Chapter 5, Problem 16PEB

How much time is required for a sound to travel 1 mile (5,280.0 ft) if the air temperature is:

a. 0.0°C?

b. 20.0°C?

c. 40.0°C?

d. 80.0°C?

To determine

The time required for the sound wave to travel $1\text{ mile}$ or $5280.0\text{ ft}$ if the temperature of the airis $0.0°\text{ C}$.

Answer to Problem 16PEB

Solution:

$4.9\text{ s}$

Explanation of Solution

Given data:

The distance travelled by the sound is $5280.0\text{ ft}$. Also, the temperature of the airis $0.0°\text{ C}$.

Formula used:

Write the expression for the time.

$t=\frac{d}{v}$

Here, $d$ and $v$ are the distance travelled and velocity respectively.

Explanation:

Refer to the table 5.1 given inchapter 5. The velocity of sound at $0.0°\text{ C}$ is given as $1087\text{ ft/s}$. Calculate the time required by the sound to travel $5280.0\text{ ft}$.

$t=\frac{d}{v_P}$

Substitute $5280.0\text{ ft}$ for $d$, and $1087\text{ ft/s}$ for $v_P$.

$\begin{array}{c}t=\frac{5280.0\text{ ft}}{1087\text{ ft/s}}\\ =4.9\text{ s}\end{array}$

Conclusion:

The time required by the sound to travel $1\text{ mile}$ at the given conditionis $4.9\text{ s}$.

To determine

The time required for the sound wave to travel $1\text{ mile}$ or $5280.0\text{ ft}$ if the temperature of the airis $20.0°\text{ C}$.

Answer to Problem 16PEB

Solution:

$4.7\text{ s}$

Explanation of Solution

Given data:

The distance travelled by the sound is $5280.0\text{ ft}$. Also, the temperature of the airis $20.0°\text{ C}$.

Formula used:

Write the expression for the velocity of sound at the given temperature.

$v_{T_P}=v_0+\left(\frac{2.00\text{ ft/s}}{°\text{ C}}\right)\left(T_P\right)$

Here, $v_0$ is the velocity of sound at $0°\text{ C}$, and $T_P$ is the given temperature.

Write the expression for the time.

$t=\frac{d}{v}$

Here, $d$ and $v$ are the distance travelled and velocity respectively.

Explanation:

The velocity of sound at $0°\text{ C}$ is $1087\text{ ft/s}$. Calculate the velocity of sound if the air temperature is $20.0°\text{ C}$.

$v_{20°\text{ C}}=v_0+\left(\frac{2.00}{°\text{ C}}\right)\left(T_P\right)$

Here, $v_{20°\text{ C}}$ is the velocity of sound at air temperature $20.0°\text{ C}$.

Substitute $1087\text{ ft/s}$ for $v_0$, and $20.0°\text{ C}$ for $T_P$.

$\begin{array}{c}{v}_{20°\text{ C}}=\left(1087\text{ ft/s}\right)+\left(\frac{2.00\text{ ft/s}}{°\text{ C}}\right)\left(20.0°\text{ C}\right)\\ =1127\text{ ft/s}\end{array}$

Calculate the time required by the sound to travel $5280.0\text{ ft}$ if the temperature of the airis $20.0°\text{ C}$.

$t=\frac{d}{v_{20°\text{ C}}}$

Substitute $5280.0\text{ ft}$ for $d$, and $1127\text{ ft/s}$ for $v_{20°\text{ C}}$.

$\begin{array}{c}t=\frac{5280.0\text{ ft}}{1127\text{ ft/s}}\\ =4.7\text{ s}\end{array}$

Conclusion:

The time required by the sound to travel $1\text{ mile}$ at the given condition is $4.7\text{ s}$.

To determine

The time required for the sound wave to travel $1\text{ mile}$ or $5280.0\text{ ft}$ if the temperature of the airis $40.0°\text{ C}$.

Answer to Problem 16PEB

Solution:

$4.5\text{ s}$

Explanation of Solution

Given data:

The distance travelled by the sound is $5280.0\text{ ft}$. Also, the temperature of the airis $40.0°\text{ C}$.

Formula used:

Write the expression for the velocity of sound at the given temperature.

$v_{T_P}=v_0+\left(\frac{2.00\text{ ft/s}}{°\text{ C}}\right)\left(T_P\right)$

Here, $v_0$ is the velocity of sound at $0°\text{ C}$, and $T_P$ is the given temperature.

Write the expression for the time.

$t=\frac{d}{v}$

Here, $d$ and $v$ are the distance travelled and velocity respectively.

Explanation:

The velocity of sound at $0°\text{ C}$ is $1087\text{ ft/s}$. Calculate the velocity of sound at an air temperature of $40.0°\text{ C}$.

$v_{40°\text{ C}}=v_0+\left(\frac{2.00}{°\text{ C}}\right)\left(T_P\right)$

Here, $v_{40°\text{ C}}$ is the velocity of sound at $40.0°\text{ C}$ of air temperature.

Substitute $1087\text{ ft/s}$ for $v_0$, and $40.0°\text{ C}$ for $T_P$.

$\begin{array}{c}{v}_{20°\text{ C}}=\left(1087\text{ ft/s}\right)+\left(\frac{2.00\text{ ft/s}}{°\text{ C}}\right)\left(40.0°\text{ C}\right)\\ =1167\text{ ft/s}\end{array}$

Calculate the time required by the sound to travel $5280.0\text{ ft}$ if the air temperature is $40.0°\text{ C}$.

$t=\frac{d}{v_{40°\text{ C}}}$

Substitute $5280.0\text{ ft}$ for $d$, and $1167\text{ ft/s}$ for $v_{40°\text{ C}}$.

$\begin{array}{c}t=\frac{5280.0\text{ ft}}{1167\text{ ft/s}}\\ =4.5\text{ s}\end{array}$

Conclusion:

The time required by the sound to travel $1\text{ mile}$ at the given conditionis $4.5\text{ s}$.

To determine

The time required for the sound wave to travel $1\text{ mile}$ or $5280.0\text{ ft}$ if the temperature of the airis $80.0°\text{ C}$.

Answer to Problem 16PEB

Solution:

$4.2\text{ s}$

Explanation of Solution

Given data:

The distance travelled by the sound is $5280.0\text{ ft}$. Also, the temperature of the airis $80.0°\text{ C}$.

Formula used:

Write the expression for the velocity of sound at the given temperature.

$v_{T_P}=v_0+\left(\frac{2.00\text{ ft/s}}{°\text{ C}}\right)\left(T_P\right)$

Here, $v_0$ is the velocity of sound at $0°\text{ C}$, and $T_P$ is the given temperature.

Write the expression for the time.

$t=\frac{d}{v}$

Here, $d$ and $v$ are the distance travelled and velocity respectively.

Explanation:

The velocity of sound at $0°\text{ C}$ is $1087\text{ ft/s}$. Calculate the velocity of sound at an air temperature of $80.0°\text{ C}$.

$v_{80°\text{ C}}=v_0+\left(\frac{2.00}{°\text{ C}}\right)\left(T_P\right)$

Here, $v_{80°\text{ C}}$ is the velocity of sound at $80.0°\text{ C}$ of air temperature.

Substitute $1087\text{ ft/s}$ for $v_0$, and $80.0°\text{ C}$ for $T_P$.

$\begin{array}{c}{v}_{80°\text{ C}}=\left(1087\text{ ft/s}\right)+\left(\frac{2.00\text{ ft/s}}{°\text{ C}}\right)\left(80.0°\text{ C}\right)\\ =1247\text{ ft/s}\end{array}$

Calculate the time required by the sound to travel $5280.0\text{ ft}$ if the air temperature is $80.0°\text{ C}$.

$t=\frac{d}{v_{80°\text{ C}}}$

Substitute $5280.0\text{ ft}$ for $d$, and $1247\text{ ft/s}$ for $v_{80°\text{ C}}$.

$\begin{array}{c}t=\frac{5280.0\text{ ft}}{1247\text{ ft/s}}\\ =4.2\text{ s}\end{array}$

Conclusion:

The time required by the sound to travel $1\text{ mile}$ at the given conditionis $4.2\text{ s}$.