Problem 50 - 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 emitted 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 tirst at 18.0 \(\mathrm{m} / \mathrm{s}\) and (a) approaching the first and (b) receding from the first?

Step-by-Step Solution

Answer

(a) Approximately 303 Hz, (b) approximately 272 Hz.

Step 1: Define the Problem

We need to calculate the frequency heard by a passenger on a second train. First, identify the given values: Original frequency \( f = 262 \, \text{Hz} \), speed of sound in air \( v = 343 \, \text{m/s} \), speed of the first train \( v_s = 30.0 \, \text{m/s} \), and speed of the second train \( v_o = 18.0 \, \text{m/s} \).

Step 2: Apply the Doppler Effect Formula for Approaching Trains

For part (a), as the second train is approaching the first train, the formula is:\[f' = \left(\frac{v + v_o}{v - v_s}\right) f\]Substitute the values:\[f' = \left(\frac{343 + 18.0}{343 - 30.0}\right) \times 262\]

Step 3: Calculate the Frequency for Approaching Trains

Calculate the frequency:\[f' = \left(\frac{361}{313}\right) \times 262 \]\[f' \approx 302.7 \, \text{Hz}\]So, the frequency heard by the passenger when approaching is approximately 303 Hz.

Step 4: Apply the Doppler Effect Formula for Receding Trains

For part (b), as the second train is moving away, the formula is:\[f' = \left(\frac{v - v_o}{v - v_s}\right) f\]Substitute the values:\[f' = \left(\frac{343 - 18.0}{343 - 30.0}\right) \times 262\]

Step 5: Calculate the Frequency for Receding Trains

Calculate the frequency:\[f' = \left(\frac{325}{313}\right) \times 262 \]\[f' \approx 271.5 \, \text{Hz}\]So, the frequency heard by the passenger when receding is approximately 272 Hz.

Key Concepts

Understanding Frequency

Frequency is a fundamental concept in wave physics. It refers to the number of occurrences of a repeating event per unit time. In the context of sound waves, frequency determines the pitch of the sound. For instance, a high frequency corresponds to a high-pitched sound, like a soprano's singing voice or a bird's chirp.

• In our train whistle problem, the original frequency of the train whistle is given as 262 Hz. This means the sound wave repeats 262 times per second.
• This frequency is altered by the relative motion of the trains due to what is known as the Doppler Effect.

Understanding frequency is crucial for solving problems involving sound waves in motion, like the train whistle scenario.

The Role of the Train Whistle

A train whistle is a device designed to create a loud sound to alert people and vehicles when a train is approaching. This sound is a result of sound waves being generated by the whistle and shot out into the air.

• In this scenario, the whistle produces a steady frequency of 262 Hz when the train is stationary or heard at rest relative to the listener.
• However, as the train moves, the sound frequency perceived by an observer changes. This is due to the relative motion between the sound source (train) and the observer (passenger on another train).

Thus, the train whistle serves as a practical example of the Doppler Effect at work, altering the perceived frequency depending on whether the train is approaching or receding.

Velocity and Their Influence in the Scenario

Velocity plays a significant role in determining how we perceive the frequency of sound waves from moving sources like trains. Both the velocity of the sound wave and the velocities of the observer and the source are crucial here.

• In the original problem, the speed of sound in air is assumed to be 343 m/s.
• The first train moves at a velocity of 30.0 m/s while the second train has a velocity of 18.0 m/s, each contributing differently when trains are approaching or receding.

For this scenario:
- When the two trains are moving towards each other, the frequencies heard become higher.
- Conversely, when they move away, the perceived frequency lowers.
This change occurs because their relative velocities alter the compression and elongation of sound waves, effectively squeezing or stretching them.

The Nature of Sound Waves

Sound waves are vibrations that travel through the air (or other mediums) and are perceived by our ears as sound. They can be described as mechanical waves, meaning they require a medium like air or water to travel.

• They propagate in the form of longitudinal waves, where particle displacement is parallel to the direction of wave propagation.
• The speed at which sound waves move through air is determined by various factors including temperature and pressure, but is assumed to be approximately 343 m/s at room temperature.

In practical applications such as the problem with the trains, sound waves help illustrate how the speed and direction of an object affect the frequency detected by an observer. The Doppler Effect leverages this principle by altering the frequency due to the motion of the sound source relative to the listener.