Question

A railroad train is traveling at 30.0 m/s in still air. The frequency of the note emitted by the train whistle is 352 Hz. What frequency is heard by a passenger on a train moving in the opposite direction to the first at 18.0 m/s and (a) approaching the first and (b) receding from the first?

Short answer

Approaching: 406.3 Hz; Receding: 365.1 Hz.

Step-by-step solution

Understand the Problem Context

We are given the frequency of the note from a whistle on a train, and two scenarios where a passenger on another train moving in the opposite direction either approaches or recedes from the first train. We need to determine the frequency heard by the passenger in both scenarios.

Identify Relevant Formula

The problem relates to the Doppler effect, which describes the change in frequency observed when a source of sound and the observer are in relative motion. The formula for the apparent frequency $f^{\prime }$ is given by:$$f^{\prime }=\left(\frac{v+v_o}{v-v_s}\right)f$$where $v$ is the speed of sound in air (assumed to be approximately 343 m/s), $v_o$ is the speed of the observer, $v_s$ is the speed of the source, and $f$ is the emitted frequency.

Calculate Frequency for the Approaching Scenario

In this scenario, the observer is approaching the source. Thus, $v_o=18.0$ m/s and $v_s=30.0$ m/s. Apply the values to the formula:$$f^{\prime }=\left(\frac{343+18}{343-30}\right)\times 352$$Simplifying:$$f^{\prime }=\left(\frac{361}{313}\right)\times 352\approx 406.3\text{Hz}$$

Calculate Frequency for the Receding Scenario

In this scenario, the observer is moving away from the source, meaning the relative velocity of the observer reduces frequency. Here, $v_o=-18.0$ m/s while $v_s=30.0$ m/s remains the same:$$f^{\prime }=\left(\frac{343-18}{343-30}\right)\times 352$$Simplifying:$$f^{\prime }=\left(\frac{325}{313}\right)\times 352\approx 365.1\text{Hz}$$

Key concepts

Frequency Calculation

Frequency calculation in the context of the Doppler Effect is essential to understand how sound waves are perceived differently due to relative motion between a source and an observer.
To determine the frequency heard by an observer moving relative to a sound source, we use the Doppler effect formula:

• $f^{\prime }=\left(\frac{v+v_o}{v-v_s}\right)f$
• Here, $f^{\prime }$ is the apparent frequency heard by the observer.
• $f$ is the original frequency of the sound emitted by the source.
• $v$ is the speed of sound in the medium, often air, which is approximately 343 m/s.
• $v_o$ is the speed of the observer moving towards or away from the source.
• $v_s$ is the speed of the source of sound.

For an approaching observer, the speed $v_o$ is added, increasing perceived frequency.
Conversely, when the observer moves away, $v_o$ becomes negative, reducing the frequency.

Relative Motion

The concept of relative motion is crucial when discussing the Doppler Effect. It examines how movement impacts observation between two entities: the source of sound and the observer.
In the problem context, both trains are moving in opposite directions, affecting the frequency of the sound heard.

• When the observer moves towards the sound source, the frequency appears higher than the actual emitted frequency. This is due to the sound waves becoming compressed.
• When moving away, the frequency seems lower, as the sound waves stretch out.

By understanding and applying the concept of relative motion, one can accurately predict and calculate how motion affects sound perception. The Doppler formula quantifies this by adjusting how the speeds of the observer and source alter the perceived frequency.

Train Whistle Sounds

The sound of a train whistle is an ideal example for exploring the Doppler Effect, due to its loud, clear, and consistent frequency.
This scenario can often be recognized in everyday life, when a train approaches and then passes by:

• As the train gets closer, the pitch of the train whistle sounds higher.
• Once it passes, the pitch drops noticeably lower.

In the exercise, the train whistle has an actual frequency of 352 Hz. But for the observer on the opposing train, this frequency changes depending on their motion: - Approaching: The frequency rises to approximately 406.3 Hz. - Receding: The frequency lowers to about 365.1 Hz. Comprehending how train whistle sounds change with movement enhances the understanding of sound perception in motion and demonstrates real-world application of the Doppler Effect.