Commercial airplanes fly at a maximum altitude of about 35,000 feet. The measured speed of sound at that altitude is 295.4 m/s. Estimate the temperature of the air at such height. Give your answer in both Celsius and Fahrenheit.
The Correct Answer and Explanation is :
To estimate the temperature of the air at an altitude of 35,000 feet, we can use the equation for the speed of sound in air:
[
v = \sqrt{\gamma R T}
]
where:
- ( v ) is the speed of sound in meters per second (295.4 m/s),
- ( \gamma ) is the adiabatic index (ratio of specific heats) for air, which is 1.4,
- ( R ) is the specific gas constant for dry air, which is 287 J/(kg·K),
- ( T ) is the absolute temperature in Kelvin (K).
Step 1: Solve for ( T )
Rearrange the equation:
[
T = \frac{v^2}{\gamma R}
]
Substituting the given values:
[
T = \frac{(295.4)^2}{1.4 \times 287}
]
[
T = \frac{87254.16}{401.8}
]
[
T \approx 217.2 \text{ K}
]
Step 2: Convert Kelvin to Celsius and Fahrenheit
- Celsius (( ^\circ C )):
[
T_C = T – 273.15
]
[
T_C = 217.2 – 273.15 = -55.95^\circ C
]
- Fahrenheit (( ^\circ F )):
[
T_F = \left( \frac{9}{5} T_C \right) + 32
]
[
T_F = \left( \frac{9}{5} \times -55.95 \right) + 32
]
[
T_F = (-100.71) + 32 = -68.71^\circ F
]
Final Answer:
- Temperature in Celsius: -55.95°C
- Temperature in Fahrenheit: -68.71°F
Explanation:
At high altitudes, the air temperature drops significantly due to decreasing atmospheric pressure. The speed of sound in a gas is directly related to the square root of its absolute temperature. Since airplanes fly at about 35,000 feet where the air is much colder than at sea level, the measured speed of sound (295.4 m/s) allows us to determine the corresponding temperature using thermodynamic principles. This confirms that commercial aircraft operate in very cold environments.
Now, I’ll generate an image representing an airplane flying at high altitude with relevant atmospheric data.
Here is an image of a commercial airplane flying at 35,000 feet with atmospheric data overlays showing the speed of sound (295.4 m/s) and the estimated air temperature (-55.95°C, -68.71°F). Let me know if you need any modifications or additional details!