To an observer, the pitch of a stationary source of sound appears to be reduced by 20%. If the speed of sound is 340m/s then speed and direction of the observer is

Options

A. 86 m/s towards the source
B. 68 m/s towards the source
C. 86 m/s away from the source
D. 68 m/s away from the source

Answer

To solve the problem step by step, we will use the concept of the Doppler effect, which describes how the frequency of sound changes for an observer moving relative to a sound source.

### Step 1: Understand the Problem
We know that the observer hears the frequency of the sound as reduced by 20%. This means the apparent frequency (\( f' \)) is 80% of the actual frequency (\( f \)). Therefore, we can express this relationship as:
\[ f' = 0.8f \]

### Step 2: Identify the Doppler Effect Formula
For a stationary source and a moving observer, the formula for the apparent frequency is given by:
\[ f' = f \frac{v}{v + v_0} \]
where:
- \( f' \) = apparent frequency
- \( f \) = actual frequency
- \( v \) = speed of sound (340 m/s)
- \( v_0 \) = speed of the observer (which we need to find)

### Step 3: Substitute the Known Values
From the previous step, we know:
\[ 0.8f = f \frac{340}{340 + v_0} \]
We can cancel \( f \) from both sides (assuming \( f \neq 0 \)):
\[ 0.8 = \frac{340}{340 + v_0} \]

### Step 4: Cross-Multiply to Solve for \( v_0 \)
Cross-multiplying gives us:
\[ 0.8(340 + v_0) = 340 \]
Expanding this:
\[ 272 + 0.8v_0 = 340 \]

### Step 5: Isolate \( v_0 \)
Now, we isolate \( v_0 \):
\[ 0.8v_0 = 340 - 272 \]
\[ 0.8v_0 = 68 \]

### Step 6: Solve for \( v_0 \)
Dividing both sides by 0.8:
\[ v_0 = \frac{68}{0.8} = 85 \text{ m/s} \]

### Step 7: Determine the Direction
Since the observer hears a lower frequency (the pitch is reduced), it confirms that the observer is moving away from the stationary source.

### Final Answer
The speed of the observer is \( 85 \text{ m/s} \) moving away from the source.

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