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   "source": [
    "# Projectile Motion Equations\n",
    "\n",
    "This notebook derives the equations for the projectile's position, velocity and acceleration as a function of the trebuchet's angle parameters"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "1",
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import Function, cos, diff, simplify, sin, symbols\n",
    "\n",
    "# Define the trebuchet angle variables (time-dependent)\n",
    "t = symbols(\"t\")\n",
    "theta = Function(\"theta\")(t)  # Arm angle\n",
    "psi = Function(\"psi\")(t)  # Sling angle\n",
    "\n",
    "# Define the trebuchet lengths\n",
    "l2, l3 = symbols(\"l2 l3\")  # Length of arm and sling\n",
    "\n",
    "# Define the position of the projectile\n",
    "px = -l2 * cos(theta) + l3 * cos(psi)\n",
    "py = -l2 * sin(theta) + l3 * sin(psi)\n",
    "\n",
    "# Derive the velocity components by differentiating position with respect to time\n",
    "vx = simplify(diff(px, t))\n",
    "vy = simplify(diff(py, t))\n",
    "\n",
    "# Derive the acceleration components by differentiating velocity with respect to time\n",
    "ax = simplify(diff(vx, t))\n",
    "ay = simplify(diff(vy, t))\n",
    "\n",
    "# Output the results\n",
    "print(\"Projectile Position:\")\n",
    "print(f\"px: {px}\")\n",
    "print(f\"py: {py}\\n\")\n",
    "print(\"Projectile Velocity:\")\n",
    "print(f\"vx: {vx}\")\n",
    "print(f\"vy: {vy}\\n\")\n",
    "print(\"Projectile Acceleration:\")\n",
    "print(f\"ax: {ax}\")\n",
    "print(f\"ay: {ay}\")"
   ]
  }
 ],
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