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Question
A source emitting a sound of frequency v is placed at a large distance from an observer. The source starts moving towards the observer with a uniform acceleration a. Find the frequency heard by the observer corresponding to the wave emitted just after the source starts. The speed of sound in the medium is v.
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Solution
Let d be the initial distance between the source and the observer.
If v is the speed of sound emitted by the observer, then the time taken by the sound to reach the observer is given by:
T1 = d/v
The source is also moving. Therefore, at t = T, it moves a distance of (s) and is given by :
\[s = 0 \times T + \frac{1}{2}a T^2\]
Time taken by the pulse to reach the observer :
\[\frac{\left( d - \frac{1}{2}a T^2 \right)}{v}\]
Time difference \[\left( ∆ t \right)\] between the two pulses :
\[\left( T + \left( \frac{d - \frac{1}{2}a T^2}{v} \right) \right) - \frac{d}{v}\]
\[T - \frac{a T^2}{2v}\]
On replacing u =\[\frac{1}{T}\],
the apparent frequency will be :
\[\frac{1}{∆ t}\] = \[\frac{2u v^2}{2uv - a}\]
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