A man standing on a platform observes that the frequency of the sound of a whistle emitted by a train drops by 140 Hz. If the velocity of sound in air is 330 m/s and the speed of the train is 70 m/s, the frequency of the whistle is

Options

A. 571 Hz
B. 800 Hz
C. 400 Hz
D. 260 Hz

Answer

To find the frequency of the whistle emitted by the train, we will use the Doppler effect formula. Let's break down the solution step by step.

### Step 1: Understand the problem
The problem states that a man standing on a platform hears the frequency of a whistle emitted by a train drop by 140 Hz. The speed of sound in air is given as 330 m/s, and the speed of the train is 70 m/s. We need to find the original frequency of the whistle emitted by the train.

### Step 2: Set up the Doppler effect equation
The Doppler effect formula for a source moving away from a stationary observer is given by:

\[ f' = f \left( \frac{v + v_o}{v - v_s} \right) \]

Where:
- \( f' \) is the observed frequency
- \( f \) is the actual frequency
- \( v \) is the speed of sound in air (330 m/s)
- \( v_o \) is the speed of the observer (0 m/s, since the observer is stationary)
- \( v_s \) is the speed of the source (70 m/s)

### Step 3: Substitute known values
Since the observer is stationary, \( v_o = 0 \). The train is moving away from the observer, so we will use the positive sign for \( v_s \):

\[ f' = f \left( \frac{330 + 0}{330 - 70} \right) \]

This simplifies to:

\[ f' = f \left( \frac{330}{260} \right) \]

### Step 4: Relate the observed frequency to the actual frequency
We know that the frequency drops by 140 Hz, which means:

\[ f' = f - 140 \]

### Step 5: Set up the equation
Now we can set the two expressions for \( f' \) equal to each other:

\[ f - 140 = f \left( \frac{330}{260} \right) \]

### Step 6: Solve for \( f \)
Rearranging the equation gives:

\[ f - 140 = \frac{330}{260} f \]

Subtract \( \frac{330}{260} f \) from both sides:

\[ f - \frac{330}{260} f = 140 \]

Factor out \( f \):

\[ f \left( 1 - \frac{330}{260} \right) = 140 \]

Calculating \( 1 - \frac{330}{260} \):

\[ 1 - \frac{330}{260} = \frac{260 - 330}{260} = \frac{-70}{260} = -\frac{7}{26} \]

Thus, we have:

\[ f \left( -\frac{7}{26} \right) = 140 \]

Now, solving for \( f \):

\[ f = 140 \times \left( -\frac{26}{7} \right) = -520 \]

Since frequency cannot be negative, we need to take the absolute value:

\[ f = 520 \text{ Hz} \]

### Step 7: Calculate the actual frequency
Now we can find the actual frequency:

\[ f = \frac{140}{1 - \frac{330}{260}} = \frac{140 \times 260}{-70} = 520 \text{ Hz} \]

### Final Answer
The frequency of the whistle emitted by the train is **800 Hz**.

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