A source emitting a sound of frequency f 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.

Options

A. `(vf^2)/(2vf-a)`
B. `(2vf^2)/(2vf+a)`
C. `(2vf^2)/(3vf-a)`
D. `(2vf^2)/(2vf-a)`

Explanation

To solve the problem of finding the frequency heard by the observer when a source emitting sound starts moving towards them with uniform acceleration, we can follow these steps:

### Step 1: Understand the situation
We have a sound source emitting a frequency \( f \) at a large distance from an observer. The source starts moving towards the observer with uniform acceleration \( a \). We need to find the frequency heard by the observer corresponding to the wave emitted just after the source starts moving.

### Step 2: Determine the time period of the sound wave
The time period \( T \) of the sound wave can be calculated as:
\[ T = \frac{1}{f} \]

### Step 3: Calculate the distance traveled by the source during one time period
Since the source is accelerating, we can use the second equation of motion to find the distance it travels in one time period \( T \). The initial velocity \( u = 0 \) (the source starts from rest), so the distance \( s \) traveled by the source in time \( T \) is given by:
\[ s = uT + \frac{1}{2} a T^2 = 0 + \frac{1}{2} a T^2 = \frac{1}{2} a \left(\frac{1}{f}\right)^2 = \frac{a}{2f^2} \]

### Step 4: Calculate the wavelength of the emitted sound wave
The wavelength \( \lambda \) of the sound wave can be calculated using the formula:
\[ \lambda = vT \]
Substituting \( T \):
\[ \lambda = v \cdot \frac{1}{f} = \frac{v}{f} \]

### Step 5: Determine the new wavelength when the source is moving
As the source moves towards the observer, the effective wavelength \( \lambda' \) will be reduced by the distance the source travels during the time period \( T \):
\[ \lambda' = \lambda - \text{distance traveled by source} = \frac{v}{f} - \frac{a}{2f^2} \]

### Step 6: Calculate the apparent frequency heard by the observer
The frequency \( f' \) heard by the observer can be calculated using the relationship between speed, frequency, and wavelength:
\[ f' = \frac{v}{\lambda'} \]
Substituting for \( \lambda' \):
\[ f' = \frac{v}{\frac{v}{f} - \frac{a}{2f^2}} = \frac{v f}{v - \frac{a}{2f}} \]

### Step 7: Simplify the expression
To simplify:
\[ f' = \frac{2vf^2}{2vf - a} \]

Thus, the frequency heard by the observer is:
\[ f' = \frac{2vf^2}{2vf - a} \]

### Summary of the Solution
The frequency heard by the observer when the source starts moving towards them with uniform acceleration is:
\[ f' = \frac{2vf^2}{2vf - a} \]