Question

A person with their head sideways in the water sees a beaver slap its tail on the lake surface 1.2 km away. What is the time difference between the sound heard in the ear listening in the air and the ear hearing the sound transmitted through the water? The speed of sound in water is 1 498 m/s and the air temperature is 30 °C.

Answer 1

Given data:

Distance between the sound source and the ear of the listener: d = 1.2 km

Speed of sound in water: V(water) = 1498 m/s

To find:

Time difference between the speed of sound in air and the speed of sound in water.

Step-by-step explanation:

The speed of sound in air at temperature T is given by:

`V(air)=331+0.59T`

Here, T is the temperature and V(air) is the speed of sound in air measured in m/s.

Substitute the given values in above equation, we get:

`\begin{gathered} V(air)=331+0.59*30 \\ V(air)=348.7\text{ m/s} \end{gathered}`

Thus, the speed of sound in air at 30°C is 348.7 m/s.

The time is taken by sound in air to travel a distance of 1.2 km is:

`t(air)=(d)/(V(air))`

Substitute the values and simplify the above equation, we get:

`\begin{gathered} t(air)=\frac{1.2\text{ km}(\frac{1000\text{ m}}{\text{km}})}{348.7\text{ m/s}} \\ t(air)=\frac{1200\text{ m}}{348.7\text{ m/s}} \\ t(air)=3.44\text{ s} \end{gathered}`

Thus, the time taken by sound to travel in air is 3.44 seconds.

Now the speed of sound in water is given. The time taken by sound to travel a distance of 1.2 km in water is:

`t(water)=(d)/(V(water))`

Substitute the values and simplify the above equation, we get:

`\begin{gathered} t(water)=\frac{1.2\text{ km}\lbrack(\frac{1000\text{ m}}{\text{km}})}{1498\text{ m/s}} \\ t(water)=\frac{1200\text{ m}}{1498\text{ m/s}} \\ t(water)=0.80\text{ s} \end{gathered}`

The time taken by sound to travel through water is 0.80 seconds.

The time difference between the speed of sound in air and speed of sound in water is:

`\begin{gathered} t=t(air)-t(water) \\ t=3.44\text{ s }-0.80\text{ s} \\ t=2.64\text{ s} \end{gathered}`

Final answer:

The difference between the time taken by sound to travel in air and the time taken by sound to travel in water is 2.64 seconds.