Question

On a dry day, sound travels a certain distance in $ 10\sec $ . What is the time required for sound to travel the same distance on a humid day. The densities of dry air and humid air are in the ratio $ 3:2 $ :
A. $ 8.165\sec $
B. $ 7.165\sec $
C. $ 5.165\sec $
D. $ 4.165\sec $

Answer 1

Hint: You can start by defining what speed of sound means. Then use the equation $ {V_s} = \sqrt {\dfrac{{\gamma p}}{\rho }} $ and compare the speed of sound in both dry air and humid air. Then use the equation $ V = \dfrac{d}{t} $ for both the dry air and humid air and compare them to calculate the time taken by sound in dry air.

Complete step-by-step answer:
The distance travelled by a sound wave in per unit time is known as the speed of sound. For an ideal gas the speed of sound only depends upon the composition of the medium and the temperature.
In the given question we are provided with the speed of sound in dry air (air slightly deviate from the ideal gas behavior)
Let, the density of dry air $ = 3\rho $
Humid air $ = 2\rho $
The distance travelled by sound $ = d $
Time taken in dry air \[ = {t_D}\]
Time taken in humid air\[ = {t_H}\]
Velocity of sound in dry air \[ = {V_D}\]
Velocity of sound in humid air\[ = {V_H}\]
Pressure\[ = p\]
We know the equation of speed of sound is
 $ {V_s} = \sqrt {\dfrac{{\gamma p}}{\rho }} $
 $ \gamma = \dfrac{{{V^2}\rho }}{p} $ (Equation 1)
For dry air equation 1 becomes
 $ \gamma = \dfrac{{{V_D}^2{\rho _D}}}{p} $ (Equation 2)
For humid air equation 1 becomes
 $ \gamma = \dfrac{{{V_H}^2{\rho _H}}}{p} $ (Equation 3)
Comparing equation 2 and 3, we get
 $ {V_D}^2{\rho _D} = {V_H}^2{\rho _H} $
 $ \Rightarrow {V_D}^23\rho = {V_H}^22\rho $
 $ \Rightarrow \dfrac{{{V_D}}}{{{V_H}}} = \sqrt {\dfrac{2}{3}} $
We know that the equation of speed is
 $ V = \dfrac{d}{t} $
 $ d = Vt $ (Equation 4)
For dry air equation 4 becomes
\[d = {V_D}{t_D}\](Equation 5)
For humid air equation 4 becomes
\[d = {V_H}{t_H}\](Equation 6)
Comparing equation 5 and 6, we get
\[{V_D}{t_D} = {V_H}{t_H}\]
\[ \Rightarrow \dfrac{{{V_D}}}{{{V_H}}} = \dfrac{{{t_H}}}{{{t_D}}}\]
\[ \Rightarrow \sqrt {\dfrac{2}{3}} = \dfrac{{{t_H}}}{{10}}\]
\[ \Rightarrow {t_H} = 10\sqrt {\dfrac{2}{3}} \]
\[ \Rightarrow {t_H} = 8.165\sec \]

Note - We discussed above that sound travels in a medium, it can be gas, liquid or solid. Most people believe that sound travels fastest in gases (air), but it is not so. Sound travels faster in solids than it does in liquids. The speed of sound is slowest in the gaseous medium. For sound to travel it has to transfer energy from one molecule to the next and since the molecules in solids are closely packed in solids than in liquids and gases the speed of sound in solids is also the fastest.