A source and a detector move away from each other, each with a speed of 10 m/s with respect to ground with no wind. If the detector detects a frequency 1650 Hz of the sound coming from the source, what is the original frequency of the source? (speed of sound = 340 m/s

Options

A. 750 Hz
B. 1750 Hz
C. 2000 Hz
D. 1800 Hz

Answer

To find the original frequency of the source when both the source and the detector are moving away from each other, we can use the Doppler effect formula for sound. Here’s a step-by-step solution:

### Step 1: Identify the given values
- Apparent frequency detected by the observer (f') = 1650 Hz
- Speed of sound in air (v) = 340 m/s
- Speed of the source (v_s) = 10 m/s
- Speed of the observer (v_o) = 10 m/s

### Step 2: Write the formula for apparent frequency
When both the source and the observer are moving away from each other, the formula for the apparent frequency (f') is given by:

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

Where:
- \( f' \) = apparent frequency
- \( f \) = original frequency of the source
- \( v \) = speed of sound
- \( v_o \) = speed of the observer
- \( v_s \) = speed of the source

### Step 3: Rearrange the formula to solve for the original frequency (f)
To find the original frequency (f), we rearrange the formula:

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

### Step 4: Substitute the known values into the formula
Now, we substitute the known values into the rearranged formula:

\[ f = 1650 \cdot \frac{340 + 10}{340 - 10} \]

### Step 5: Calculate the values in the equation
Calculate the numerator and denominator:

- Numerator: \( 340 + 10 = 350 \)
- Denominator: \( 340 - 10 = 330 \)

Now substitute these values back into the equation:

\[ f = 1650 \cdot \frac{350}{330} \]

### Step 6: Perform the multiplication and division
Now, calculate the value:

\[ f = 1650 \cdot \frac{350}{330} = 1650 \cdot 1.060606 \approx 1750 \text{ Hz} \]

### Step 7: Conclusion
Thus, the original frequency of the source is approximately **1750 Hz**.

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