Problem 20 - Physics

Question

A celebrity holds a press conference, which is televised live. A television viewer hears the sound picked up by a microphone directly in front of the celebrity. This viewer is seated \(2.3 \mathrm{~m}\) from the television set. A reporter at the press conference is located \(4.1 \mathrm{~m}\) from the microphone and hears the words directly at the very same instant that the television viewer hears them. Using a value of \(343 \mathrm{~m} / \mathrm{s}\) for the speed of sound, determine the maximum distance between the television viewer and the celebrity.

Step-by-Step Solution

Answer

The maximum distance is 4.116 meters.

Step 1: Understand the Problem

We need to measure the maximum distance between a television viewer and a celebrity who is speaking during a press conference. The viewer hears the sound from the TV after it travels a certain distance from the microphone in front of the celebrity. The viewer must be farther than the microphone from the sound origin so that the time taken for sound to reach both the viewer and a reporter is the same.

Step 2: Calculate the Time for Sound to Travel to the Reporter

The distance from the celebrity (through the microphone) to the reporter is given as \[ d_r = 4.1 \text{ m} \]The speed of sound is \[ v = 343 \text{ m/s} \]The time taken for sound to reach the reporter is therefore \[ t_r = \frac{d_r}{v} = \frac{4.1}{343} \approx 0.012 \text{ seconds} \]

Step 3: Calculate the Time for Sound to Travel to the TV

Assume the distance from the celebrity to the TV is \[ d_m \] Since sound first travels from the celebrity to the TV microphone and then from the TV to the viewer, the total distance is \[ d_t = d_m + 2.3 \text{ m} \]The time taken for sound to travel this distance: \[ t_v = \frac{d_m + 2.3}{343} \]

Step 4: Equate Times to Solve for Maximum Distance

Since the viewer and the reporter hear the sound at the same time: \[ t_v = t_r \]Substitute the expressions:\[ \frac{d_m + 2.3}{343} = 0.012 \]Solve for \( d_m \):\[ d_m + 2.3 = 0.012 \times 343 \]\[ d_m + 2.3 = 4.116 \]\[ d_m = 4.116 - 2.3 = 1.816 \text{ m} \]

Step 5: Calculate Maximum Distance to the Viewer

The maximum distance from the viewer to the celebrity is the total path of the sound from the celebrity to where the viewer is seated:\[ d_v = d_m + 2.3 = 1.816 + 2.3 = 4.116 \text{ m} \]

Step 6: Solution Conclusion

The maximum distance between the television viewer and the celebrity is \( 4.116 \text{ meters} \).

Key Concepts

Speed of Sound

Understanding the speed of sound is crucial in solving problems involving sound wave travel. Speed of sound refers to how fast sound waves can travel through a medium, like air. Typically, the speed of sound in air at room temperature is about 343 m/s. This speed can change depending on variables such as temperature and pressure.
The faster the speed, the quicker the sound reaches an observer.
Knowing this speed allows you to calculate how long it takes for a sound to travel a certain distance.

• Measured in meters per second (m/s).
• Depends on environmental factors (e.g., temperature, humidity).
• Understanding this allows for precise calculations in physics problems.

Distance Calculation

Calculating distances using the speed of sound involves using the fundamental relationship between time, speed, and distance. The formula to find the distance sound travels in a certain time is:
\[ d = v \times t \] where \( d \) is the distance, \( v \) is the speed of sound, and \( t \) is the time the sound takes to travel.
This formula can be rearranged to find time and speed if the other variables are known.
In the given problem, the distances that sound travels to different listeners help determine the relative positions and maximum distances involved:

• Direct path to a reporter.
• Total path to a television viewer through indirect travel by a microphone and TV.

These distances are linked with the speed of sound to calculate the time taken for these travels.

Sound Wave Propagation

Sound wave propagation refers to how sound travels through a medium like air. When a sound is generated, it creates vibrations that move molecules in the medium, transferring energy from one to the next.
The sound wave continues to travel until it reaches the ear or a microphone, converting it into an audible message.
This process is instantaneous, but the time taken for the sound to travel varies with distance and the speed of sound.
Insights into sound wave propagation can solve problems like determining if two observers hear the same sound simultaneously, as seen in this exercise:

• Identifying paths of travel (direct or indirect).
• Ensuring equal time of reception for simultaneous hearing.
• Analyzing how waves interact with their surroundings.

Acoustics in Physics

Acoustics involves studying sound and how it behaves in different environments. This field helps understand sound's production, control, transmission, and effects.
Acoustics provides essential knowledge in solving real-world problems like sound distribution in a room or concert hall.
In the problem at hand, acoustics principles allow us to determine how the sound travels from a source to various listeners while considering factors such as distance and barriers:

• Identifying direct versus indirect paths for sound.
• Understanding how sound is picked up and relayed by a microphone.
• Solving practical problems with theoretical insights.

By applying acoustic principles, we can predict where and when the sound will reach listeners accurately.