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 train A blows a horn of frequency 450 Hz, we can use the Doppler effect formula for sound. Here are the steps to arrive at the solution:

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

### Step 2: Understand the scenario
Both trains are moving in the same direction, with train A moving faster than train B. Since train B is behind train A, it will hear a different frequency due to the relative motion.

### Step 3: Use the Doppler effect formula
The apparent frequency \( f' \) heard by an observer (train B) when the source (train A) is moving away from the observer can be calculated using the formula:

\[ f' = f \left( \frac{v + v_B}{v + v_A} \right) \]

Where:
- \( f' \) is the apparent frequency heard by train B.
- \( f \) is the actual frequency emitted by the source (train A).
- \( v \) is the speed of sound.
- \( v_B \) is the speed of the observer (train B).
- \( v_A \) is the speed of the source (train A).

### Step 4: Substitute the values into the formula
Substituting the known values into the formula:

\[ f' = 450 \left( \frac{330 + 10}{330 + 30} \right) \]

### Step 5: Simplify the expression
Calculating the numerator and denominator:

\[ f' = 450 \left( \frac{340}{360} \right) \]

### Step 6: Calculate the apparent frequency
Now, simplify the fraction:

\[ f' = 450 \times \frac{34}{36} \]

Calculating this gives:

\[ f' = 450 \times 0.9444 \approx 423.99 \, \text{Hz} \]

### Step 7: Round the result
Rounding to two decimal places, the apparent frequency heard by train B is approximately:

\[ f' \approx 424 \, \text{Hz} \]

### Final Answer
The apparent frequency heard by train B is approximately **424 Hz**.

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