Question

At some point on its trip a Concorde aircraft flies 520 m/s at an altitude where the temperature is -56C.


Image courtesy of Dan Davison, CC - BY

Determine the Mach angle (in degrees) of its shockwaves at that instance.

Answer 1

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.