Question

How long will it take for sound waves to travel a distance $l$ between points $A$ and $B$. If air temperature between them varies linearly from $T_1$ to $T_2$. The velocity of sound in air is given by$v=a\sqrt{T}$ where $a$ is a constant :

• A. $\frac{2l}{a\sqrt{T_1{T}_2}}$
• B. $al\sqrt{{}^{T_1}/T_2}$
• C. $al\sqrt{T_1+T_2}$
• D. $\frac{2l}{a(\sqrt{T_1}+\sqrt{T_2})}$

Answer 1

The correct option is D $\frac{2l}{a(\sqrt{T_1}+\sqrt{T_2})}$

Please see figure,
$v=\frac{dx}{dt}=\left(\frac{dx}{dT}\right)\frac{dT}{dt}=\frac{1}{\left(\frac{dT}{dx}\right)}\frac{dT}{dt}..................................\left(i\right)$
$\text{Given},v=a\sqrt{T}.........................................\left(ii\right)$
From $\left(i\right)$ and $\left(ii\right),$
$a\sqrt{T}=\frac{1}{\left(\frac{dT}{dx}\right)}\frac{dT}{dt}$
$\text{From the figure},\frac{dT}{dx}=\frac{T_2-T_1}{l}$
$\therefore a\sqrt{T}=\frac{l}{T_2-T_1}\frac{dT}{dt}$
$\therefore dt=\frac{l}{a(T_2-T_1)}\frac{dT}{\sqrt{T}}$
$\text{Integrating from A to B}$
${\int }_A^{B}dt=t_{AB}=\frac{l}{a(T_2-T_1)}{\int }_{A\left(T_1\right)}^{B\left(T_2\right)}\frac{1}{\sqrt{T}}dT$
$t_{AB}=\frac{2l}{a(T_2-T_1)}{\left[\sqrt{T}\right]}_{T_1}^{T_2}$
$t_{AB}=\frac{2l}{a(\sqrt{T_2}-\sqrt{T_1})(\sqrt{T_2}+\sqrt{T_1})}[\sqrt{T_2}-\sqrt{T_1}]$
$t_{AB}=\frac{2l}{a(\sqrt{T_2}+\sqrt{T_1})}$
Option "D" is correct.