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.
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