{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "74112073",
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import symbols, Rational, Eq, solveset, simplify"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "6bc9efc7",
   "metadata": {},
   "outputs": [],
   "source": [
    "m1, m2, u1, u2, v1, v2 = symbols(\"m_1 m_2 u_1 u_2 v_1 v_2\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "3e6364f1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle m_{1} u_{1} + m_{2} u_{2} = m_{1} v_{1} + m_{2} v_{2}$"
      ],
      "text/plain": [
       "Eq(m_1*u_1 + m_2*u_2, m_1*v_1 + m_2*v_2)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "momentum = Eq(m1*u1 + m2*u2, m1*v1 + m2*v2)\n",
    "momentum"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "9ded4789",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{m_{1} u_{1}^{2}}{2} + \\frac{m_{2} u_{2}^{2}}{2} = \\frac{m_{1} v_{1}^{2}}{2} + \\frac{m_{2} v_{2}^{2}}{2}$"
      ],
      "text/plain": [
       "Eq(m_1*u_1**2/2 + m_2*u_2**2/2, m_1*v_1**2/2 + m_2*v_2**2/2)"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "energy = Eq(Rational(1,2)*m1*u1**2 + Rational(1,2)*m2*u2**2, Rational(1,2)*m1*v1**2 + Rational(1,2)*m2*v2**2)\n",
    "energy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "cf10849e",
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{m_{1} u_{1} + m_{2} u_{2} - m_{2} v_{2}}{m_{1}}$"
      ],
      "text/plain": [
       "(m_1*u_1 + m_2*u_2 - m_2*v_2)/m_1"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v1subs = solveset(momentum, v1).args[0]\n",
    "v1subs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "7ddd7217",
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{m_{1} u_{1} - m_{1} v_{1} + m_{2} u_{2}}{m_{2}}$"
      ],
      "text/plain": [
       "(m_1*u_1 - m_1*v_1 + m_2*u_2)/m_2"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v2subs = solveset(momentum, v2).args[0]\n",
    "v2subs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "91ed38f7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left\\{u_{1}, \\frac{m_{1} u_{1} - m_{2} u_{1} + 2 m_{2} u_{2}}{m_{1} + m_{2}}\\right\\}$"
      ],
      "text/plain": [
       "{u_1, (m_1*u_1 - m_2*u_1 + 2*m_2*u_2)/(m_1 + m_2)}"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solveset(energy.subs(v2, v2subs), v1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "68dc2807",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left\\{u_{2}, \\frac{2 m_{1} u_{1} - m_{1} u_{2} + m_{2} u_{2}}{m_{1} + m_{2}}\\right\\}$"
      ],
      "text/plain": [
       "{u_2, (2*m_1*u_1 - m_1*u_2 + m_2*u_2)/(m_1 + m_2)}"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solveset(energy.subs(v1, v1subs), v2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "f173f290",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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  "kernelspec": {
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  "language_info": {
   "codemirror_mode": {
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