Chapter 10.1, Problem 7R

To determine

the time taken by a sound wave to travel $1.0\text{km}$ through air at temperature of $20°\text{C}$ .

Answer to Problem 7R

The required answer is $\boxed{t=2.925\text{ seconds}}$ .

Explanation of Solution

Given:

Wavelength, $\lambda =1.0\text{ km}$

Temperature, $T={20}^0\text{ C}$ and

Speed of sound in air, $v=330\text{ m/s}$

v = 330 m/s

Formula used:

For conversion of $T°\text{C}$ (degree Celsius) to $T\text{K}$ ,

$T=T^0\text{ C}+273\text{ K}$ …… (1)

The expression for speed/velocity with temperature is,

$v\text{'}=v\sqrt{\frac{T}{273K}}$…… (2)

Here, $v^{\prime }$ is the speed of sound at temperature $T$ , $v$is the speed of sound in air, $T°\text{C}$ is the temperature at $20°\text{C}$

The expression for speed with wavelength and time is,

$v\text{'}=\frac{\lambda }{t}$ …… (3)

Here, $\lambda$ is the wavelength or distance of travel by sound, $t$ is the time required.

Calculation:

Substitute $20°\text{C}$ for $T°\text{C}$ in equation (1)

$\begin{array}{l}T={20}^0\text{ C}+273\text{ K}\\ \text{T=293 K}\end{array}$

Substitute $330\text{ m/s}$ for $v$ , $293\text{K}$ for $T$ in equation (2),

$\begin{array}{l}v\text{'}=(330\text{ m/s)}\sqrt{\frac{293}{273}}\\ v\text{'}=341.87\text{ m/s}\end{array}$

Substitute $341.87\text{m/s}$ for $v$ and $1.0\text{km}$ for $\lambda$ in equation (3)

$\begin{array}{l}341.87=\frac{1.0\times 1000}{t}\\ t=\frac{1000}{341.87}=2.925\\ t=2.925\text{ seconds}\end{array}$

Conclusion:

Hence, the required time is, $\boxed{t=2.925\text{ seconds}}$ .