{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "a09370ca",
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy as smp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "fcd9ee5a",
   "metadata": {},
   "outputs": [],
   "source": [
    "m1, m2, v1i, v2i, v1f, v2f = smp.symbols(\"m_1 m_2 v_{1i} v_{2i} v_{1f} v_{2f}\", real=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0e449d85",
   "metadata": {},
   "source": [
    "Create an expression for the kinetic energy of a two body problem\n",
    "\n",
    "$v_{1i}$ is initial velocity of mass $m_1$  \n",
    "$v_{2i}$ is initial velocity of mass $m_2$  \n",
    "$v_{1f}$ is final velocity of mass $m_1$  \n",
    "$v_{2f}$ is final velocity of mass $m_2$\n",
    "\n",
    "I am also assuming that $m_2$ >> $m_1$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "8ec19eeb",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\frac{m_{1} v_{1f}^{2}}{2} + \\frac{m_{1} v_{1i}^{2}}{2} - \\frac{m_{2} v_{2f}^{2}}{2} + \\frac{m_{2} v_{2i}^{2}}{2} = 0$"
      ],
      "text/plain": [
       "Eq(-m_1*v_{1f}**2/2 + m_1*v_{1i}**2/2 - m_2*v_{2f}**2/2 + m_2*v_{2i}**2/2, 0)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eqn1a = m1*v1i**2/2 + m2*v2i**2/2 - m1*v1f**2/2 - m2*v2f**2/2\n",
    "smp.Eq(eqn1a,0)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a5d8f7cb",
   "metadata": {},
   "source": [
    "However the initial velocity $v_{2i}$, the wedge, is zero."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "b58aa7e3",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\frac{m_{1} v_{1f}^{2}}{2} + \\frac{m_{1} v_{1i}^{2}}{2} - \\frac{m_{2} v_{2f}^{2}}{2} = 0$"
      ],
      "text/plain": [
       "Eq(-m_1*v_{1f}**2/2 + m_1*v_{1i}**2/2 - m_2*v_{2f}**2/2, 0)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eqn1 = eqn1a.subs(v2i, 0)\n",
    "smp.Eq(eqn1,0)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d299a75c",
   "metadata": {},
   "source": [
    "Now create and expression for the momentum of the two body problem"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "835f7764",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - m_{1} v_{1f} + m_{1} v_{1i} - m_{2} v_{2f} + m_{2} v_{2i} = 0$"
      ],
      "text/plain": [
       "Eq(-m_1*v_{1f} + m_1*v_{1i} - m_2*v_{2f} + m_2*v_{2i}, 0)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq2a = m1*v1i + m2*v2i - m1*v1f - m2*v2f\n",
    "smp.Eq(eq2a,0)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4f1c7b08",
   "metadata": {},
   "source": [
    "Again the initial velocity $v_{2i}$, the wedge, is zero."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "fb9d5a79",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - m_{1} v_{1f} + m_{1} v_{1i} - m_{2} v_{2f} = 0$"
      ],
      "text/plain": [
       "Eq(-m_1*v_{1f} + m_1*v_{1i} - m_2*v_{2f}, 0)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eqn2 = eq2a.subs(v2i, 0)\n",
    "smp.Eq(eqn2, 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0690d7dd",
   "metadata": {},
   "source": [
    "Make $v_{2f}$ the subject and substitute back in to equation 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "649e6927",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle v_{2f} = - \\frac{m_{1} v_{1f} - m_{1} v_{1i}}{m_{2}}$"
      ],
      "text/plain": [
       "Eq(v_{2f}, -(m_1*v_{1f} - m_1*v_{1i})/m_2)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v2fs = smp.solveset(eqn2, v2f).args[0]\n",
    "smp.Eq(v2f, v2fs)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "0b284c3e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\frac{m_{1} v_{1f}^{2}}{2} + \\frac{m_{1} v_{1i}^{2}}{2} - \\frac{\\left(m_{1} v_{1f} - m_{1} v_{1i}\\right)^{2}}{2 m_{2}} = 0$"
      ],
      "text/plain": [
       "Eq(-m_1*v_{1f}**2/2 + m_1*v_{1i}**2/2 - (m_1*v_{1f} - m_1*v_{1i})**2/(2*m_2), 0)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq3a = eqn1.subs(v2f, v2fs)\n",
    "smp.Eq(eq3a, 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "00e9be73",
   "metadata": {},
   "source": [
    "Now solve for $v_{1f}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "8f601a39",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left\\{v_{1i}, \\frac{m_{1} v_{1i} - m_{2} v_{1i}}{m_{1} + m_{2}}\\right\\}$"
      ],
      "text/plain": [
       "{v_{1i}, (m_1*v_{1i} - m_2*v_{1i})/(m_1 + m_2)}"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq3b = smp.solveset(eq3a, v1f)\n",
    "eq3b"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ee70d203",
   "metadata": {},
   "source": [
    "There are two answers two this quadratic equation but if m2 >> m1 (which is what I am assuming) then the second term is negative which will be shown explicitly below.  Since mass must be positive that can only mean that the $v_{1i} $ is negative with respect to $ v_{1f} $ which looks like the solution we are looking for.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "47942641",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{v_{1i} \\left(m_{1} - m_{2}\\right)}{m_{1} + m_{2}}$"
      ],
      "text/plain": [
       "v_{1i}*(m_1 - m_2)/(m_1 + m_2)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq3c = eq3b.args[1].factor()\n",
    "eq3c"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cd233b73",
   "metadata": {},
   "source": [
    "Now take the limit as $ m_2 \\rightarrow \\infty $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "a2ec4275",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - v_{1i}$"
      ],
      "text/plain": [
       "-v_{1i}"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "smp.limit(eq3c, m2, smp.oo)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5032584b",
   "metadata": {},
   "source": [
    "This proves that $ v_{1f} = -v_{1i} $ in the case for when $ m_2 $ is fixed."
   ]
  }
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