A person is observing two trains, one is approaching him with a velocity of of 4 m/s while the other is receding from him with the same velocity. IF both the trains blow their respective whistles of frequency 240 hertz, the beat frequency heard by the observer will be (speed of sound in air = 320 m/s)

Options

A. 6
B. 3
C. zero
D. 12

Explanation

To solve the problem step by step, we will use the Doppler effect formula to find the apparent frequencies of the sound from both trains and then calculate the beat frequency.

### Step 1: Identify the given values
- Frequency of the whistles (N) = 240 Hz
- Velocity of sound in air (V) = 320 m/s
- Velocity of the approaching train (Vs) = 4 m/s
- Velocity of the receding train (Vs) = 4 m/s

### Step 2: Calculate the apparent frequency from the approaching train
Using the Doppler effect formula for an approaching source:
\[ N_A = \frac{N \cdot V}{V - V_s} \]
Where:
- \(N_A\) = apparent frequency from the approaching train
- \(N\) = actual frequency = 240 Hz
- \(V\) = speed of sound = 320 m/s
- \(V_s\) = speed of the source (approaching train) = 4 m/s

Substituting the values:
\[ N_A = \frac{240 \cdot 320}{320 - 4} \]
\[ N_A = \frac{76800}{316} \]
\[ N_A \approx 243.67 \text{ Hz} \]

### Step 3: Calculate the apparent frequency from the receding train
Using the Doppler effect formula for a receding source:
\[ N_R = \frac{N \cdot V}{V + V_s} \]
Where:
- \(N_R\) = apparent frequency from the receding train

Substituting the values:
\[ N_R = \frac{240 \cdot 320}{320 + 4} \]
\[ N_R = \frac{76800}{324} \]
\[ N_R \approx 237.04 \text{ Hz} \]

### Step 4: Calculate the beat frequency
The beat frequency (f_beat) is the difference between the two apparent frequencies:
\[ f_{beat} = |N_A - N_R| \]
Substituting the values:
\[ f_{beat} = |243.67 - 237.04| \]
\[ f_{beat} \approx 6.63 \text{ Hz} \]

### Final Answer
The beat frequency heard by the observer is approximately **6.63 Hz**.

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