To determine the altitude where a commercial aircraft flies at a Mach number of 0.67 with an air temperature of -19.15 °C, we need to relate temperature to altitude based on the International Standard Atmosphere (ISA) model.
Step 1: Determine the altitude based on the temperature
In the ISA model, the temperature decreases linearly with altitude in the troposphere. The temperature at sea level (0 m) is typically 15 °C and decreases at about 6.5 °C per 1000 meters until the tropopause.
Let's denote the altitude \( h \) in meters, and the formula relating temperature and altitude in the troposphere is: \[ T = T_0 - L \cdot h \] Where:
- \( T_0 \) is the sea-level standard temperature (15 °C = 288.15 K),
- \( L \) is the temperature lapse rate (6.5 °C per 1000 m or 0.0065 °C per m),
- \( T \) is the air temperature in °C at altitude \( h \).
We can convert the equations into Kelvin: \[ T = 288.15 - (6.5 \times 0.001) \cdot h \quad \text{(and recall: \( 1 °C = 1 K\) for the purpose of lapse rates)} \]
Given \( T = -19.15 °C = 288.15 - 19.15 = 269.00 K \), we can equate and solve for \( h \): \[ 269.00 = 288.15 - 0.0065 \cdot h \] \[ 0.0065 \cdot h = 288.15 - 269.00 \] \[ 0.0065 \cdot h = 19.15 \] \[ h = \frac{19.15}{0.0065} \] \[ h ≈ 2944.62 \text{ m} \]
Step 2: Calculate equivalent airspeed (EAS) in feet per second
Next, we can calculate the equivalent airspeed (EAS). The formula to convert Mach number to equivalent airspeed is:
\[ EAS = M \times a \]
Where:
- \( M \) is the Mach number (0.67),
- \( a \) is the speed of sound at the given temperature.
The speed of sound can be calculated using the formula: \[ a = \sqrt{\gamma \cdot R \cdot T} \] Where:
- \( \gamma \) (ratio of specific heats) is approximately \( 1.4 \) for air,
- \( R \) (specific gas constant for air) is approximately \( 287.05 , \text{J/(kg·K)} \),
- \( T \) is in Kelvin (269.00 K).
Now calculating \( a \): \[ a = \sqrt{1.4 \cdot 287.05 \cdot 269.00} \] \[ a \approx \sqrt{1.4 \cdot 287.05 \cdot 269.00} \approx \sqrt{107,062.51} \approx 327.08 , \text{m/s} \]
Now calculating the equivalent airspeed (EAS): \[ EAS = 0.67 \times 327.08 \approx 219.14 , \text{m/s} \]
Convert EAS to feet per second
To convert from m/s to feet per second, use the conversion factor \( 1 , \text{m/s} \approx 3.28084 , \text{ft/s} \): \[ EAS \approx 219.14 , \text{m/s} \times 3.28084 \approx 719.79 , \text{ft/s} \]
Summary of Results
- Altitude: approximately \( 2944.62 , \text{m} \)
- Equivalent Airspeed: approximately \( 719.79 , \text{ft/s} \)
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