Question

How long will it take sound waves to travel a distance 1 between points $A$ and $B$ if the air temperature between them varies linearly from $T_1$ to $T_2?$ (The velocity of sound in air at temperature $T$ is given by $v=\alpha \sqrt{T}$, where $\alpha$ is a constant)

• A. $\frac{2l}{\alpha \sqrt{T_1{T}_2}}$
• B. $\alpha l\sqrt{\frac{T_1}{T_2}}$
• C. $\sqrt{T_1+T_2}\cdot \alpha l$
• D. $\frac{2l}{\alpha \left(\sqrt{T_2}+\sqrt{T_1}\right)}$

Answer 1

Answer: (D)

$<v>=\frac{v_1+v_2}{2}=\frac{\alpha \sqrt{T_1}+\alpha \sqrt{T_2}}{2}$
$\Rightarrow$ Time taken $=\frac{2l}{\alpha \left(\sqrt{T_1}+\sqrt{T_2}\right)}$
Alternate Solution:
$\frac{dx}{dt}=V=\alpha \sqrt{T_1+\left(\frac{T_2-T_1}{l}\right)x}$
$\underset{x=0}{\overset{x=l}{\int }}\frac{dx}{\sqrt{T_1+\left(\frac{T_2-T_1}{l}\right)x}}=\underset{0}{\overset{t}{\int }}\alpha dt$
on solving we get
$t=\frac{2l}{\alpha \left(\sqrt{T_1}+\sqrt{T_2}\right)}$