source and observer both start moving simultaneously from origion one along y - axis with speed of source = 2 (speed of observer ). The graph between the apparent frequency observed by observer (f) and time (t) would be

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Answer

To solve the problem step by step, we need to analyze the situation where both the source and the observer are moving away from each other, and we need to determine the relationship between the apparent frequency observed by the observer and time.

### Step 1: Define the Variables
Let:
- \( v_0 \) = speed of the observer
- \( v_s = 2v_0 \) = speed of the source (twice the speed of the observer)
- \( v \) = speed of sound

### Step 2: Determine the Direction of Motion
- The observer is moving along the y-axis.
- The source is moving along the x-axis.
- Since both are moving away from each other, we need to consider their velocities in the direction of the line joining them.

### Step 3: Calculate the Components of Velocity
- The component of the observer's velocity in the direction of the line joining them can be expressed as:
\[ v'_{o} = v_0 \cos(\beta) \]
- The component of the source's velocity in the direction of the line joining them can be expressed as:
\[ v'_{s} = v_s \cos(\alpha) = 2v_0 \cos(\alpha) \]

### Step 4: Use the Doppler Effect Formula
The apparent frequency (\( f' \)) can be calculated using the Doppler effect formula:
\[ f' = f \frac{v - v'_{o}}{v + v'_{s}} \]
Substituting the values:
\[ f' = f \frac{v - \frac{v_0}{\sqrt{5}}}{v + \frac{4v_0}{\sqrt{5}}} \]

### Step 5: Analyze the Frequency Change
Since \( v_s = 2v_0 \), we can substitute this into the formula:
\[ f' = f \frac{v - \frac{v_0}{\sqrt{5}}}{v + \frac{4v_0}{\sqrt{5}}} \]
This shows that \( f' \) is dependent on \( v_0 \) and \( v \), but the original frequency \( f \) remains constant.

### Step 6: Determine the Relationship Over Time
- As both the source and observer are moving away from each other, the apparent frequency \( f' \) will be less than the original frequency \( f \).
- There is no time dependence in the expression for \( f' \) since it does not change with time.

### Step 7: Graphical Representation
- Since \( f' < f \) and there is no time dependence, the graph of apparent frequency \( f' \) against time \( t \) will be a horizontal line below the original frequency \( f \).

### Conclusion
The graph between the apparent frequency observed by the observer and time \( t \) would be a horizontal line indicating that the apparent frequency remains constant but is less than the original frequency.