Source and observer start moving simulatneously along x and y-axis respectively. The speed of source is twice the speed of observer `V_0`. If the ratio of observer frequency to the frequency of the source is `0.75`, find the velocity of sound.

Options

A. (a)`(11)/(sqrt(5))V_0`
B. (b)`(17)/(sqrt5)V_0`
C. (c)`(16)/(sqrt5)V_0`
D. (d)`(19)/(sqrt5)V_0`

Explanation

To solve the problem step by step, we will use the concepts of the Doppler effect and the relationship between the velocities of the source and observer.

### Step 1: Define the variables
Let:
- \( V_0 \) = speed of the observer
- \( V_s = 2V_0 \) = speed of the source (since the speed of the source is twice that of the observer)
- \( V \) = velocity of sound
- \( f_0 \) = frequency of the source
- \( f' \) = frequency observed by the observer

### Step 2: Use the frequency ratio
We are given that the ratio of the observer frequency to the frequency of the source is \( 0.75 \):
\[ \frac{f'}{f_0} = 0.75 \]
This implies:
\[ f' = 0.75 f_0 \]

### Step 3: Apply the Doppler effect formula
The apparent frequency \( f' \) when both the source and observer are moving can be expressed as:
\[ f' = f_0 \frac{V + V_0 \cos \theta}{V - V_s \cos \phi} \]
where:
- \( \theta \) is the angle of the observer's motion with respect to the line of the source's motion.
- \( \phi \) is the angle of the source's motion with respect to the line of the observer's motion.

In this case, since the source moves along the x-axis and the observer moves along the y-axis, we can assume \( \theta = 90^\circ \) and \( \phi = 0^\circ \). Thus, \( \cos \theta = 0 \) and \( \cos \phi = 1 \).

Substituting these values into the formula:
\[ f' = f_0 \frac{V + 0}{V - 2V_0} \]
This simplifies to:
\[ f' = f_0 \frac{V}{V - 2V_0} \]

### Step 4: Set up the equation using the frequency ratio
Now we can set up the equation using the frequency ratio:
\[ 0.75 f_0 = f_0 \frac{V}{V - 2V_0} \]
Dividing both sides by \( f_0 \) (assuming \( f_0 \neq 0 \)):
\[ 0.75 = \frac{V}{V - 2V_0} \]

### Step 5: Solve for \( V \)
Cross-multiplying gives:
\[ 0.75(V - 2V_0) = V \]
Expanding this:
\[ 0.75V - 1.5V_0 = V \]
Rearranging terms:
\[ 0.75V - V = 1.5V_0 \]
\[ -0.25V = 1.5V_0 \]
Dividing both sides by -0.25:
\[ V = -\frac{1.5V_0}{-0.25} = 6V_0 \]

### Conclusion
The velocity of sound \( V \) is:
\[ V = 6V_0 \]

### Final Answer
Thus, the velocity of sound is \( 6V_0 \).