wave_mechanics mock_test_waves_and_acoustics Question 19
How long will it take sound waves to travel a distance 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)
Options:
A) $ \frac{2l}{\alpha \sqrt{T _{1}T _{2}}} $
B) $ \alpha l\sqrt{\frac{T _{1}}{T _{2}}} $
C) $ \sqrt{T _{1}+T _{2}}.\alpha l $
D) $ \frac{2l}{\alpha (\sqrt{T _{2}+\sqrt{T _{1}}})} $
Show Answer
Answer:
Correct Answer: D
Solution:
[d]
$ \Longrightarrow $
Time taken for the reaction to occur
$ =\frac{2l}{\alpha (\sqrt{T _{1}}+\sqrt{T _{2}})}$
Alternate Solution:
$ \frac{dx}{dt}=V=\alpha \sqrt{T _{1}+( \frac{T _{2}-T _{1}}{l} )x} $
$ \int _{x=0}^{x=1}{\frac{dx}{\sqrt{T _{1}+( \frac{T _{2}-T _{1}}{l} )x}}}=\int _{0}^{t}{\alpha dt} $
On solving we get $ t=\frac{2l}{\alpha (\sqrt{T _{1}}+\sqrt{T _{2}})} $
BETA
