A train is moving with a constant speed along a circular track. The engine of the train emits a sound of frequency f. The frequency heard by the guard at the rear end of the train.

Options

A. is less than f
B. is greter than f
C. is equal to f
D. may be greater than , less than or equal to f denpending on the factors like speed of train, legth of train and radius of circular track

Answer

To solve the problem of determining the frequency heard by the guard at the rear end of a train moving along a circular track, we can follow these steps:

### Step 1: Understand the Doppler Effect
The Doppler Effect describes how the frequency of a wave changes for an observer moving relative to the source of the wave. In this case, the source of sound is the engine of the train, and the observer is the guard at the rear end of the train.

### Step 2: Define Variables
Let:
- \( f \) = frequency emitted by the train's engine
- \( V \) = speed of sound in air
- \( U \) = speed of the train
- \( \theta \) = angle between the direction of the sound and the direction of the train's motion

### Step 3: Apply the Doppler Effect Formula
The formula for the frequency heard by an observer moving relative to a source is given by:

\[ f' = f \cdot \frac{V + U \cos \theta}{V - U \cos \theta} \]

Where:
- \( f' \) = frequency heard by the guard
- \( V + U \cos \theta \) = effective speed of sound towards the observer
- \( V - U \cos \theta \) = effective speed of sound away from the source

### Step 4: Determine the Angle \( \theta \)
In this scenario, since the guard is at the rear of the train, the angle \( \theta \) will be 180 degrees (or \( \pi \) radians). Thus, \( \cos(180^\circ) = -1 \).

### Step 5: Substitute Values into the Formula
Substituting \( \cos(180^\circ) = -1 \) into the Doppler Effect formula:

\[ f' = f \cdot \frac{V - U}{V + U} \]

### Step 6: Analyze the Result
Since the guard is moving away from the source of sound (the engine), the frequency \( f' \) will be less than the emitted frequency \( f \). Therefore, the frequency heard by the guard at the rear end of the train is:

\[ f' < f \]

### Conclusion
The frequency heard by the guard at the rear end of the train is less than the frequency emitted by the engine of the train.