Problem 48 - University Physics with Modern Physics

Question

A railroad train is traveling at 30.0 \(\mathrm{m} / \mathrm{s}\) in still air. The frequency of the note cmitted by the train whistle is 262 \(\mathrm{Hz}\) . What frequency is heard by a passenger on a train moving in the opposite direction to the first at 18.0 \(\mathrm{m} / \mathrm{s}\) and (a) approaching the first; and (b) receding from the first?

Step-by-Step Solution

Answer

Approaching: ~302.1 Hz. Receding: ~271.9 Hz.

Step 1: Understand the Problem

We need to determine the frequency heard by a passenger on a train moving in the opposite direction of a train emitting a whistle. Two scenarios are considered: when the passenger is approaching and receding from the emitting train. We will use the Doppler effect for sound to solve this.

Step 2: Identify the Known Variables

The speed of sound in air is approximately \( v = 343 \, \mathrm{m/s} \), frequency of the whistle \( f_0 = 262 \, \mathrm{Hz} \), speed of the first train \( v_s = 30.0 \, \mathrm{m/s} \), and speed of the second train (passenger's train) \( v_o = 18.0 \, \mathrm{m/s} \).

Step 3: Apply Doppler Effect Formula for Approaching Trains

The frequency heard \( f \) when the trains are approaching is given by:\[f = f_0 \cdot \frac{v + v_o}{v - v_s}\]Substitute the known values: \( f = 262 \cdot \frac{343 + 18}{343 - 30} \).

Step 4: Calculate Frequency for Approaching Trains

Calculate the frequency:\[f = 262 \cdot \frac{361}{313} = 262 \cdot 1.153\]\[f \approx 302.1 \, \mathrm{Hz}\]

Step 5: Apply Doppler Effect Formula for Receding Trains

The frequency heard when the trains are receding is given by:\[f = f_0 \cdot \frac{v - v_o}{v - v_s}\]Substitute the known values: \( f = 262 \cdot \frac{343 - 18}{343 - 30} \).

Step 6: Calculate Frequency for Receding Trains

Calculate the frequency:\[f = 262 \cdot \frac{325}{313} = 262 \cdot 1.038\]\[f \approx 271.9 \, \mathrm{Hz}\]

Step 7: Conclusion: Frequency Heard by Passenger

- When approaching: approximately \( 302.1 \, \mathrm{Hz} \).- When receding: approximately \( 271.9 \, \mathrm{Hz} \).

Key Concepts

Sound Frequency

The concept of sound frequency refers to the number of sound wave cycles that occur in one second. It is measured in Hertz (Hz), and it determines the pitch of the sound we hear. A higher frequency means a higher pitch, like a whistle, whereas a lower frequency corresponds to a deeper sound, like a drum beat.
When an object like a train whistle emits sound, it produces waves that move through the air at a specific frequency. In the problem at hand, the train whistle emits a sound at 262 Hz.
It's essential to understand that this frequency might not be the frequency heard by an observer or passenger if they are moving relative to the sound source. This variation in perceived frequency due to relative motion is explained by the Doppler effect. It can cause the sound to seem higher in pitch (or frequency) when you're moving towards the source, and lower when moving away.

Relative Motion

Relative motion refers to the movement of an object as observed from another moving or stationary object. In exercises involving the Doppler effect, like the one we are discussing, understanding relative motion is crucial.
In this scenario, you have two trains moving in opposite directions, which makes their relative motion significant in determining the frequency of sound perceived by each other. The first train is moving at 30.0 m/s, and the passenger's train is moving at 18.0 m/s.
When these trains are moving towards each other, their combined speed affects how sound waves are compressed, resulting in a higher frequency detected by the passenger. Conversely, when the trains are moving away, those waves are stretched, causing a lower frequency to be heard.
The relative speed of these trains dictates this apparent change, applying the Doppler effect to find the frequency shifts in both approaching and receding scenarios.

Speed of Sound

The speed of sound is the rate at which sound waves travel through a medium, such as air. It is a critical component in the Doppler effect calculations because it serves as the baseline speed for sound wave propagation.
In air at room temperature, this speed is approximately 343 m/s. However, it can vary due to factors like temperature, air pressure, and humidity.

• When incorporating speed of sound into calculations, it serves as the denominator or divisor when using the Doppler effect formula.
• It helps in determining how much the sound's frequency is shifted due to the relative speeds of the source and the observer.

In our given problem, the speed of sound is used along with the trains' speeds to calculate the frequencies heard by the passenger when approaching and receding from the sound source. Therefore, recognizing how the speed of sound interacts with other velocities allows you to accurately assess the change in perceived frequency.