When a siren of a fire engine approaches an observer at a speed of `10"m s"^(-1)`, taking the velocity of sound is `340"m s"^(-1)`, if the frequency of the siren is 600 Hz, the frequency heard by the observer is

Options

A. 580 Hz
B. 620 Hz
C. 600 Hz
D. none of these

Explanation

To solve the problem of finding the frequency heard by an observer when a siren approaches, we can use the Doppler effect formula for sound. Here’s a step-by-step breakdown of the solution:

### Step 1: Identify the given values
- Speed of sound (V) = 340 m/s
- Speed of the source (Vs) = 10 m/s (since the fire engine is approaching, this is positive)
- Frequency of the siren (Fs) = 600 Hz

### Step 2: Understand the formula for the Doppler effect
The formula to calculate the frequency heard by the observer (Fl) when the source is moving towards the observer is given by:

\[ Fl = \frac{V + Vl}{V - Vs} \times Fs \]

Where:
- Fl = frequency heard by the listener
- V = speed of sound
- Vl = speed of the listener (0 m/s, since the observer is at rest)
- Vs = speed of the source

### Step 3: Substitute the values into the formula
Since the listener is at rest, Vl = 0. Therefore, the formula simplifies to:

\[ Fl = \frac{V}{V - Vs} \times Fs \]

Substituting the known values:

\[ Fl = \frac{340}{340 - 10} \times 600 \]

### Step 4: Calculate the denominator
Calculate \(340 - 10\):

\[ 340 - 10 = 330 \]

### Step 5: Substitute back into the equation
Now substitute back into the equation:

\[ Fl = \frac{340}{330} \times 600 \]

### Step 6: Calculate the fraction
Now calculate the fraction:

\[ \frac{340}{330} = \frac{34}{33} \]

### Step 7: Multiply by the frequency of the siren
Now multiply by the frequency of the siren:

\[ Fl = \frac{34}{33} \times 600 \]

### Step 8: Perform the multiplication
Calculate \( \frac{34 \times 600}{33} \):

\[ Fl = \frac{20400}{33} \approx 618.18 \text{ Hz} \]

### Step 9: Round to the nearest option
The frequency heard by the observer is approximately 618.18 Hz, which can be rounded to 620 Hz.

### Final Answer
The frequency heard by the observer is approximately **620 Hz**.

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