When a train is at a distance of 2 km , its engine sounds a whistle . A man near the railway track hears the whistle directly and by placing his ear against the track of the train. If the two sounds are heard at an internal of 5.2 s, the speed of the sound in iron ( material of the rail track ) is : (Given that velocity of sound in air is 330 m `s ^(_1)` )

Options

A. `2,325.6 ms^(-1)`
B. `1,932.4 ms^(-1)`
C. `2,132.5 ms^(-1)`
D. `3,213.2 ms ^(-1)`

Answer

To solve the problem step by step, we can follow these calculations:

### Step 1: Understand the problem
We have a train that is 2 km away from a man near the railway track. The man hears the whistle of the train in two ways: directly through the air and through the iron track. The time interval between the two sounds is given as 5.2 seconds.

### Step 2: Convert distance to meters
The distance of the train from the man is given as 2 km. We need to convert this distance into meters for our calculations.
\[ \text{Distance} = 2 \text{ km} = 2000 \text{ m} \]

### Step 3: Calculate the time taken for sound to travel through air
The speed of sound in air is given as 330 m/s. We can calculate the time taken for the sound to travel through air using the formula:
\[ t_a = \frac{d}{v_a} \]
Where:
- \( d = 2000 \text{ m} \)
- \( v_a = 330 \text{ m/s} \)

Substituting the values:
\[ t_a = \frac{2000 \text{ m}}{330 \text{ m/s}} \approx 6.06 \text{ s} \]

### Step 4: Calculate the time taken for sound to travel through iron
Let \( t_i \) be the time taken for the sound to travel through the iron. According to the problem, the difference between the time taken for sound in air and sound in iron is 5.2 seconds:
\[ t_a - t_i = 5.2 \text{ s} \]
Substituting \( t_a \):
\[ 6.06 \text{ s} - t_i = 5.2 \text{ s} \]
Rearranging gives:
\[ t_i = 6.06 \text{ s} - 5.2 \text{ s} = 0.86 \text{ s} \]

### Step 5: Calculate the speed of sound in iron
Now we can calculate the speed of sound in iron using the formula:
\[ v_i = \frac{d}{t_i} \]
Where:
- \( d = 2000 \text{ m} \)
- \( t_i = 0.86 \text{ s} \)

Substituting the values:
\[ v_i = \frac{2000 \text{ m}}{0.86 \text{ s}} \approx 2325.58 \text{ m/s} \]

### Step 6: Final answer
The speed of sound in iron is approximately:
\[ v_i \approx 2325.58 \text{ m/s} \]

### Conclusion
The speed of sound in iron is approximately **2325.58 m/s**.

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