A train blowintg its whistle moves with constant - speed on a straight tack towards obsever and then crosses him . If the ratio and difference between the actual and apparent frequencies be 3:2 in the two cases , then the speed of train is [v is speed of sound ]

Options

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

Answer

To solve the problem, we'll use the Doppler effect for sound waves. The apparent frequency of a sound depends on the relative motion between the source of sound and the observer.

### Step-by-Step Solution:

1. **Identify the Frequencies**:
- Let \( f_r \) be the actual frequency of the whistle.
- Let \( f_1 \) be the apparent frequency when the train is approaching the observer.
- Let \( f_2 \) be the apparent frequency when the train is moving away from the observer.

2. **Doppler Effect Formulas**:
- When the source (train) is moving towards the observer:
\[ f_1 = f_r \frac{v}{v - v_t} \]
- When the source is moving away from the observer:
\[ f_2 = f_r \frac{v}{v + v_t} \]
Here, \( v \) is the speed of sound, and \( v_t \) is the speed of the train.

3. **Calculate the Differences**:
- The difference between the actual frequency and the apparent frequency when approaching:
\[ f_1 - f_r = f_r \frac{v}{v - v_t} - f_r = f_r \left( \frac{v}{v - v_t} - 1 \right) = f_r \left( \frac{v - (v - v_t)}{v - v_t} \right) = f_r \left( \frac{v_t}{v - v_t} \right) \]
- The difference when receding:
\[ f_r - f_2 = f_r - f_r \frac{v}{v + v_t} = f_r \left( 1 - \frac{v}{v + v_t} \right) = f_r \left( \frac{v + v_t - v}{v + v_t} \right) = f_r \left( \frac{v_t}{v + v_t} \right) \]

4. **Set Up the Ratio**:
- According to the problem, the ratio of the differences is given as:
\[ \frac{f_1 - f_r}{f_r - f_2} = \frac{3}{2} \]
- Substituting the expressions for the differences:
\[ \frac{f_r \left( \frac{v_t}{v - v_t} \right)}{f_r \left( \frac{v_t}{v + v_t} \right)} = \frac{3}{2} \]
- Simplifying this gives:
\[ \frac{v + v_t}{v - v_t} = \frac{3}{2} \]

5. **Cross Multiply and Solve**:
- Cross multiplying gives:
\[ 2(v + v_t) = 3(v - v_t) \]
- Expanding and rearranging:
\[ 2v + 2v_t = 3v - 3v_t \implies 5v_t = v \implies v_t = \frac{v}{5} \]

### Final Answer:
The speed of the train \( v_t \) is \( \frac{v}{5} \).

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