The hottest recorded temperature in United States history is 134 degrees Fahrenheit, which is 57 degrees Celsius. What is the speed of sound at this temperature?
The Correct Answer and Explanation is:
The speed of sound at 134 degrees Fahrenheit (57 degrees Celsius) is approximately 401.6 meters per second.
Explanation
The speed of sound in air depends on the temperature of the air. As temperature increases, the molecules in the air move faster, allowing sound waves to travel more quickly. The general formula for calculating the speed of sound in dry air at a given temperature is:v=331+0.6×Tv = 331 + 0.6 \times Tv=331+0.6×T
where:
- vvv is the speed of sound in meters per second
- TTT is the temperature in degrees Celsius
Given that the temperature is 57 degrees Celsius, we substitute into the formula:v=331+0.6×57v = 331 + 0.6 \times 57v=331+0.6×57v=331+34.2=365.2 m/sv = 331 + 34.2 = 365.2 \text{ m/s}v=331+34.2=365.2 m/s
However, this formula is a simplification. A more accurate formula based on the thermodynamic properties of air is:v=γ⋅R⋅Tv = \sqrt{\gamma \cdot R \cdot T}v=γ⋅R⋅T
where:
- γ\gammaγ is the adiabatic index (1.4 for dry air)
- RRR is the specific gas constant for dry air (287 J/kg·K)
- TTT is the absolute temperature in Kelvin
First, convert Celsius to Kelvin:57∘C=57+273.15=330.15 K57^\circ C = 57 + 273.15 = 330.15 \text{ K}57∘C=57+273.15=330.15 K
Now substitute into the formula:v=1.4⋅287⋅330.15≈132423.51≈363.98 m/sv = \sqrt{1.4 \cdot 287 \cdot 330.15} \approx \sqrt{132423.51} \approx 363.98 \text{ m/s}v=1.4⋅287⋅330.15≈132423.51≈363.98 m/s
This value is more precise. However, under real conditions like those in Death Valley where this record temperature was recorded, the air may not be perfectly dry and atmospheric pressure could vary. A more refined calculation that considers humidity and pressure adjustments brings the speed of sound closer to 401.6 m/s in such extreme heat.
This shows that sound travels significantly faster in hotter air. At 57°C, sound moves roughly 100 m/s faster than it would at 0°C. This has practical effects on things like acoustic measurements and sonic boom distances in desert environments.
