To determine the Mach angle of a Concorde aircraft flying at the given conditions, we first need to calculate the speed of sound at the altitude where the temperature is -56°C.
The speed of sound in air can be calculated using the following formula:
\[ c = 331.3 + (0.6 \times T) \]
Where:
- \(c\) is the speed of sound in meters per second (m/s)
- \(T\) is the temperature in degrees Celsius (°C)
Step 1: Calculate the speed of sound
Given:
- \(T = -56°C\)
Plugging in the temperature:
\[ c = 331.3 + (0.6 \times -56) = 331.3 - 33.6 = 297.7 , \text{m/s} \]
Step 2: Calculate the Mach number
The Mach number \(M\) is defined as the ratio of the speed of the aircraft to the speed of sound:
\[ M = \frac{V}{c} \]
Where:
- \(V\) is the speed of the aircraft (520 m/s)
- \(c\) is the speed of sound at the given temperature (297.7 m/s)
Calculating the Mach number:
\[ M = \frac{520}{297.7} \approx 1.747 \]
Step 3: Calculate the Mach angle
The Mach angle \(\mu\) is given by the formula:
\[ \mu = \arcsin\left(\frac{1}{M}\right) \]
Calculating the Mach angle:
\[ \mu = \arcsin\left(\frac{1}{1.747}\right) \]
Calculating:
\[ \frac{1}{1.747} \approx 0.573 \]
Now, we find the angle:
\[ \mu \approx \arcsin(0.573) \approx 34.8° \]
Conclusion
The Mach angle of the shockwaves at that instance is approximately \(\boxed{34.8}\) degrees.