The engine of a train sound a whistle at frequency v, the frequency heard by a passenger is

Options

A. `gtv`
B. `ltv`
C. `=(1)/(v)`
D. `=v`

Explanation

To solve the problem of the frequency heard by a passenger when a train sounds its whistle, we can apply the principles of the Doppler effect. Here’s a step-by-step solution:

### Step 1: Understand the scenario
The train is moving with a certain velocity \( v \) and it emits a sound at a frequency \( f \). The passenger is also moving with the same velocity \( v \).

### Step 2: Identify 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. When both the source and observer are moving at the same speed, the relative motion between them is zero.

### Step 3: Apply the Doppler effect formula
The formula for the observed frequency \( f' \) when both the source and observer are moving is given by:

\[ f' = f \frac{v + v_o}{v - v_s} \]

Where:
- \( f' \) = observed frequency
- \( f \) = source frequency
- \( v \) = speed of sound in air
- \( v_o \) = speed of the observer (passenger)
- \( v_s \) = speed of the source (train)

### Step 4: Substitute the values
In our case, since the passenger and the train are moving at the same speed, we have:
- \( v_o = v \) (speed of the observer)
- \( v_s = v \) (speed of the source)

Substituting these into the formula gives:

\[ f' = f \frac{v + v}{v - v} = f \frac{2v}{0} \]

### Step 5: Analyze the result
The denominator becomes zero, which indicates that the observed frequency is undefined in this case. This means that when both the source and observer are moving at the same speed, the passenger does not perceive any change in frequency due to the Doppler effect.

### Conclusion
Thus, the frequency heard by the passenger is the same as the frequency emitted by the train, which is \( f \).