A source of sound and an observer are approaching each other with the same speed which is equal to `(1)/(10)` times the speed of sound. The apparent change in the frequency of the source is

Options

A. `22.2%` increase
B. `22.2%` decrease
C. `18.2%` decrease
D. `18.2%` decrease

Explanation

To solve the problem of finding the apparent change in frequency when a source of sound and an observer are approaching each other with the same speed, we can follow these steps:

### Step 1: Understand the given parameters
Let:
- \( v \) = speed of sound
- Speed of the source \( v_s = \frac{v}{10} \)
- Speed of the observer \( v_o = \frac{v}{10} \)
- Actual frequency emitted by the source \( n \)

### Step 2: Use the Doppler Effect formula
The formula for the apparent frequency \( n' \) when both the source and observer are moving towards each other is given by:

\[ n' = n \left( \frac{v + v_o}{v - v_s} \right) \]

### Step 3: Substitute the values into the formula
Substituting \( v_o \) and \( v_s \):

\[ n' = n \left( \frac{v + \frac{v}{10}}{v - \frac{v}{10}} \right) \]

### Step 4: Simplify the equation
Calculating the numerator and denominator:

- Numerator:
\[ v + \frac{v}{10} = \frac{10v + v}{10} = \frac{11v}{10} \]

- Denominator:
\[ v - \frac{v}{10} = \frac{10v - v}{10} = \frac{9v}{10} \]

Now substituting these back into the formula:

\[ n' = n \left( \frac{\frac{11v}{10}}{\frac{9v}{10}} \right) \]

### Step 5: Cancel out common terms
The \( v \) and \( 10 \) cancel out:

\[ n' = n \left( \frac{11}{9} \right) \]

### Step 6: Calculate the change in frequency
The change in frequency \( \Delta n \) can be calculated as:

\[ \Delta n = n' - n = n \left( \frac{11}{9} \right) - n = n \left( \frac{11}{9} - 1 \right) = n \left( \frac{11 - 9}{9} \right) = n \left( \frac{2}{9} \right) \]

### Step 7: Calculate the percentage change in frequency
The percentage change in frequency is given by:

\[ \text{Percentage Change} = \left( \frac{\Delta n}{n} \right) \times 100 = \left( \frac{n \left( \frac{2}{9} \right)}{n} \right) \times 100 = \frac{2}{9} \times 100 = \frac{200}{9} \approx 22.22\% \]

### Final Answer
The apparent change in frequency of the source is approximately **22.22%**.

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