two trains move towards each other sith the same speed. The speed of sound is 340 m/s. If the height of the tone of the whistle of one of them heard on the other changes 9/8 times, then the speed of each train should be

Options

A. 20 m/s
B. 2 m/s
C. 200 m/s
D. 2000 m/s

Answer

To solve the problem, we will use the Doppler effect formula for sound. Let's break down the steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have two trains moving towards each other with the same speed \( V_1 \).
- The speed of sound \( v \) is given as 340 m/s.
- The frequency of the whistle heard by the observer changes by a factor of \( \frac{9}{8} \).

2. **Doppler Effect Formula**:
- The apparent frequency \( f' \) can be calculated using the formula:
\[ f' = f \cdot \frac{v + v_o}{v - v_s} \]
- Here, \( f \) is the original frequency, \( v \) is the speed of sound, \( v_o \) is the speed of the observer, and \( v_s \) is the speed of the source.

3. **Assigning Values**:
- Let \( v_o = V_1 \) (the speed of the observer, which is the speed of the train moving towards the source).
- Let \( v_s = V_1 \) (the speed of the source, which is the speed of the train producing the whistle).
- Since both trains are moving towards each other, \( v_o \) will be positive and \( v_s \) will be negative in the formula.

4. **Setting Up the Equation**:
- Since \( f' = \frac{9}{8} f \), we can write:
\[ \frac{f'}{f} = \frac{9}{8} = \frac{340 + V_1}{340 - V_1} \]

5. **Cross-Multiplying**:
- Cross-multiplying gives us:
\[ 9(340 - V_1) = 8(340 + V_1) \]

6. **Expanding Both Sides**:
- Expanding the equation:
\[ 3060 - 9V_1 = 2720 + 8V_1 \]

7. **Rearranging the Equation**:
- Moving all terms involving \( V_1 \) to one side and constant terms to the other side:
\[ 3060 - 2720 = 8V_1 + 9V_1 \]
\[ 340 = 17V_1 \]

8. **Solving for \( V_1 \)**:
- Dividing both sides by 17:
\[ V_1 = \frac{340}{17} = 20 \text{ m/s} \]

### Final Answer:
The speed of each train is \( 20 \text{ m/s} \).

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