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When a train is at a distance of 2 km , ...

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)` )

A

`2,325.6 ms^(-1)`

B

`1,932.4 ms^(-1)`

C

`2,132.5 ms^(-1)`

D

`3,213.2 ms ^(-1)`

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to determine the speed of sound in iron given the distance of the train, the speed of sound in air, and the time interval between the two sounds heard by the man. ### Step-by-Step Solution: 1. **Identify the Given Data:** - Distance of the train from the man, \( d = 2 \text{ km} = 2000 \text{ m} \) - Speed of sound in air, \( v_{air} = 330 \text{ m/s} \) - Time interval between the two sounds, \( \Delta t = 5.2 \text{ s} \) 2. **Calculate the Time Taken for Sound to Travel in Air:** - The time taken for sound to travel through air, \( t_1 \): \[ t_1 = \frac{d}{v_{air}} = \frac{2000 \text{ m}}{330 \text{ m/s}} \approx 6.06 \text{ s} \] 3. **Set Up the Equation for Time Taken for Sound to Travel in Iron:** - Let the speed of sound in iron be \( v_{iron} \). - The time taken for sound to travel through iron, \( t_2 \): \[ t_2 = \frac{d}{v_{iron}} = \frac{2000 \text{ m}}{v_{iron}} \] 4. **Relate the Two Times Using the Given Time Interval:** - According to the problem, the difference in time between the two sounds is: \[ t_1 - t_2 = \Delta t \] Substituting the values we have: \[ 6.06 \text{ s} - \frac{2000 \text{ m}}{v_{iron}} = 5.2 \text{ s} \] 5. **Rearranging the Equation:** - Rearranging gives: \[ \frac{2000}{v_{iron}} = 6.06 - 5.2 \] \[ \frac{2000}{v_{iron}} = 0.86 \] 6. **Solve for \( v_{iron} \):** - Rearranging to find \( v_{iron} \): \[ v_{iron} = \frac{2000 \text{ m}}{0.86} \approx 2325.58 \text{ m/s} \] 7. **Final Answer:** - The speed of sound in iron is approximately \( 2325.58 \text{ m/s} \). ### Summary: The speed of sound in iron is approximately \( 2325.58 \text{ m/s} \).
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