An observer starts moving with unifrom acceleration a towards a stationary sound soure of frequency `f_(o)`. As the observer approaches the source ,the apparent frequency f heard by the observer varies with time t as

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Answer

To solve the problem of how the apparent frequency \( f \) heard by an observer moving with uniform acceleration \( a \) towards a stationary sound source of frequency \( f_0 \) varies with time \( t \), we can follow these steps:

### Step 1: Understand the Doppler Effect
The apparent frequency \( f \) is given by the formula:
\[ f' = \frac{V + V_0}{V} f_0 \]
where:
- \( f' \) is the apparent frequency heard by the observer,
- \( V \) is the speed of sound,
- \( V_0 \) is the speed of the observer,
- \( f_0 \) is the frequency of the source.

### Step 2: Determine the Speed of the Observer
Since the observer is moving with uniform acceleration \( a \), we can express the speed of the observer \( V_0 \) as:
\[ V_0 = U + at \]
Given that the observer starts from rest, \( U = 0 \), so:
\[ V_0 = at \]

### Step 3: Substitute the Observer's Speed into the Frequency Formula
Substituting \( V_0 \) into the apparent frequency formula gives:
\[ f' = \frac{V + at}{V} f_0 \]
This can be simplified to:
\[ f' = f_0 \left(1 + \frac{at}{V}\right) \]

### Step 4: Analyze the Equation
The equation \( f' = f_0 \left(1 + \frac{at}{V}\right) \) indicates that the apparent frequency \( f' \) increases linearly with time \( t \). The term \( \frac{a}{V} \) acts as the slope of this linear relationship.

### Step 5: Identify the Graph
Since the relationship is linear, we expect a straight line graph where:
- The y-intercept is \( f_0 \) (the frequency of the source when \( t = 0 \)),
- The slope is \( \frac{a}{V} f_0 \).

Thus, the correct graph will show a straight line starting from \( f_0 \) and increasing with time.

### Conclusion
The correct graph representing the variation of the apparent frequency \( f \) with time \( t \) is the one that shows a linear increase starting from \( f_0 \).

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