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

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Explanation

To solve the problem, we need to determine the velocity of sound given the conditions of the source and observer's motion and the ratio of their frequencies. Here’s a step-by-step breakdown of the solution:

### Step 1: Define the Variables
Let:
- \( V_0 \) = speed of the observer
- \( V_s \) = speed of the source = \( 2V_0 \)
- \( v \) = velocity of sound
- \( n \) = actual frequency of the source
- \( n_a \) = apparent frequency observed

### Step 2: Calculate the Distances
In time \( t \):
- Distance moved by the observer along the y-axis:
\[ y = V_0 \cdot t \]
- Distance moved by the source along the x-axis:
\[ x = V_s \cdot t = 2V_0 \cdot t \]

### Step 3: Determine the Angles
Using the distances moved, we can find the hypotenuse of the triangle formed by the movements of the source and observer:
\[ \text{Hypotenuse} = \sqrt{(2V_0 t)^2 + (V_0 t)^2} = \sqrt{4V_0^2 t^2 + V_0^2 t^2} = \sqrt{5V_0^2 t^2} = V_0 t \sqrt{5} \]

### Step 4: Calculate the Cosines of the Angles
- For the angle \( \alpha \) (source's angle):
\[ \cos \alpha = \frac{2V_0 t}{\sqrt{5} V_0 t} = \frac{2}{\sqrt{5}} \]
- For the angle \( \beta \) (observer's angle):
\[ \cos \beta = \frac{V_0 t}{\sqrt{5} V_0 t} = \frac{1}{\sqrt{5}} \]

### Step 5: Apply the Doppler Effect Formula
The ratio of the apparent frequency to the actual frequency is given by:
\[ \frac{n_a}{n} = \frac{v - V_0 \cos \beta}{v + V_s \cos \alpha} \]
Substituting \( V_s = 2V_0 \):
\[ \frac{n_a}{n} = \frac{v - V_0 \cdot \frac{1}{\sqrt{5}}}{v + 2V_0 \cdot \frac{2}{\sqrt{5}}} \]

### Step 6: Substitute Known Values
Given that \( \frac{n_a}{n} = 0.75 \):
\[ 0.75 = \frac{v - \frac{V_0}{\sqrt{5}}}{v + \frac{4V_0}{\sqrt{5}}} \]
This can be rewritten as:
\[ \frac{3}{4} = \frac{v - \frac{V_0}{\sqrt{5}}}{v + \frac{4V_0}{\sqrt{5}}} \]

### Step 7: Cross Multiply and Solve for \( v \)
Cross multiplying gives:
\[ 3(v + \frac{4V_0}{\sqrt{5}}) = 4(v - \frac{V_0}{\sqrt{5}}) \]
Expanding both sides:
\[ 3v + \frac{12V_0}{\sqrt{5}} = 4v - \frac{4V_0}{\sqrt{5}} \]
Rearranging gives:
\[ v = \frac{16V_0}{\sqrt{5}} \]

### Final Answer
The velocity of sound \( v \) is:
\[ v = \frac{16V_0}{\sqrt{5}} \]