To solve this problem, we need to analyze how sound travels through different mediums and how the distances traveled by sound in each medium relate to the time it takes for Ravi to hear the sounds. Let's break this down step by step.
Understanding Sound Travel in Different Mediums
When Ram strikes the metal pipe, sound waves travel through the pipe, through the air, and through the water. Each medium has a different speed of sound:
- Velocity of sound in air: 300 m/s
- Velocity of sound in water: 600 m/s
- Velocity of sound in the metal pipe: 1200 m/s
Calculating the Time for Each Sound
Ravi hears two distinct sounds: one traveling through the metal pipe and the other through the water. We need to calculate the time it takes for each sound to reach Ravi.
1. Sound Traveling Through the Metal Pipe
Let’s denote the length of the pipe as L. The time taken for the sound to travel through the pipe can be calculated using the formula:
Time = Distance / Velocity
Thus, the time for sound in the pipe is:
T_pipe = L / 1200
2. Sound Traveling Through Water
Since the pipe is half immersed in water, the distance sound travels in water is L/2. The time taken for sound to travel through the water is:
T_water = (L/2) / 600 = L / 1200
3. Sound Traveling Through Air
The distance sound travels in air is also L/2. The time taken for sound to travel through the air is:
T_air = (L/2) / 300 = L / 600
Finding the Length of the Pipe
Ravi hears the sound from the pipe first, followed by the sound from the water. Therefore, the time for the sound in the pipe must be less than the time for the sound in the air:
T_pipe < T_air
Substituting the expressions we derived:
L / 1200 < L / 600
To solve this inequality, we can multiply both sides by 1200L (assuming L is positive):
L < 2L
This inequality is always true for any positive L, meaning Ravi will always hear the sound from the pipe before the sound from the air. Now, we need to find the actual length of the pipe based on the time difference.
Determining if Ravi Can Hear a Third Sound
For Ravi to hear a third sound, we need to consider if there’s a delay between the sounds he hears. Since the sound in the pipe reaches him first, followed by the sound in the water, we can check if the time difference is significant enough for him to perceive a third sound.
From our calculations:
T_pipe = L / 1200
T_water = L / 1200
T_air = L / 600
The time difference between the sound in the pipe and the sound in the air is:
T_air - T_pipe = (L / 600) - (L / 1200) = L / 1200
Since the sound in the pipe and the sound in water arrive at the same time, Ravi will not hear a third sound. He will only hear the sound from the pipe and the sound from the water simultaneously.
Final Thoughts
In summary, the length of the pipe can be any positive value, and Ravi will not hear a third distinct sound because the sounds from the pipe and the water arrive at the same time. This example illustrates how sound travels differently through various mediums and how these differences can affect our perception of sound.
