Two trains A and B are moving in the same direction with velocities 30 m/s and 10 m/s respectively, B is behind from A, blows a horn of frequency 450 Hz. Then the apparent frequency heard by B is (The velocity of sound is 330 m/s) :-

Options

A. 425 Hz
B. 300 Hz
C. 450 Hz
D. 350 Hz

Explanation

To solve the problem of finding the apparent frequency heard by train B when it blows a horn while moving, we can use the Doppler effect formula. Here’s a step-by-step solution:

### Step 1: Identify the given values
- Frequency of the horn (f) = 450 Hz
- Velocity of sound (v) = 330 m/s
- Velocity of train A (v_A) = 30 m/s
- Velocity of train B (v_B) = 10 m/s

### Step 2: Understand the scenario
Train A is moving faster than train B, and both are moving in the same direction. Train B is behind train A and is the observer in this scenario.

### Step 3: Use the Doppler effect formula
The apparent frequency (f') heard by an observer moving in the same direction as the source is given by the formula:
\[ f' = f \times \frac{v + v_B}{v + v_A} \]
Where:
- \(f'\) = apparent frequency
- \(f\) = actual frequency of the source
- \(v\) = speed of sound
- \(v_B\) = speed of the observer (train B)
- \(v_A\) = speed of the source (train A)

### Step 4: Substitute the values into the formula
Substituting the known values into the formula:
\[ f' = 450 \times \frac{330 + 10}{330 + 30} \]

### Step 5: Calculate the values
1. Calculate the numerator:
\[ 330 + 10 = 340 \]
2. Calculate the denominator:
\[ 330 + 30 = 360 \]
3. Substitute these values back into the equation:
\[ f' = 450 \times \frac{340}{360} \]

### Step 6: Simplify the fraction
\[ \frac{340}{360} = \frac{34}{36} = \frac{17}{18} \]
Thus,
\[ f' = 450 \times \frac{17}{18} \]

### Step 7: Perform the multiplication
\[ f' = 450 \times 0.9444 \approx 425 \text{ Hz} \]

### Step 8: Conclusion
The apparent frequency heard by train B is approximately 425 Hz.

### Final Answer
The apparent frequency heard by train B is **425 Hz**.

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