A source of sound is travelling towards a stationary observer. The frequency of sound heard by the observer is 25% more that the actual frequency. If the speed of sound is v, that of the source is

Options

A. `(v)/(5)`
B. `(v)/(4)`
C. `(v)/(3)`
D. `(v)/(2)`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Given Information
We know that:
- The actual frequency of the sound emitted by the source is \( F_0 \).
- The frequency heard by the observer is 25% more than the actual frequency, which can be expressed as:
\[ F' = F_0 + 0.25F_0 = \frac{5}{4}F_0 \]
- The speed of sound in the medium is \( v \).

### Step 2: Use the Doppler Effect Formula
The formula for the frequency heard by a stationary observer when the source is moving towards the observer is given by:
\[ F' = F_0 \frac{v}{v - v_s} \]
where \( v_s \) is the speed of the source.

### Step 3: Substitute the Known Values
Substituting \( F' \) with \( \frac{5}{4}F_0 \) in the Doppler effect formula, we have:
\[ \frac{5}{4}F_0 = F_0 \frac{v}{v - v_s} \]

### Step 4: Cancel \( F_0 \)
Since \( F_0 \) is common on both sides and is non-zero, we can cancel it out:
\[ \frac{5}{4} = \frac{v}{v - v_s} \]

### Step 5: Cross Multiply to Solve for \( v_s \)
Cross multiplying gives:
\[ 5(v - v_s) = 4v \]
Expanding this:
\[ 5v - 5v_s = 4v \]

### Step 6: Rearrange the Equation
Rearranging the equation to isolate \( v_s \):
\[ 5v - 4v = 5v_s \]
This simplifies to:
\[ v = 5v_s \]

### Step 7: Solve for \( v_s \)
Now, divide both sides by 5 to find \( v_s \):
\[ v_s = \frac{v}{5} \]

### Conclusion
The speed of the source \( v_s \) is:
\[ \boxed{\frac{v}{5}} \]