To solve the problem, we will follow these steps:
### Step 1: Write down the given data
- Mass of the aircraft, \( m = 4 \times 10^5 \, \text{kg} \)
- Total wing area, \( A = 500 \, \text{m}^2 \)
- Speed of the aircraft, \( v = 720 \, \text{km/h} \)
- Density of air, \( \rho = 1.2 \, \text{kg/m}^3 \)
- Acceleration due to gravity, \( g = 10 \, \text{m/s}^2 \)
### Step 2: Convert the speed from km/h to m/s
To convert the speed from kilometers per hour to meters per second, we use the conversion factor:
\[
1 \, \text{km/h} = \frac{1}{3.6} \, \text{m/s}
\]
Thus,
\[
v = 720 \, \text{km/h} = \frac{720}{3.6} = 200 \, \text{m/s}
\]
### Step 3: Calculate the weight of the aircraft
The weight \( W \) of the aircraft can be calculated using the formula:
\[
W = mg
\]
Substituting the values:
\[
W = 4 \times 10^5 \, \text{kg} \times 10 \, \text{m/s}^2 = 4 \times 10^6 \, \text{N}
\]
### Step 4: Calculate the pressure difference using the weight and area
The pressure difference \( \Delta P \) that balances the weight of the aircraft is given by:
\[
\Delta P = \frac{W}{A}
\]
Substituting the values:
\[
\Delta P = \frac{4 \times 10^6 \, \text{N}}{500 \, \text{m}^2} = 8000 \, \text{N/m}^2
\]
### Step 5: Apply Bernoulli's equation
According to Bernoulli's equation, we have:
\[
P_1 + \frac{1}{2} \rho v_1^2 = P_2 + \frac{1}{2} \rho v_2^2
\]
Rearranging gives us:
\[
P_1 - P_2 = \frac{1}{2} \rho v_2^2 - \frac{1}{2} \rho v_1^2
\]
This can be expressed as:
\[
\Delta P = \frac{1}{2} \rho (v_2^2 - v_1^2)
\]
### Step 6: Substitute for \( v_2 - v_1 \)
Using the identity \( a^2 - b^2 = (a-b)(a+b) \), we can express:
\[
\Delta P = \frac{1}{2} \rho (v_2 - v_1)(v_2 + v_1)
\]
Let \( v_{\text{avg}} = \frac{v_1 + v_2}{2} \). Then, we can write:
\[
v_2 - v_1 = \frac{2 \Delta P}{\rho (v_2 + v_1)}
\]
### Step 7: Calculate \( v_{\text{avg}} \)
Since \( v_{\text{avg}} \) is approximately equal to the speed of the aircraft:
\[
v_{\text{avg}} = 200 \, \text{m/s}
\]
### Step 8: Substitute values to find \( v_2 - v_1 \)
Now substituting the values into the equation:
\[
v_2 - v_1 = \frac{2 \Delta P}{\rho v_{\text{avg}}}
\]
Substituting \( \Delta P = 8000 \, \text{N/m}^2 \), \( \rho = 1.2 \, \text{kg/m}^3 \), and \( v_{\text{avg}} = 200 \, \text{m/s} \):
\[
v_2 - v_1 = \frac{2 \times 8000}{1.2 \times 200} = \frac{16000}{240} = 66.67 \, \text{m/s}
\]
### Step 9: Calculate the fractional increase in speed
The fractional increase in speed is given by:
\[
\text{Fractional increase} = \frac{v_2 - v_1}{v_1}
\]
Assuming \( v_1 \) is approximately \( 200 \, \text{m/s} \):
\[
\text{Fractional increase} = \frac{66.67}{200} = 0.3335
\]
### Final Answer
The fractional increase in the speed of the air on the upper surface of its wings relative to the lower surface is approximately \( 0.3335 \).
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