chapter_3/simulation_of_exterior_ballistics.ipynb

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    "1.Start with the initial values: \n",
    "velocity (v₀), angle of elevation (el), position (x₀, y₀), and initialize other parameters such as k, m, and n.\n",
    "\n",
    "2.Set a small time interval Δt for simulation accuracy.\n",
    "\n",
    "3.Inside a loop until the projectile touches the ground (y ≤ 0), perform the following calculations:\n",
    "\n",
    "3.1. Calculate the change in velocity (Δv) using the equation:\n",
    "Δv = -g * sin(theta) - (k * v^n) / m\n",
    "\n",
    "3.2. Calculate the change in angle (Δtheta) using the equation:\n",
    "Δtheta = -g * cos(theta) / v\n",
    "\n",
    "3.3. Calculate the change in position in the x-direction (Δx) using the equation:\n",
    "Δx = v * cos(theta) * Δt\n",
    "\n",
    "3.4. Calculate the change in position in the y-direction (Δy) using the equation:\n",
    "Δy = v * sin(theta) * Δt\n",
    "\n",
    "3.5. Update the values of velocity, angle, and position:\n",
    "v = v + Δv\n",
    "theta = theta + Δtheta\n",
    "x = x + Δx\n",
    "y = y + Δy\n",
    "\n",
    "4.Repeat steps 3 until the projectile touches the ground (y ≤ 0), which means the simulation has ended."
   ]
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  {
   "cell_type": "code",
   "execution_count": null,
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    "import math\n",
    "\n",
    "def simulate_projectile(v0, el, k, m, n):\n",
    "    # Convert the angle of elevation from degrees to radians\n",
    "    theta = math.radians(el)\n",
    "    \n",
    "    # Initialize variables\n",
    "    x = 0\n",
    "    y = 0\n",
    "    g = 9.8  # Acceleration due to gravity\n",
    "    \n",
    "    # Set a small time interval for simulation accuracy\n",
    "    dt = 0.01\n",
    "    \n",
    "    # Lists to store the trajectory points\n",
    "    trajectory_x = [x]\n",
    "    trajectory_y = [y]\n",
    "    \n",
    "    # Perform simulation until the projectile touches the ground\n",
    "    while y >= 0:\n",
    "        # Calculate the change in velocity\n",
    "        dv = -g * math.sin(theta) - (k * math.pow(v0, n) / m)\n",
    "        \n",
    "        # Calculate the change in angle\n",
    "        dtheta = -g * math.cos(theta) / v0\n",
    "        \n",
    "        # Calculate the change in position\n",
    "        dx = v0 * math.cos(theta) * dt\n",
    "        dy = v0 * math.sin(theta) * dt\n",
    "        \n",
    "        # Update velocity, angle, and position\n",
    "        v0 += dv * dt\n",
    "        theta += dtheta * dt\n",
    "        x += dx\n",
    "        y += dy\n",
    "        \n",
    "        # Add the current position to the trajectory\n",
    "        trajectory_x.append(x)\n",
    "        trajectory_y.append(y)\n",
    "    \n",
    "    return trajectory_x, trajectory_y\n",
    "\n",
    "# Example usage\n",
    "v0 = 100  # Initial velocity in m/s\n",
    "el = 45  # Angle of elevation in degrees\n",
    "k = 0.1  # Drag coefficient\n",
    "m = 30  # Mass of the projectile in kg\n",
    "n = 0.001  # Power of velocity for air resistance\n",
    "\n",
    "trajectory_x, trajectory_y = simulate_projectile(v0, el, k, m, n)\n",
    "\n",
    "# Print the trajectory points\n",
    "for i in range(len(trajectory_x)):\n",
    "    print(f\"Point {i+1}: x = {trajectory_x[i]}, y = {trajectory_y[i]}\")\n"
   ]
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    "The program can be further modified to find different parameter of the projectile, as the maximum height reached, the distance traveled and the time of flight. A range table for the above projectile, with 0 varying from 30° to 80° and velocity varying from 100 to 150 m/sec and range in meters is given below:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import math\n",
    "\n",
    "def projectile_trajectory():\n",
    "    # Constants and initial values\n",
    "    m = 30\n",
    "    c = 0.001\n",
    "    delt = 0.01\n",
    "    g = 9.81\n",
    "    pi = math.pi\n",
    "    \n",
    "    print(\"\\nRange Table\")\n",
    "    print(\"Mass of Projectile = 30.00 kg Release Velocity (m/sec)\")\n",
    "    print(\"\\nRelease Angle\\t100\\t\\t110\\t\\t120\\t\\t130\\t\\t140\\t\\t150\")\n",
    "\n",
    "    for angle in range(30, 81, 5):\n",
    "        print(angle, end=\"\\t\\t\")\n",
    "        \n",
    "        for v in range(100, 151, 10):\n",
    "            x = 0\n",
    "            y = 0.0001\n",
    "            theta = math.radians(angle)\n",
    "            max_height = 0\n",
    "            distance = 0\n",
    "            time_of_flight = 0\n",
    "            \n",
    "            while y >= 0:\n",
    "                # Calculate changes in velocity, angle, and position\n",
    "                delv = (-g * math.sin(theta) - c * v**2 / m) * delt\n",
    "                dtheta = -g * math.cos(theta) * delt / v\n",
    "                delx = v * math.cos(theta) * delt\n",
    "                dely = v * math.sin(theta) * delt\n",
    "                \n",
    "                # Update velocity, angle, and position\n",
    "                v += delv\n",
    "                theta += dtheta\n",
    "                x += delx\n",
    "                y += dely\n",
    "                \n",
    "                # Update maximum height\n",
    "                if y > max_height:\n",
    "                    max_height = y\n",
    "\n",
    "                # Update distance traveled\n",
    "                distance = x\n",
    "\n",
    "                # Update time of flight\n",
    "                time_of_flight += delt\n",
    "\n",
    "            print(f\"{distance:.2f}\", end=\"\\t\\t\")\n",
    "\n",
    "        print()\n",
    "\n",
    "projectile_trajectory()\n"
   ]
  }
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