Question

A commercial aircraft flies at a Mach number of 0.67 at an altitude where the air temperature is -19.15 C. Assume ISA conditions.

Determine the altitude (in metres) this aircraft flies at.

Answer 1

To determine the altitude at which the aircraft flies given a Mach number of 0.67 and an air temperature of -19.15 °C, we can use the relationship between altitude, temperature, and the speed of sound. The speed of sound in air can be calculated using the formula:

\[
a = \sqrt{\gamma \cdot R \cdot T}
\]

Where:
- \( a \) is the speed of sound (m/s),
- \( \gamma \) is the specific heat ratio (approximately 1.4 for air),
- \( R \) is the specific gas constant for dry air (approximately \( 287 \, \text{J/(kg·K)} \)),
- \( T \) is the absolute temperature in Kelvin.

First, we need to convert the temperature from Celsius to Kelvin:

\[
T(K) = -19.15 + 273.15 = 253.00 \, K
\]

Next, we can calculate the speed of sound at this temperature:

\[
a = \sqrt{1.4 \cdot 287 \cdot 253.00} \approx \sqrt{101,787.4} \approx 318.71 \, m/s
\]

Next, we can find the true airspeed (\( v \)) of the aircraft using its Mach number:

\[
v = M \cdot a = 0.67 \cdot 318.71 \approx 213.64 \, m/s
\]

Now, we need to determine the altitude where the temperature is -19.15 °C under ISA conditions. Under ISA, the standard temperature decreases with altitude at a rate of approximately 6.5 °C per kilometer up to about 11,000 meters. The formula for the temperature as a function of altitude \( h \) in meters is given by:

\[
T(h) = T_0 - L \cdot h
\]

Where:
- \( T_0 \) is the standard temperature at sea level (approximately 288.15 K or 15 °C),
- \( L \) is the lapse rate (approximately 0.0065 K/m).

The temperature at the altitude \( h \) can be set equal to the given temperature of -19.15 °C:

\[
-19.15 + 273.15 = 288.15 - 0.0065h
\]

Simplifying:

\[
253 = 288.15 - 0.0065h
\]

Now we can rearrange to solve for \( h \):

\[
0.0065h = 288.15 - 253 \\
0.0065h = 35.15 \\
h = \frac{35.15}{0.0065} \approx 5400 \, m
\]

Therefore, the altitude at which the aircraft flies is approximately **5400 meters**.