouLy/C8gRKKmVRDM5dl8l0o5grFTKqhmcvfmYC1mspnamY5DLUHKO71uhiwz6UsxhiT7WSWYvEtUFpESopILqAVsMDlTMYYk21kipseqep5EekOLAPCgGmqutnlWMYYk21kimIBoKpLgCVu5zDGmOwosxyGMsYY4yIrFsYYY9JlxcIYY0y6rFgYY4xJV6b4BXdGiMghYGcGZ78aOOzHOP4QipkgNHNZJt+FYq5QzAShmcvfma5X1f+kNCLLFotLISJrNbWfvLskFDNBaOayTL4LxVyhmAlCM1cwM9lhKGOMMemyYmGMMSZdVixSNtntACkIxUwQmrksk+9CMVcoZoLQzBW0THbOwhhjTLpsz8IYY0y6rFgYY4xJlxULLyJST0S2ish2Eenjdp4kIvKbiPwgIutFZK1LGaaJyEER2eTVVkhEPhGRbc5jwRDJFS0ie53ttV5EGgQ5U3ERiRORLSKyWUSectpd215pZHJ7W10uImtEZIOT60WnvaSIfONsq3ecrgnczjRDRH712lYVg5XJK1uYiHwvIouc18HbTqn1ipTdBjy3Pt8B3ADkAjYA5dzO5WT7Dbja5QzVgNuBTV5trwJ9nOd9gGEhkisa+J+L26owcLvzPB/wM1DOze2VRia3t5UA4c7zy4BvgLuAd4FWTvtEoGsIZJoBPODWtnLyPAvMBhY5r4O2nWzP4m93ANtV9RdVPQvEAk1dzhQyVPUL4Giy5qbAm87zN4FmQQ1Fqrlcpar7VfU75/lJYAtQFBe3VxqZXKUe8c7Ly5xBgZrAXKc92NsqtUyuEpFiQENgivNaCOJ2smLxt6LAbq/XewiBPyaHAh+LyDoRiXI7jJcIVd0Png8j4BqX83jrLiIbncNUQT88lkRESgC34fl2GhLbK1kmcHlbOYdW1gMHgU/w7OEfV9XzziRB/1tMnklVk7bVYGdbjRaR3MHMBIwBegOJzuurCOJ2smLxN0mhzfVvE457VPV2oD7QTUSquR0oxMUANwIVgf3ASDdCiEg4MA94WlVPuJEhuRQyub6tVPWCqlYEiuHZwy+b0mRuZhKR8kBf4CagMlAIeC5YeUSkEXBQVdd5N6cwacC2kxWLv+0Binu9LgbscynLP6jqPufxIPA+nj+oUPC7iBQGcB4PupwHAFX93fljTwTewIXtJSKX4flQnqWq851mV7dXSplCYVslUdXjwAo85wcKiEhST56u/S16ZarnHMpTVT0DTCe42+oeoImI/IbnEHlNPHsaQdtOViz+9i1Q2rm6IBfQCljgciZEJK+I5Et6DtQBNqU9V9AsANo7z9sDH7qY5S9JH8iO5gR5eznHkqcCW1R1lNco17ZXaplCYFv9R0QKOM/zALXxnE+JAx5wJgv2tkop009ehV7wnBsI2rZS1b6qWkxVS+D5bFquqm0I5nZy88x+qA1AAzxXiewAnnc7j5PpBjxXZm0ANruVC5iD5zDFOTx7YZ3wHDP9DNjmPBYKkVxvAz8AG/F8QBcOcqaqeA4HbATWO0MDN7dXGpnc3lYVgO+d9W8CXnDabwDWANuB94DcIZBpubOtNgEzca6YCvYAVOfvq6GCtp3sdh/GGGPSZYehjDHGpMuKhTHGmHRZsTDGGJMuKxbGGGPSZcXCGGNMuqxYGOMDESkgIk94vS4iInPTmucS1tVMRF5IY/wtIjIjEOs2JjV26awxPnDup7RIVcsHYV1fAU1U9XAa03wKPKqquwKdxxiwPQtjfDUUuNHpx2C4iJRI6kNDRDqIyAcistDp76C7iDzr9DuwWkQKOdPdKCJLnRtCrhSRm5KvRETKAGeSCoWIPCgim5y+Fb7wmnQhnl/yGhMUViyM8U0fYIeqVlTVXimMLw88jOd+QYOB06p6G/A18IgzzWSgh6pGAv8DXk9hOfcA33m9fgGoq6q3Ak282tcC/3cJ78eYi5Iz/UmMMT6IU08/ESdF5A883/zBc3uICs7dXu8G3vPcWgiAlG5xXRg45PX6S2CGiLwLzPdqPwgU8WN+Y9JkxcIY/zjj9TzR63Uinr+zHHj6HkivK84EIH/SC1XtIiJ34un0Zr2IVFTVI8DlzrTGBIUdhjLGNyfxdEeaIerpO+JXEXkQPHcuFZFbU5h0C1Aq6YWI3Kiq36jqC8Bh/r6NfhlC5+7DJhuwYmGMD5xv8186J5uHZ3AxbYBOIpJ0B+GUuu39ArhN/j5WNVxEfnBOpn+B5+7DADWAxRnMYcxFs0tnjQkxIjIWWKiqn6YyPjfwOVBV/+5S05iAsj0LY0LPK8AVaYy/DuhjhcIEk+1ZGGOMSZftCJvGbwAAACZJREFUWRhjjEmXFQtjjDHpsmJhjDEmXVYsjDHGpMuKhTHGmHT9P0QRDu+3eHqAAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "def x(x0, v0, t, g=9.81):\n",
    "    return x0 + v0*t - g*t**2/2 \n",
    "\n",
    "x0 = 0 # initial position\n",
    "v0 = 200 # initial velocity\n",
    "\n",
    "# x = 0 when rock at top of volcano\n",
    "# (-9.8t**2)/2 + 200t = 0\n",
    "# t = 0s or 40.8s\n",
    "#choosing this time span for graph purposes\n",
    "\n",
    "time = np.arange(0, 40.8, 0.1) \n",
    "\n",
    "#try different time spans to check if the max value changes\n",
    "\n",
    "position = x(x0, v0, time)\n",
    "    \n",
    "print(\"Max height reached = %.f m\" % (max(position))) \n",
    "result = np.where(position == max(position)) # find index of maximum position\n",
    "print(\"Time needed for the body to reach that height = %.2fs\" % (time[result[0][0]]))\n",
    "\n",
    "\n",
    "# plot position over time\n",
    "fig = plt.figure(figsize=(6,4))\n",
    "\n",
    "plt.plot(time, position, 'k')\n",
    "plt.plot(time[result[0][0]], max(position), 'ro')\n",
    "plt.xlabel('time (s)')\n",
    "plt.ylabel('height (m)')\n",
    "plt.title('Motion of the rock over time', fontsize=14)\n",
    "plt.grid(True)\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### More about contact forces\n",
    "\n",
    "**Newton's third law of motion** states that when two bodies interact, when a force is exerted from a body A to body B, a force **equal in magnitude and opposite in direction** is exerted by body B to body A. \n",
    "\n",
    "Therefore, on the surface of contact between two bodies, there is contact force and a **normal force \\\\(N\\\\)** with the same magnitude and in opposite direction to that of the contact force.\n",
    "\n",
    "### Friction\n",
    "\n",
    "Friction is a contact force that always acts **tangential** to the contact surface between two bodies, and acts in a direction that **prevents relative motion** between those bodies. \n",
    "\n",
    "The maximum friction that can be exerted is equal to \\\\(\\mu_dN\\\\) where \\\\(\\mu_d\\\\) is the **coefficient of friction**. If the force exerted in tangential direction is less than \\\\(\\mu_dN\\\\), friction is not overcome and there won't be relative motion.\n",
    "\n",
    "When tangential force \\\\(F\\\\) is larger than \\\\(\\mu_dN\\\\), net force on the body in the tangential direction is \\\\((F-\\mu_dN)N\\\\)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Tutorial Problem 1.3\n",
    "\n",
    "Consider a stone of mass \\\\(m = 24kg\\\\) sitting on a flat valley floor, being hit by a gust of wind that exerts a horizontal force \\\\(F = 5N\\\\) on it for a period of time \\\\(T = 5s\\\\). Assuming the coefficient of friction between the rock and the ground is \\\\(\\mu = 0.01\\\\), find the distance travelled by the rock before it stops sliding."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Total distance travelled = 2.93m\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x432 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "m = 24 # mass (kg)\n",
    "F = 5 # force of wind (N)\n",
    "T = 5 # time wind exert force on rock (s)\n",
    "u = 0.01 # coefficient of friction\n",
    "g = 9.81 # m/s2\n",
    "\n",
    "time1 = np.arange(0., 5.01, 0.01) # time blown by wind\n",
    "\n",
    "time2 = np.arange(0.01, 5.63, 0.01) # time before stopping\n",
    "\n",
    "time = np.arange(0, 10.63, 0.01) # total amount of time\n",
    "\n",
    "x = np.zeros(len(time))\n",
    "\n",
    "v = np.zeros(len(time))\n",
    "\n",
    "# Newton's second law to find acceleration\n",
    "# net force = F - uN\n",
    "# N = mg\n",
    "# a = (F-umg)/m\n",
    "\n",
    "# x = x0 + v0t + at**2/2\n",
    "x[:501] = ((F-u*m*g)*time1**2)/(2*m) # distance over time travelled when blown by wind\n",
    "\n",
    "# v = v0 + at\n",
    "v[:501] = (F-u*m*g)*time1/m # velocity over time when blown by wind\n",
    "\n",
    "# net force = -umg\n",
    "# a = -umg/m = -ug\n",
    "# x0 = last element calculated in x\n",
    "# v0 = last element calculated in v\n",
    "\n",
    "# x = x0 + v0t - at**2/2\n",
    "x[501:] = x[500] + v[500]*time2 - u*g*time2**2/2 # distance over time without wind\n",
    "\n",
    "# v = v0 + at\n",
    "v[501:] = v[500] - u*g*time2 # velocity over time without wind\n",
    "\n",
    "print(\"Total distance travelled = %.2fm\" % (x[-1]))\n",
    "\n",
    "\n",
    "# plot figure of distance and velocity of the rock over time\n",
    "\n",
    "fig = plt.figure(figsize=(12,6))\n",
    "\n",
    "ax1 = fig.add_subplot(121)\n",
    "ax1.plot(time, x, 'r')\n",
    "ax1.set_xlabel('time (s)')\n",
    "ax1.set_ylabel('distance (m)')\n",
    "ax1.set_ylim(0, 3)\n",
    "ax1.set_title('Position of rock over time', fontsize=14)\n",
    "ax1.grid(True)\n",
    "\n",
    "ax2 = fig.add_subplot(122)\n",
    "ax2.plot(time, v, 'b')\n",
    "ax2.set_xlabel('time (s)')\n",
    "ax2.set_ylabel('velocity (m/s)')\n",
    "ax2.set_ylim(0, 0.6)\n",
    "ax2.set_title('Velocity of rock over time', fontsize=14)\n",
    "ax2.grid(True)\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Viscous drag force\n",
    "\n",
    "![](images/fluid.png)\n",
    "\n",
    "Resistance from fluid arise from its **viscosity** \\\\(\\mu\\\\), and it is called the viscous drag force.\n",
    "\n",
    "For a spherical solid particle moving through a stationary fluid, or a fluid flowing past a stationary body, Stoke's law states that:\n",
    "\n",
    "\\\\[F=-6\\pi \\mu Rv\\\\]\n",
    "\n",
    "where \\\\(R\\\\) is the radius of the particle, and \\\\(v\\\\) is the relative velocity between the body and the fluid. The minus sign implies it opposes the particle's motion.\n",
    "\n",
    "Weight of the grain \\\\(F_w\\\\) is given by:\n",
    "\n",
    "\\\\[F_w=m_sg=\\rho_sVg=\\rho_s\\frac{4}{3}\\pi R^{3}g\\\\]\n",
    "\n",
    "where \\\\(m_s\\\\) is mass of the particle, \\\\(\\rho_s\\\\) is its density, and \\\\(V\\\\) its volume.\n",
    "\n",
    "Buoyancy force \\\\(F_b\\\\) is the weight of water displaced:\n",
    "\n",
    "\\\\[F_b=m_wg=\\rho_wVg=\\rho_w\\frac{4}{3}\\pi R^3g\\\\]\n",
    "\n",
    "Accounting all these forces into Newton's second law:\n",
    "\n",
    "\\\\[\\frac{4}{3}\\pi\\rho_sR^3\\frac{d^2x}{dt^2}=\\rho_s\\frac{4}{3}\\pi R^3g-\\rho_w\\frac{4}{3}\\pi R^3g-6\\pi \\mu Rv\\\\]\n",
    "\n",
    "Rearranging this gives an ODE:\n",
    "\n",
    "\\\\[\\rho_s\\frac{dv}{dt}=(\\rho_s-\\rho_w)g-\\frac{9}{2}\\frac{\\mu v}{R^2}\\\\]\n",
    "\n",
    "which can be solved analytically."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Tutorial Problem 1.5\n",
    "\n",
    "Find the terminal velocity of spherical particles with densities \\\\(\\rho_s=2.65\\times10^3kgm^{-3}\\\\), and radii \\\\(10\\mu m\\\\), \\\\(100\\mu m\\\\), \\\\(1mm\\\\), and \\\\(1cm\\\\). \n",
    "\n",
    "Density of water \\\\(\\rho_w=1000kgm^{-3}\\\\), viscosity of water \\\\(\\mu=10^{-3}\\\\), and the velocity of a sphere from rest through a viscous fluid is given by:\n",
    "\n",
    "\\\\[v=\\frac{2(\\rho_s-\\rho_w)gR^2}{9\\mu}(1-e^{\\frac{-9\\mu t}{2\\rho_sR^2}})\\\\]\n",
    "\n",
    "As \\\\(t\\to \\infty\\\\), \\\\(v\\to \\frac{2(\\rho_s-\\rho_w)gR^2}{9\\mu}\\\\), which is the terminal velocity."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Terminal velocity of 1um grain is 3.60e-07 m/s\n",
      "Terminal velocity of 10um grain is 3.60e-05 m/s\n",
      "Terminal velocity of 1mm grain is 0.36 m/s\n",
      "Terminal velocity of 1cm grain is 35.96 m/s\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1152x864 with 4 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "def velocity(t, R, ps=2650, pw=1000, g=9.81, mu=10e-3):\n",
    "    return (2*(ps-pw)*g*R**2)/(9*mu) * (1-np.exp((-9*mu*t)/(2*ps*R**2))) # analytical solution to ode\n",
    "\n",
    "def t_v(radius, start, end, interval):\n",
    "    t = np.arange(start, end, interval)\n",
    "    v = np.zeros(len(t))\n",
    "    for i in range(len(t)):\n",
    "        v[i] = velocity(t[i], radius)\n",
    "    return t, v # obtain list of time and velocity for particles with different radius over a time interval\n",
    "\n",
    "t_1cm, v_1cm = t_v(1e-2, 0, 50, 0.05)\n",
    "t_1mm, v_1mm = t_v(1e-3, 0, 0.5, 0.0005)\n",
    "t_10um, v_10um = t_v(10e-6, 0, 1e-4, 1e-7)\n",
    "t_1um, v_1um = t_v(1e-6, 0, 5e-7, 5e-10)\n",
    "\n",
    "print(\"Terminal velocity of 1um grain is %.2e m/s\" % (v_1um[-1]))\n",
    "print(\"Terminal velocity of 10um grain is %.2e m/s\" % (v_10um[-1]))\n",
    "print(\"Terminal velocity of 1mm grain is %.2f m/s\" % (v_1mm[-1]))\n",
    "print(\"Terminal velocity of 1cm grain is %.2f m/s\" % (v_1cm[-1]))\n",
    "\n",
    "# plot figures of velocity over time for grains of different radii\n",
    "\n",
    "fig = plt.figure(figsize=(16,12))\n",
    "\n",
    "ax1 = fig.add_subplot(221)\n",
    "ax1.plot(t_1um, v_1um, 'r')\n",
    "ax1.set_xlabel('time (s)')\n",
    "ax1.set_ylabel('velocity (m/s)')\n",
    "ax1.set_title('Velocity of sand grains with radius 1um over time')\n",
    "ax1.grid(True)\n",
    "\n",
    "ax2 = fig.add_subplot(222)\n",
    "ax2.plot(t_10um, v_10um, 'y')\n",
    "ax2.set_xlabel('time (s)')\n",
    "ax2.set_ylabel('velocity (m/s)')\n",
    "ax2.set_title('Velocity of sand grain with radius 10um over time')\n",
    "ax2.grid(True)\n",
    "\n",
    "ax3 = fig.add_subplot(223)\n",
    "ax3.plot(t_1mm, v_1mm, 'g')\n",
    "ax3.set_xlabel('time (s)')\n",
    "ax3.set_ylabel('velocity (m/s)')\n",
    "ax3.set_title('Velocity of sand grain with radius 1mm over time')\n",
    "ax3.grid(True)\n",
    "\n",
    "ax4 = fig.add_subplot(224)\n",
    "ax4.plot(t_1cm, v_1cm, 'b')\n",
    "ax4.set_xlabel('time (s)')\n",
    "ax4.set_ylabel('velocity (m/s)')\n",
    "ax4.set_title('Velocity of sand grain with radius 1cm over time')\n",
    "ax4.grid(True)\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Elastic Spring Forces\n",
    "\n",
    "According to Hooke's law, the force applied to the spring \\\\(F\\\\) is directly proportional to its extension \\\\(x\\\\). Mathematically:\n",
    "\n",
    "\\\\[F=kx\\\\]\n",
    "\n",
    "where k is the spring constant in \\\\(Nm^{-1}\\\\).\n",
    "\n",
    "#### Coupled Oscillators\n",
    "\n",
    "Consider two masses, \\\\(m_1\\\\) and \\\\(m_2\\\\), connected by a spring with spring constant \\\\(k\\\\). Let the position of \\\\(m_1\\\\) be \\\\(x_1\\\\), and the position of \\\\(m_2\\\\) be \\\\(x_2\\\\).\n",
    "\n",
    "From this, the length of the spring is \\\\((x_2-x_1)\\\\), and the change in length is \\\\(\\Delta(x_2-x_1)=\\Delta x_2-\\Delta x_1\\\\).\n",
    "\n",
    "Applying Hooke's law into Newton's second law to \\\\(m_1\\\\):\n",
    "\n",
    "\\\\[k(\\Delta x_2-\\Delta x_1)=m_1\\frac{d^2x_1}{dt^2}\\\\]\n",
    "\n",
    "Applying the same laws for \\\\(m_2\\\\), and considering Newton's third law:\n",
    "\n",
    "\\\\(-k(\\Delta x_2-\\Delta x_1)=m_2\\frac{d^2x_2}{dt^2}\\\\)\n",
    "\n",
    "These two equations are coupled and must be solved simultaneously."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Tutorial Problem 1.6\n",
    "\n",
    "![](images/problem1_6.png)\n",
    "\n",
    "The oscillatory solutions to the problem are of the form \\\\(x_1=A_1e^{i\\omega t}\\\\), \\\\(x_2=A_2e^{i\\omega t}\\\\). Substitute these expressions into the coupled equations, and solve the eigenvalue problem, finding the two values of \\\\(\\omega\\\\) which are the eigenvalues, and the ratio \\\\(\\frac{A_2}{A_1}\\\\) that corresponds to each eigenvalue.\n",
    "\n",
    "Discuss your results.\n",
    "\n",
    "Hint: solving the problem analytically, you should obtain the following:\n",
    "\n",
    "When \\\\(\\omega_1=0\\\\), \\\\(A_1=A_2\\\\)\n",
    "\n",
    "When \\\\(\\omega_2=\\sqrt{\\frac{2k}{m}}\\\\), \\\\(A_1=-A_2\\\\)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "scrolled": false
   },
   "outputs": [],
   "source": [
    "import cmath\n",
    "\n",
    "w = np.pi\n",
    "\n",
    "t = np.arange(0, 2*np.pi, 0.01)\n",
    "X1 = np.zeros(len(t))\n",
    "X2 = np.zeros(len(t))\n",
    "X3 = np.zeros(len(t))\n",
    "X4 = np.zeros(len(t))\n",
    "\n",
    "for i in range(len(t)):\n",
    "    # w1 = 0\n",
    "    # A1 = A2\n",
    "    z1 = cmath.exp(1j*w*t[i])+1\n",
    "    X1[i] = z1.real\n",
    "    z2 = cmath.exp(1j*w*t[i])-1\n",
    "    X2[i] = z2.real\n",
    "    \n",
    "    # w2 = (2k/m)**0.5\n",
    "    # A1 = -A2\n",
    "    # starting at different initial positions\n",
    "    z3 = cmath.exp(1j*w*t[i])+1\n",
    "    X3[i] = z3.real\n",
    "    z4 = -(cmath.exp(1j*w*t[i]))-1\n",
    "    X4[i] = z4.real"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "scrolled": false,
    "tags": [
     "remove-output"
    ]
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 504x360 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# now we make a nice animation\n",
    "nframes = len(t)\n",
    "\n",
    "# Plot background axes\n",
    "fig, axes = plt.subplots(2,1, figsize=(7,5))\n",
    "\n",
    "line1, = axes[0].plot([], [], 'ro', lw=2)\n",
    "line2, = axes[0].plot([], [], 'go', lw=2)\n",
    "line3, = axes[1].plot([], [], 'yo', lw=2)\n",
    "line4, = axes[1].plot([], [], 'bo', lw=2)\n",
    "\n",
    "for ax in axes:\n",
    "    ax.set_xlim(-2,2)\n",
    "    ax.set_ylim(-0.1,0.1)\n",
    "    \n",
    "axes[0].set_title('A1 = A2')\n",
    "axes[1].set_title('A1 = -A2')\n",
    "    \n",
    "lines = [line1, line2, line3, line4]\n",
    "    \n",
    "plt.subplots_adjust(hspace=0.5)\n",
    "\n",
    "# Plot background for each frame\n",
    "def init():\n",
    "    for line in lines:\n",
    "        line.set_data([], [])\n",
    "    return lines\n",
    "\n",
    "# Set what data to plot in each frame\n",
    "def animate(i):\n",
    "    \n",
    "    x1 = X1[i]\n",
    "    y1 = 0\n",
    "    lines[0].set_data(x1, y1)\n",
    "    \n",
    "    x2 = X2[i]\n",
    "    y2 = 0\n",
    "    lines[1].set_data(x2, y2)\n",
    "    \n",
    "    x3 = X3[i]\n",
    "    y3=0\n",
    "    lines[2].set_data(x3, y3)\n",
    "    \n",
    "    x4 = X4[i]\n",
    "    y4 = 0\n",
    "    lines[3].set_data(x4, y4)\n",
    "    \n",
    "    return lines\n",
    "\n",
    "# Call the animator\n",
    "anim = animation.FuncAnimation(fig, animate, init_func=init,\n",
    "                               frames=nframes, interval=10, blit=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
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    "### References\n",
    "\n",
    "Course notes from Lecture 1 of the module ESE 95011 Mechanics"
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