Vinay fires two bullets from the same place at an interval of 12 minutes but Raju sitting in a train approaching the place hears the second report 11 minutes 30 seconds after the first. What is the approximate speed of train (if sound travels at the speed of 330 metre per second)?

Options

A. 660/23 m/s
B. 220/7 m/s
C. 330/23 m/s
D. 110/23 m/s

Answer

To solve the problem, we need to determine the speed of Raju's train based on the information provided about the firing of the bullets and the time it takes for Raju to hear the second bullet.

### Step-by-Step Solution:

1. **Understanding the Time Intervals**:
- The first bullet is fired at time \( t = 0 \) minutes.
- The second bullet is fired at \( t = 12 \) minutes.
- Raju hears the first bullet at \( t = 0 \) minutes and the second bullet at \( t = 11 \) minutes and 30 seconds.

2. **Calculating the Time Difference**:
- The time difference between hearing the first and second bullet is \( 11 \) minutes and \( 30 \) seconds = \( 11.5 \) minutes.
- The second bullet is fired \( 12 \) minutes after the first bullet, so the time taken for Raju to hear the second bullet after it was fired is:
\[ 12 \text{ minutes} - 11.5 \text{ minutes} = 0.5 \text{ minutes} = 30 \text{ seconds} \]

3. **Understanding the Sound Travel**:
- The speed of sound is given as \( 330 \) meters per second.
- The distance sound travels in \( 30 \) seconds is:
\[ \text{Distance} = \text{Speed} \times \text{Time} = 330 \text{ m/s} \times 30 \text{ s} = 9900 \text{ meters} \]

4. **Setting Up the Relative Motion**:
- Let \( v \) be the speed of the train (Raju's speed).
- In \( 30 \) seconds, while the sound travels \( 9900 \) meters, Raju also travels a distance of \( v \times 30 \) seconds.

5. **Establishing the Equation**:
- The total distance covered by sound is the distance Raju travels plus the distance the sound travels:
\[ 9900 = v \times 30 + 9900 \]
- Since Raju is moving towards the source of the sound, we can set up the equation as:
\[ 9900 = 9900 - v \times 30 \]
- Rearranging gives:
\[ v \times 30 = 9900 - 9900 + 9900 \]
- Therefore, we can simplify this to:
\[ v \times 30 = 9900 \]

6. **Solving for Raju's Speed**:
- Dividing both sides by \( 30 \):
\[ v = \frac{9900}{30} = 330 \text{ meters per second} \]

7. **Adjusting for the Speed of Sound**:
- The speed of Raju's train is actually the speed of sound divided by the ratio of the time taken for sound to reach Raju and the time taken for Raju to hear the sound.
- The ratio of the time taken by sound to the time taken by Raju is \( 30 \) seconds to \( 690 \) seconds (since \( 12 \) minutes = \( 720 \) seconds):
\[ \text{Ratio} = \frac{30}{690} = \frac{1}{23} \]
- Therefore, Raju's speed is:
\[ v = \frac{330}{23} \approx 14.35 \text{ meters per second} \]

### Final Answer:
The approximate speed of Raju's train is \( \frac{330}{23} \) meters per second.