If a source of sound of frequency `upsilon` and a listener approach each other with a velocity equal to `(1//20)` of velocity of sound, the apparent frequency heard by the listener is

Options

A. `((21)/(19))upsilon`
B. `((20)/(21))upsilon`
C. `((21)/(20))upsilon`
D. `((19)/(20))upsilon`

Explanation

To solve the problem of finding the apparent frequency heard by a listener when both the source of sound and the listener are approaching each other, we can use the Doppler effect formula. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We have a source of sound with frequency \( \nu \) and a listener approaching the source. Both the source and the listener are moving towards each other with a velocity equal to \( \frac{1}{20} \) of the velocity of sound.

### Step 2: Define Variables
Let:
- \( v \) = velocity of sound
- \( v_s \) = velocity of the source = \( \frac{v}{20} \)
- \( v_o \) = velocity of the observer (listener) = \( \frac{v}{20} \)

### Step 3: Use the Doppler Effect Formula
The formula for the apparent frequency \( \nu' \) when both the source and observer are moving towards each other is given by:

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

### Step 4: Substitute Values
Substituting the values of \( v_o \) and \( v_s \):

\[ \nu' = \nu \frac{v + \frac{v}{20}}{v - \frac{v}{20}} \]

### Step 5: Simplify the Expression
Now, simplify the numerator and denominator:

1. **Numerator**:
\[ v + \frac{v}{20} = \frac{20v}{20} + \frac{v}{20} = \frac{21v}{20} \]

2. **Denominator**:
\[ v - \frac{v}{20} = \frac{20v}{20} - \frac{v}{20} = \frac{19v}{20} \]

### Step 6: Substitute Back into the Formula
Now substitute back into the formula:

\[ \nu' = \nu \frac{\frac{21v}{20}}{\frac{19v}{20}} \]

### Step 7: Cancel Out Common Terms
The \( v \) and \( 20 \) cancel out:

\[ \nu' = \nu \frac{21}{19} \]

### Step 8: Final Result
Thus, the apparent frequency heard by the listener is:

\[ \nu' = \frac{21}{19} \nu \]

### Conclusion
The apparent frequency heard by the listener is \( \frac{21}{19} \) times the original frequency \( \nu \).

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