An explosion occurs at the end of a pier. The sound reaches the other end of the pier by traveling through three media: air, fresh water, and a slender metal handrail. The speeds of sound in air, water, and the handrail are 356, 1460, and 5400 m/s, respectively. The sound travels a distance of 166 m in each medium. (a) After the first sound arrives, how much later does the second sound arrive? (b) After the first sound arrives, how much later does the third sound arrive?
To solve this problem, we need to calculate the time it takes for the sound to travel through each medium using the formula:
Time = Distance / Speed
(a) To find the time it takes for the sound to travel through water and reach the other end of the pier, we can use the speed of sound in water (1460 m/s) and the distance traveled in water (166 m).
Time in water = 166 m / 1460 m/s = 0.11369863 s
Since the second sound does not need to travel through air, we do not need to calculate the time it takes for the sound to travel through air.
(b) To find the time it takes for the sound to travel through the slender metal handrail and reach the other end of the pier, we can use the speed of sound in the handrail (5400 m/s) and the distance traveled in the handrail (166 m).
Time in handrail = 166 m / 5400 m/s = 0.03074074 s
To find the time difference between the first and second sound arrivals, we subtract the time for the first sound from the time for the second sound:
Time difference = Time in water - Time in air = 0.11369863 s
To find the time difference between the first and third sound arrivals, we subtract the time for the first sound from the time for the third sound:
Time difference = Time in handrail - Time in air = 0.03074074 s
Therefore, (a) the second sound arrives approximately 0.1137 seconds later after the first sound, and (b) the third sound arrives approximately 0.0307 seconds later after the first sound.
To calculate the time it takes for each sound to travel through each medium and arrive at the other end of the pier, we can use the formula:
Time = Distance / Speed
Let's calculate the time it takes for each sound to travel through each medium and determine the time differences.
(a) After the first sound arrives, how much later does the second sound arrive?
The speed of sound in air is 356 m/s, and the distance traveled through air is 166 m. Using the formula, we can calculate the time it takes for the sound to travel through air:
Time in air = Distance in air / Speed in air
= 166 m / 356 m/s
= 0.466 seconds
The speed of sound in water is 1460 m/s, and the distance traveled through water is also 166 m. Using the formula, we can calculate the time it takes for the sound to travel through water:
Time in water = Distance in water / Speed in water
= 166 m / 1460 m/s
= 0.114 seconds
The time difference between the arrival of the second sound and the arrival of the first sound is:
Time difference = Time in water - Time in air
= 0.114 seconds - 0.466 seconds
= -0.352 seconds
Therefore, the second sound arrives 0.352 seconds earlier than the first sound.
(b) After the first sound arrives, how much later does the third sound arrive?
The speed of sound in the handrail is 5400 m/s, and the distance traveled through the handrail is also 166 m. Using the formula, we can calculate the time it takes for the sound to travel through the handrail:
Time in handrail = Distance in handrail / Speed in handrail
= 166 m / 5400 m/s
= 0.031 seconds
The time difference between the arrival of the third sound and the arrival of the first sound is:
Time difference = Time in handrail - Time in air
= 0.031 seconds - 0.466 seconds
= -0.435 seconds
Therefore, the third sound arrives 0.435 seconds later than the first sound.