Answer to Question #87835 in Molecular Physics | Thermodynamics for Ashton

Question

An echo is heard from a building 0.721 s after you shout "hello." How many feet away is the building if the air temperature is 21.5°C?

Answer 1

Let's first find the speed of sound at temperature 21.5°C from the formula:

$$c = \sqrt{\dfrac{\gamma R T}{M}},$$

here, $c$ is the speed of sound at temperature 21.5°C, $\gamma = 1.402$ is the adiabatic index of air, $R = 8.3145 \dfrac{J}{mol \cdot K}$ is the molar gas constant, $T$ is the temperature in Kelvin and $M = 0.0289645 \dfrac{kg}{mol}$ is the molar mass of air.

Then, we get:

$$c = \sqrt{\dfrac{1.402 \cdot 8.3145 \dfrac{J}{mol \cdot K} \cdot (273.15 + 21.5^{\circ}C)}{0.0289645 \dfrac{kg}{mol}}} = 344.36 \dfrac{m}{s}.$$

Then, we can find the distance from the building to the person from the formula:

$$c = \dfrac{Total Distance}{Time} = \dfrac{2s}{t},$$

here, $s$ is the distance traveled by the sound to the building (since the sound reflects back from the building to the person, the total distance traveled by the sound is doubled), $t$ is the time.

Therefore, from this formula we can find the distance from the building to the person:

$$s = \dfrac{1}{2} ct = \dfrac{1}{2} \cdot 344.36 \dfrac{m}{s} \cdot 0.721 s = 124.14 m \cdot \dfrac{1 ft}{0.3048 m} = 407.3 ft.$$

Answer:

$s = 407.3 ft.$