Question

Duration: 3:00:00 Time Left : 2:42:48
How long will it take sound waves to
travel the distance 'l' between two
points A and B if the air temperature
between them varies linearly from Tito
T2 ? The velocity of sound propagation
in air is given by v=avT, where a is
a constant.
(A) la VTT,​

Answer 1

$\bigstar \: \: { \underline{\underline{ \large \rm{Requiréd \: Question : - }}}}$

How long will it take sound waves to travel the distance 'l' between two points A and B if the air temperature between them varies linearly from $\rm{{T_1} to {T_2}}$ ? The velocity of sound propagation in air is given by v=a√T, where a is a constant.

$\bigstar \: \: { \underline{\underline{ \large \rm{Answér : - }}}}$

$➣ \red{ \: \: \: \: \: \sf\: t_{AB} \: = \frac{21}{a( \sqrt{T_2} + \sqrt{ T_1} )}}$

$\bigstar \: \: { \underline{\underline{ \large \rm{Solutíon : - }}}}$

We know that,

$➩ \: \: \: \: \: \: \: \: \: \sf \: v \: \: \: = \: \: \frac{dx}{dt} \: \: = \: \: (\frac{dx}{dT}) \frac{dT}{dt} \: \: = \: \: \frac{1}{ (\frac{dt}{dx}) } \frac{dT}{dT} ....................(1)$

Given, v = a√T ..........................(2)

Now,

From 1 and 2,

$\: \: \: \longrightarrow \: \: \: \sf a \: = (\frac{dx}{dT}) \frac{dT}{dt}$

From the figure,$\sf{\frac{dT}{dx} = \frac{T _2 - T _1 }{1}}$

$\: \: \: \therefore \: \: \: \sf a \sqrt{T} = \frac{1}{T _2 - T _1 } \: \frac{dT}{dt}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \therefore \: \: \: \sf dt \: = \frac{1}{a(T _2 - T _1) } \: \frac{dT}{ \sqrt{T} }$

Integrating from A to B ,

$\sf \: \: \int_A ^{B_A} \: \: dt \: = t_{AB} \: = \frac{1}{a(T_2 - T_1)} \: \: \int_{A_{(T_2)}} ^{B_{(T_2)}} \frac{1}{ \sqrt{T} } = dT$

$➨ \: \: \: \sf \: t_{AB} \: \: = \frac{21}{a(T_2- T_1)} [ \sqrt{T} ] _{T_1} ^{T_2}$

$➨ \: \: \: \sf \: t_{AB} \: \: = \frac{21}{a(T_2- T_1) (T_2 + T_1)} \: \: \: \: [ \sqrt{T_2} - \sqrt T_1]$

$➯ \blue{\underline\blue { \: \boxed{\green{{ \sf\: t_{AB} \: = \frac{21}{a( \sqrt{T_2} + \sqrt{ T_1} )}}} }}}$