Math Problem Statement

Calculate the air temperature (in degrees Kelvin) if the speed of sound is 353.8 m/s (at standard air pressure).

Solution

The speed of sound in air can be calculated using the following formula:

v=γRTv = \sqrt{\gamma \cdot R \cdot T}

where:

  • vv is the speed of sound,
  • γ\gamma is the adiabatic index (ratio of specific heats, which is 1.4 for air),
  • RR is the specific gas constant for dry air (287 J/(kg·K)),
  • TT is the absolute temperature in Kelvin.

Rearranging the formula to solve for TT:

T=v2γRT = \frac{v^2}{\gamma \cdot R}

Substituting the given values:

v=353.8m/sv = 353.8 \, \text{m/s} γ=1.4\gamma = 1.4 R=287J/(kg\cdotpK)R = 287 \, \text{J/(kg·K)}

T=(353.8)21.4287T = \frac{(353.8)^2}{1.4 \cdot 287}

Calculating this:

T=125173.44401.8T = \frac{125173.44}{401.8}

T311.52KT \approx 311.52 \, \text{K}

Therefore, the air temperature is approximately 311.52K311.52 \, \text{K}.

Would you like any further details or have any questions?

Here are 5 questions you might be interested in:

  1. How does humidity affect the speed of sound in air?
  2. What is the relationship between air pressure and the speed of sound?
  3. How can you calculate the speed of sound at different altitudes?
  4. What are the typical ranges for the speed of sound in various mediums?
  5. How does temperature inversion affect sound propagation?

Tip: When solving physics problems, always double-check unit consistency to ensure accuracy in your calculations.

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Math Problem Analysis

Mathematical Concepts

Physics
Thermodynamics

Formulas

Speed of sound formula: v = sqrt(gamma * R * T)
Rearranged speed of sound formula: T = v^2 / (gamma * R)

Theorems

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Suitable Grade Level

Advanced High School