To determine the velocity at point A on the airfoil, we can use Bernoulli's equation, which for incompressible flows can be expressed as:
\[ P_1 + \frac{1}{2} \rho v_1^2 = P_2 + \frac{1}{2} \rho v_2^2 \]
Where:
- \(P_1\) and \(P_2\) are the pressures at initial and final points (upstream and point A),
- \(\rho\) is the air density,
- \(v_1\) and \(v_2\) are the velocities at the respective points.
Given data are:
- Upstream pressure \(P_1 = 1.01 \times 10^5 , \text{N/m}^2\)
- Upstream velocity \(v_1 = 150 , \text{km/h} = \frac{150 \times 1000, \text{m}}{3600, \text{s}} = 41.67 , \text{m/s}\)
- Air density \(\rho = 1.225 , \text{kg/m}^3\)
- Pressure at point A \(P_2 = 9.95 \times 10^4 , \text{N/m}^2\)
We want to calculate the velocity at point A, denoted as \(v_2\).
Using Bernoulli's equation, we rearrange it to solve for \(v_2\):
\[ P_1 + \frac{1}{2} \rho v_1^2 = P_2 + \frac{1}{2} \rho v_2^2 \]
Substituting the known values:
\[ 1.01 \times 10^5 + \frac{1}{2} \cdot 1.225 \cdot (41.67)^2 = 9.95 \times 10^4 + \frac{1}{2} \cdot 1.225 \cdot v_2^2 \]
Calculating \(\frac{1}{2} \cdot 1.225 \cdot (41.67)^2\):
- Calculate \((41.67)^2 \approx 1736.89\),
- Then calculate \(\frac{1}{2} \cdot 1.225 \cdot 1736.89 \approx 1065.16 , \text{N/m}^2\).
Now substitute this back into the equation:
\[ 1.01 \times 10^5 + 1065.16 = 9.95 \times 10^4 + \frac{1}{2} \cdot 1.225 \cdot v_2^2 \]
Calculating the left side:
\[ 1.01 \times 10^5 + 1065.16 \approx 102155.16 , \text{N/m}^2 \]
Calculating the right side:
\[ 9.95 \times 10^4 = 99500 , \text{N/m}^2 \]
Putting everything together:
\[ 102155.16 = 99500 + \frac{1}{2} \cdot 1.225 \cdot v_2^2 \]
Subtract \(99500\) from both sides:
\[ 102155.16 - 99500 = \frac{1}{2} \cdot 1.225 \cdot v_2^2 \]
\[ 1655.16 = \frac{1.225}{2} \cdot v_2^2 \]
Now, calculating \(\frac{1.225}{2} \approx 0.6125\):
\[ 1655.16 = 0.6125 \cdot v_2^2 \]
Now solve for \(v_2^2\):
\[ v_2^2 = \frac{1655.16}{0.6125} \approx 2702.43 \]
Taking the square root to find \(v_2\):
\[ v_2 \approx \sqrt{2702.43} \approx 52.0 , \text{m/s} \]
Therefore, the velocity at point A is approximately:
\[ \boxed{52.0 , \text{m/s}} \]
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