# Question 1 – Range of Boeing 777 300ER Let's solve **part (a)** step by step, using the given data and the Breguet Range Equation for jet aircraft: --- ## **Given Data** - **Maximum Takeoff Mass (\(W_{i}\))**: 351,000 kg - **Operating Empty Mass**: 169,000 kg - **Maximum Payload Mass**: 68,500 kg - **Thrust Specific Fuel Consumption (\(c\))**: \(155 \times 10^{-6}\) N/N-s - **Zero Lift Drag Coefficient (\(C_{D}\))**: .013 - **Spanwise Efficiency Factor (\(e\))**: .72 - **Wing Area (\(S\))**: 436.8 m² - **Wing Span (\(b\))**: 64.8 m - **Design Flight Altitude**: 12 km - **Lift Slope**: .091 1/deg - **Zero Lift Angle of Attack**: -2.8° - **Gravity (\(g\))**: 9.81 m/s² --- ## **Step 1: Calculate Maximum Fuel Mass** \[ \text{Maximum Fuel Mass} = \text{Maximum Takeoff Mass} - \text{Operating Empty Mass} - \text{Maximum Payload Mass} \] \[ = 351,000\,\text{kg} - 169,000\,\text{kg} - 68,500\,\text{kg} = 113,500\,\text{kg} \] --- ## **Step 2: Initial and Final Weights** - **Initial Weight (\(W_i\))** = Maximum Takeoff Mass = **351,000 kg** - **Final Weight (\(W_f\))** = Operating Empty Mass + Maximum Payload Mass = **169,000 kg + 68,500 kg = 237,500 kg** --- ## **Step 3: Breguet Range Equation (Jet Aircraft, Cruise only)** \[ R = \frac{V}{c} \frac{L}{D} \ln\left(\frac{W_i}{W_f}\right) \] Where: - \(V\) = Cruise speed (to be calculated) - \(c\) = Thrust Specific Fuel Consumption (\(155 \times 10^{-6}\) 1/s) - \(L/D\) = Lift-to-Drag Ratio (to be calculated) - \(W_i\), \(W_f\) as above --- ## **Step 4: Estimate Cruise Speed (\(V\))** At 12 km altitude, the **speed of sound** is about 295 m/s, and typical cruise Mach for 777-300ER is ~.84: \[ V = M \cdot a = .84 \times 295 \approx 248\,\text{m/s} \] --- ## **Step 5: Calculate \(L/D\) (Lift-to-Drag Ratio)** Maximum \(L/D\) occurs at: \[ \left(\frac{L}{D}\right)_{\text{max}} = \frac{1}{2} \sqrt{\frac{\pi e AR}{C_{D}}} \] Where: - \(AR = \frac{b^2}{S} = \frac{(64.8)^2}{436.8} \approx 9.62\) Plugging in values: \[ \left(\frac{L}{D}\right)_{\text{max}} = \frac{1}{2} \sqrt{\frac{\pi \times .72 \times 9.62}{.013}} \] \[ = \frac{1}{2} \sqrt{\frac{21.76}{.013}} \] \[ = \frac{1}{2} \sqrt{1673.8} \approx \frac{1}{2} \times 40.92 \approx 20.46 \] But this is the theoretical maximum. In cruise, **actual \(L/D\)** is typically around 19-20 for the 777-300ER; we'll use **19** for a realistic estimate. --- ## **Step 6: Calculate \(\ln(W_i/W_f)\)** \[ \ln\left(\frac{W_i}{W_f}\right) = \ln\left(\frac{351,000}{237,500}\right) = \ln(1.478) \approx .391 \] --- ## **Step 7: Plug Values into Breguet Range Equation** \[ R = \frac{248}{155 \times 10^{-6}} \times 19 \times .391 \] Calculate step by step: \[ \frac{248}{155 \times 10^{-6}} = \frac{248}{.000155} \approx 1,600,000 \] \[ R = 1,600,000 \times 19 \times .391 \] \[ R = 30,400,000 \times .391 \] \[ R \approx 11,886,400 \text{ meters} \] \[ R \approx 11,886 \text{ km} \] --- ## **Final Answer (Part a)** ### **Estimated Maximum Cruise Range for Boeing 777 300ER (Max Payload):** \[ \boxed{11,900\, \text{km} \quad (\text{rounded to two significant digits})} \] --- ### **Assumptions:** - Only cruise segment considered (per instructions). - Standard atmosphere at 12 km altitude. - Actual \(L/D\) in cruise is assumed to be 19. --- **If you need parts (b)-(d), please provide the questions or instructions for those parts.**