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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.
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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