An observer moves towards a stationary source of sound, with a velocity one-fifth of the velocity of sound. What is the percentage increase in the apparent frequency?

Options

A. (a) `0.5%`
B. (b) zero
C. ( c) `20%`
D. (d) `5%`

Explanation

To solve the problem of finding the percentage increase in the apparent frequency when an observer moves towards a stationary source of sound, we can follow these steps:

### Step 1: Understand the Given Information
- Let \( v \) be the velocity of sound.
- The velocity of the observer \( v_o = \frac{v}{5} \).
- The velocity of the source \( v_s = 0 \) (since it is stationary).

### Step 2: Use the Doppler Effect Formula
The formula for the apparent frequency \( f' \) when the observer is moving towards a stationary source is given by:
\[ f' = f \left( \frac{v + v_o}{v} \right) \]
where \( f \) is the original frequency.

### Step 3: Substitute the Values
Substituting \( v_o = \frac{v}{5} \) into the formula:
\[ f' = f \left( \frac{v + \frac{v}{5}}{v} \right) \]
This simplifies to:
\[ f' = f \left( \frac{v + 0.2v}{v} \right) = f \left( \frac{1.2v}{v} \right) = f \times 1.2 \]

### Step 4: Calculate the Apparent Frequency Ratio
Now, we can express the ratio of the apparent frequency to the original frequency:
\[ \frac{f'}{f} = 1.2 \]

### Step 5: Determine the Percentage Increase
To find the percentage increase in frequency, we use the formula:
\[ \text{Percentage Increase} = \left( \frac{f' - f}{f} \right) \times 100 \]
Substituting the values we found:
\[ \text{Percentage Increase} = \left( \frac{1.2f - f}{f} \right) \times 100 = \left( \frac{0.2f}{f} \right) \times 100 = 20\% \]

### Conclusion
The percentage increase in the apparent frequency is **20%**.

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