An aircraft has the following drag polar: 𝐶𝐷 = 0.02 + 0.03𝐶2𝐿 . Determine the minimum drag to lift ratio and the lift coefficient required to achieve it.

To determine the minimum drag to lift ratio (\(\frac{D}{L}\)) and the lift coefficient (\(C_L\)) required to achieve it, we need to analyze the given drag polar equation:

\[ C_D = 0.02 + 0.03C_L^2 \]

Here, \(C_D\) is the drag coefficient and \(C_L\) is the lift coefficient.

### Step-by-Step Solution:

1. **Express the Drag-to-Lift Ratio:**

   The drag-to-lift ratio, \(\frac{D}{L}\), can be expressed in terms of the drag coefficient (\(C_D\)) and lift coefficient (\(C_L\)) as follows:

   \[
   \frac{D}{L} = \frac{C_D}{C_L}
   \]

2. **Substitute \(C_D\) into the Equation:**

   Using the given drag polar equation, substitute \(C_D\):

   \[
   \frac{D}{L} = \frac{0.02 + 0.03C_L^2}{C_L}
   \]

3. **Differentiate \(\frac{D}{L}\) with Respect to \(C_L\):**

   To find the minimum \(\frac{D}{L}\), we need to take the derivative of \(\frac{D}{L}\) with respect to \(C_L\) and set it to zero:

   \[
   \frac{d}{dC_L} \left( \frac{0.02 + 0.03C_L^2}{C_L} \right) = 0
   \]

4. **Differentiate and Simplify:**

   Let's differentiate:

   \[
   \frac{d}{dC_L} \left( \frac{0.02 + 0.03C_L^2}{C_L} \right) = \frac{d}{dC_L} \left( 0.02C_L^{-1} + 0.03C_L \right)
   \]

   Use the power rule for differentiation:

   \[
   \frac{d}{dC_L} \left( 0.02C_L^{-1} + 0.03C_L \right) = -0.02C_L^{-2} + 0.03
   \]

5. **Set the Derivative to Zero:**

   Set the derivative equal to zero to find the critical points:

   \[
   -0.02C_L^{-2} + 0.03 = 0
   \]

6. **Solve for \(C_L\):**

   \[
   0.03 = 0.02C_L^{-2}
   \]

   \[
   C_L^{-2} = \frac{0.03}{0.02} = 1.5
   \]

   \[
   C_L^2 = \frac{1}{1.5} = \frac{2}{3}
   \]

   \[
   C_L = \sqrt{\frac{2}{3}} = \sqrt{\frac{2}{3}} = \sqrt{\frac{2}{3}} \approx 0.816
   \]

7. **Find the Minimum Drag-to-Lift Ratio:**

   Substitute \(C_L\) back into the drag polar equation to find \(C_D\):

   \[
   C_D = 0.02 + 0.03 \left( 0.816 \right)^2 = 0.02 + 0.03 \left( \frac{2}{3} \right) = 0.02 + 0.02 = 0.04
   \]

   Now calculate the minimum drag-to-lift ratio:

   \[
   \frac{D}{L}_{\text{min}} = \frac{C_D}{C_L} = \frac{0.04}{0.816} \approx 0.049
   \]

### Summary:

- The minimum drag-to-lift ratio \(\left( \frac{D}{L} \right)_{\text{min}} \approx 0.049\).
- The lift coefficient required to achieve this ratio is \(C_L \approx 0.816\).

Would you like more details or have any questions about this process?

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Here are 5 numbered questions you might ask next:

1. How does the drag polar equation impact aircraft performance?
2. What is the significance of minimizing the drag-to-lift ratio in aviation?
3. Can you explain how different wing shapes affect the drag polar?
4. How do you calculate the drag coefficient in real-life scenarios?
5. What other factors besides lift and drag should be considered in aircraft design?

**Tip:** Understanding how to minimize the drag-to-lift ratio can significantly improve an aircraft's efficiency and performance, leading to reduced fuel consumption and extended range.

Learn how to determine the minimum drag-to-lift ratio and the required lift coefficient using the drag polar equation CD = 0.02 + 0.03CL^2. Detailed step-by-step solution provided.

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