Two trains `A and B` moving with speeds `20m//s` and `30m//s` respectively in the same direction on the same straight track, with `B` ahead of `A`. The engines are at the front ends. The engine of train `A` blows a long whistle.
Assume that the sound of the whistle is composed of components varying in frequency from `f_(1) = 800 Hz` to `f_(2) = 1120 Hz`, as shown in the figure. the spread in the frequency (highest frequency - lowest frequency) is thus `320 Hz`. the speed of sound in still air is `340 m//s`.
The spread of frequency as observed by the passenger in train `B` is

Options

A. 310 Hz
B. 330 Hz
C. 350 Hz
D. 290 Hz

Explanation

To solve the problem step-by-step, we will use the Doppler effect formula to find the apparent frequencies observed by the passenger in train B.

### Step 1: Identify the given values
- Speed of train A (source, \( v_s \)) = 20 m/s
- Speed of train B (observer, \( v_o \)) = 30 m/s
- Speed of sound in air (v) = 340 m/s
- Lowest frequency (\( f_1 \)) = 800 Hz
- Highest frequency (\( f_2 \)) = 1120 Hz

### Step 2: Calculate the apparent frequency for the lowest frequency (\( f_1 \))
Using the Doppler effect formula for the apparent frequency (\( f' \)):
\[ f' = f \cdot \frac{v + v_o}{v - v_s} \]
Substituting the values for \( f_1 \):
\[ f'_{1} = 800 \cdot \frac{340 + 30}{340 - 20} \]
Calculating the numerator and denominator:
\[ f'_{1} = 800 \cdot \frac{370}{320} \]
Now, simplifying:
\[ f'_{1} = 800 \cdot 1.15625 \approx 925 Hz \]

### Step 3: Calculate the apparent frequency for the highest frequency (\( f_2 \))
Now, we will do the same for \( f_2 \):
\[ f'_{2} = 1120 \cdot \frac{340 + 30}{340 - 20} \]
Substituting the values:
\[ f'_{2} = 1120 \cdot \frac{370}{320} \]
Now, simplifying:
\[ f'_{2} = 1120 \cdot 1.15625 \approx 1290 Hz \]

### Step 4: Calculate the spread of frequency
The spread of frequency is given by:
\[ \text{Spread} = f'_{2} - f'_{1} \]
Substituting the values:
\[ \text{Spread} = 1290 - 925 = 365 Hz \]

### Step 5: Final Answer
Thus, the spread of frequency as observed by the passenger in train B is **365 Hz**.

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