Consider the vehicle emitting sound wave of frequency 700 Hz moving towards an observer at a speed `22 m s^(-1)` . Assuming the observer to be at rest, and speed of sound to be `330 m s^(-1)` , the frequency of sound as measured by the observer is

Options

A. `(2525)/4 Hz`
B. `(1960)/3Hz`
C. `(2240)/3Hz`
D. `750 Hz`

Answer

To solve the problem of finding the frequency of sound as measured by the observer when a vehicle emitting sound waves is moving towards them, we can use the Doppler effect formula. Here’s a step-by-step solution:

### Step 1: Identify the given values
- Frequency of the source (f) = 700 Hz
- Speed of sound (v) = 330 m/s
- Speed of the source (vs) = 22 m/s (since the vehicle is moving towards the observer)
- Speed of the observer (vo) = 0 m/s (the observer is at rest)

### Step 2: Write the Doppler effect formula
The formula for the observed frequency (f') when the source is moving towards a stationary observer is given by:

\[ f' = \frac{v + vo}{v - vs} \times f \]

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

\[ f' = \frac{v}{v - vs} \times f \]

Substituting the known values:

\[ f' = \frac{330 \, \text{m/s}}{330 \, \text{m/s} - 22 \, \text{m/s}} \times 700 \, \text{Hz} \]

### Step 4: Calculate the denominator
Calculate \( v - vs \):

\[ v - vs = 330 \, \text{m/s} - 22 \, \text{m/s} = 308 \, \text{m/s} \]

### Step 5: Substitute back into the formula
Now substitute this value back into the equation for f':

\[ f' = \frac{330 \, \text{m/s}}{308 \, \text{m/s}} \times 700 \, \text{Hz} \]

### Step 6: Calculate the fraction
Calculate the fraction:

\[ \frac{330}{308} \approx 1.0714 \]

### Step 7: Calculate the observed frequency
Now multiply this fraction by the original frequency:

\[ f' \approx 1.0714 \times 700 \, \text{Hz} \approx 750 \, \text{Hz} \]

### Final Answer
Thus, the frequency of sound as measured by the observer is approximately **750 Hz**.

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