A train is moving on a straight track with speed `20ms^(-1)`. It is blowing its whistle at the frequency of `1000 Hz`. The percentage change in the frequency heard by a person standing near the track as the train passes him is (speed of sound `= 320 ms^(-1)`) close to :

Options

A. (a) `18%`
B. (b) `24%`
C. ( c) `6%`
D. (d) `12%`

Explanation

To solve the problem, we need to calculate the percentage change in frequency heard by a person standing near the track as the train passes him. We will use the Doppler effect formula for sound.

### Step 1: Identify the given values
- Speed of the train (source), \( v_s = 20 \, \text{m/s} \)
- Frequency of the whistle (source frequency), \( f = 1000 \, \text{Hz} \)
- Speed of sound, \( v = 320 \, \text{m/s} \)

### Step 2: Calculate the apparent frequency when the train is approaching the observer
When the source of sound (the train) is moving towards a stationary observer, the apparent frequency \( f' \) can be calculated using the formula:

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

Where:
- \( v_o \) is the speed of the observer (which is 0 since the observer is stationary).

Substituting the values:

\[ f' = 1000 \left( \frac{320 + 0}{320 - 20} \right) = 1000 \left( \frac{320}{300} \right) = \frac{320000}{300} \approx 1066.67 \, \text{Hz} \]

### Step 3: Calculate the apparent frequency when the train is moving away from the observer
When the source of sound is moving away from the observer, the apparent frequency \( f'' \) is given by:

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

Again, since the observer is stationary, \( v_o = 0 \):

\[ f'' = 1000 \left( \frac{320 + 0}{320 + 20} \right) = 1000 \left( \frac{320}{340} \right) = \frac{320000}{340} \approx 941.18 \, \text{Hz} \]

### Step 4: Calculate the percentage change in frequency
The percentage change in frequency is calculated using the formula:

\[ \text{Percentage Change} = \left( \frac{f' - f''}{f'} \right) \times 100 \]

Substituting the values we calculated:

\[ \text{Percentage Change} = \left( \frac{1066.67 - 941.18}{1066.67} \right) \times 100 \]

Calculating the numerator:

\[ 1066.67 - 941.18 = 125.49 \]

Now substituting back:

\[ \text{Percentage Change} = \left( \frac{125.49}{1066.67} \right) \times 100 \approx 11.76\% \]

### Step 5: Conclusion
The percentage change in frequency heard by the person standing near the track as the train passes him is approximately **12%**.

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