Considering the Bernoulli's effect, air must flow over the upper surface of the wing at a speed of 129.1m/s if the plane is to stay in the air.
Given the data in the question;
- Mass of airplane; [tex]m = 2 * 10^6 kg[/tex]
- Velocity ( air flows past the power surface of the wings); [tex]v = 100m/s[/tex]
- Surface area; [tex]A = 1200m^2[/tex]
We know that the pressure on wings due to gravity must be the same as the diffidence in pressure above and below the wings if the plane is to stay in the air.
That is; Pressure below - Pressure above = Weight/Area
[tex]P - P' = mg/A[/tex]
Now, using Bernoulli's principle
[tex]P + \frac{1}{2} dv^2 = P' + \frac{1}{2} dv'^2[/tex]
We re-arrange
[tex]P-P' = \frac{1}{2} dv'^2 - \frac{1}{2} dv^2[/tex]
so
[tex]\frac{mg}{A} = \frac{1}{2} dv'^2 - \frac{1}{2} dv^2[/tex]
[tex]\frac{mg}{A} = \frac{1}{2} dv'^2 - \frac{1}{2} dv^2\\\\\frac{mg}{A} = \frac{1}{2} d( v'^2 - v^2)[/tex]
We know that density of air; [tex]d = 1.225 kg/m^3[/tex] and acceleration due to gravity; [tex]g = 9.8m/s^2[/tex]
We substitute our values into the equation
[tex]\frac{(2*10^6kg)*9.8m/s^2}{1200m^2} = \frac{1}{2} * 1.225kg/m^3( v'^2 - (100m/s)^2)\\\\\frac{1.96*10^7kg.m/s^2}{1200m^2} = 0.6125kg/m^3 ( v'^2 - (100m/s)^2)\\\\16333.33kg/ms^2 = 0.6125kg/m^3 ( v'^2 - (100m/s)^2)\\\\( v'^2 - (100)^2) = \frac{16333.33kg/ms^2}{0.6125kg/m^3} \\\\( v'^2 - 10000m^2/s^2) = 26666.66m^2/s^2\\\\v'^2 = 26666.66m^2/s^2 - 10000m^2/s^2\\\\v'^2 = 16666.66m^2/s^2\\\\v' = \sqrt{16666.66m^2/s^2}\\\\v' = 129.1 m/s[/tex]
Therefore, considering the Bernoulli's effect, air must flow over the upper surface of the wing at a speed of 129.1m/s if the plane is to stay in the air.
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