Problem 37 - Physics

Question

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 \(343,1482,\) and \(5040 \mathrm{~m} / \mathrm{s},\) respectively. The sound travels a distance of \(125 \mathrm{~m}\) in each medium. (a) Through which medium does the sound arrive first, second, and third? (b) After the first sound arrives, how much later do the second and third sounds arrive?

Step-by-Step Solution

Answer

The sound arrives first through the metal handrail, then water, and finally air. The second sound arrives 0.059 s later, and the third sound arrives 0.339 s later.

Step 1: Calculate the Time for Sound Through Air

Use the formula for time, \[\text{Time} (t) = \frac{\text{Distance} (d)}{\text{Speed} (v)}\]Insert the given values: distance \(d = 125 \text{ m}\), and speed of sound in air \(v = 343 \text{ m/s}\):\[t_{\text{air}} = \frac{125 \text{ m}}{343 \text{ m/s}} \approx 0.364 \text{ seconds}\]

Step 2: Calculate the Time for Sound Through Water

Use the same formula for time:\[t_{\text{water}} = \frac{125 \text{ m}}{1482 \text{ m/s}} \approx 0.084 \text{ seconds}\]

Step 3: Calculate the Time for Sound Through Metal Handrail

Use the same formula for time:\[t_{\text{metal}} = \frac{125 \text{ m}}{5040 \text{ m/s}} \approx 0.025 \text{ seconds}\]

Step 4: Determine the Arrival Order

Compare the calculated times for each medium:- Metal handrail: \(0.025 \text{ s}\)- Water: \(0.084 \text{ s}\)- Air: \(0.364 \text{ s}\)The sound first arrives through the metal handrail, then through water, and finally through air.

Step 5: Calculate Time Delay After First Arrival

Calculate the differences between the times for sound to arrive through each medium compared to the metal handrail (first arrival):- Time delay for water: \[t_{\text{water delay}} = t_{\text{water}} - t_{\text{metal}} = 0.084 \text{ s} - 0.025 \text{ s} = 0.059 \text{ s}\]- Time delay for air:\[t_{\text{air delay}} = t_{\text{air}} - t_{\text{metal}} = 0.364 \text{ s} - 0.025 \text{ s} = 0.339 \text{ s}\]

Key Concepts

Speed of Sound

The speed of sound refers to the rate at which sound waves travel through a medium. Sound propagates by molecular collisions, so the speed of sound depends on how quickly these collisions occur, which is influenced by the medium's properties.
Factors that affect the speed of sound include:

• Density: Generally, sound travels faster in solids than in liquids, and faster in liquids than in gases.
• Elasticity: More elastic media allow sound waves to travel faster.
• Temperature: Higher temperatures usually increase the speed of sound in gases, as molecules move more swiftly.

Knowing the speed of sound in different environments is crucial for calculations involving distance and time, especially when dealing with events like explosions or using technologies like sonar.

Medium of Transmission

The medium of transmission is the material through which sound waves travel. It includes the usual states of matter: solid, liquid, and gas. For example, in the given exercise, sound travels through air, fresh water, and a slender metal handrail. Each of these media has unique properties affecting sound transmission.
Consider the mediums from the problem:

• Air: Sound speed is slowest here at 343 m/s due to lower density and elasticity.
• Fresh Water: Sound travels faster at 1482 m/s. Water's molecules are closer together, facilitating quicker transmission than in air.
• Metal Handrail: The fastest medium in the scenario, with a sound speed of 5040 m/s. Metal is dense and very elastic, enabling rapid sound conduction.

Understanding mediums helps in predicting how sound will behave, crucial for applications such as underwater communication and structural health monitoring.

Time Calculation

To determine the time it takes for sound to travel through a medium, you can use the simple formula: \[ t = \frac{d}{v} \] where:

• \(t\) is the time,
• \(d\) is the distance,
• \(v\) is the speed of sound in the medium.

In our scenario:

• Air: \( t_{\text{air}} = \frac{125 \text{ m}}{343 \text{ m/s}} \approx 0.364 \text{ s} \)
• Water: \( t_{\text{water}} = \frac{125 \text{ m}}{1482 \text{ m/s}} \approx 0.084 \text{ s} \)
• Metal: \( t_{\text{metal}} = \frac{125 \text{ m}}{5040 \text{ m/s}} \approx 0.025 \text{ s} \)

By calculating these times, we can understand the delay in sound reaching different points or observers, which is essential in sound recording, sonar detection, and other audio-related fields.

Sound Travel Through Different Media

When sound travels through different media, its velocity changes due to differences in each medium's density and elasticity. This behavior explains why sound arrives at different times once it travels through air, water, and metals, as shown in the exercise.

In the scenario with the explosion:

• The fastest arrival was through the metal handrail because metals are dense and elastic, making sound travel swiftly.
• The next arrival was through water, as it offers less resistance than air, although more than metal.
• Lastly, sound reached through air, the slowest medium due to its lower density and elasticity.

This helps us understand practical phenomena like echoes in different environments and clarifies why underwater sounds might seem closer. Applications extend to designing concert halls for better acoustics or engineering communication devices that work in specific environments.