Problem 91 - Pearson Physics

Question

A particular jet engine produces a tone of \(495 \mathrm{~Hz}\). Suppose that one jet is at rest on the tarmac while a second identical jet flies overhead at \(82.5 \%\) of the speed of sound. The pilot of each jet listens to the sound produced by the engine of the other jet. (a) Calculate the frequency heard by the pilot in the moving jet. (b) Calculate the frequency heard by the pilot in the stationary jet, assuming the flying jet is moving toward it.

Step-by-Step Solution

Answer

(a) 903.375 Hz (b) 2828.57 Hz

Step 1: Identify Given Information

We have two jets: one stationary and one moving at a speed of \( 0.825c \), where \( c \) is the speed of sound. The stationary jet produces a tone of 495 Hz.

Step 2: Doppler Effect Formula for Moving Observer

When the observer (the moving jet) is moving towards a stationary source, the frequency heard \( f' \) is given by the formula \[ f' = f \left( \frac{c + v_o}{c} \right) \], where \( f \) is the source frequency, \( c \) is the speed of sound, and \( v_o \) is the observer's speed.

Step 3: Calculate Frequency for Moving Observer

Substitute \( f = 495 \text{ Hz} \), \( v_o = 0.825c \), and solve for \( f' \). \[ f' = 495 \left( \frac{c + 0.825c}{c} \right) = 495 \times 1.825 = 903.375 \text{ Hz} \]. This is the frequency heard by the moving jet pilot.

Step 4: Doppler Effect Formula for Stationary Observer

When the source (the moving jet) moves towards a stationary observer, the frequency heard \( f'' \) is given by the formula \[ f'' = f \left( \frac{c}{c - v_s} \right) \], where \( v_s \) is the source's speed.

Step 5: Calculate Frequency for Stationary Observer

Substitute \( f = 495 \text{ Hz} \), \( v_s = 0.825c \), and solve for \( f'' \). \[ f'' = 495 \left( \frac{c}{c - 0.825c} \right) = 495 \times \frac{1}{0.175} \approx 2828.57 \text{ Hz} \]. This is the frequency heard by the stationary jet pilot.

Key Concepts

Frequency Calculation

The Doppler Effect is an essential concept when calculating the frequencies heard by observers in motion relative to a sound source. The key to understanding this concept is recognizing how the speed of the observer and the source affects the frequency of the sound wave perceived. In a typical scenario involving the Doppler Effect for sound, we can calculate the apparent frequency detected by a moving observer or an observer at rest when the source of the sound is in motion. For a moving observer approaching a stationary sound source, the formula to use is:

• \( f' = f \left( \frac{c + v_o}{c} \right) \)

Here, \( f \) is the frequency of the source, \( c \) is the speed of sound, and \( v_o \) is the observer's speed. By substituting the known values into the equation, the frequency perceived by the moving observer can be determined, providing insights into how relative motion alters sound perception. When the source moves towards a stationary observer, the formula becomes:

• \( f'' = f \left( \frac{c}{c - v_s} \right) \)

In this equation, \( v_s \) represents the speed of the source. Understanding these formulas and applying them accurately enables one to clarify how frequencies shift due to relative motion.

Jet Engine Sound

A jet engine produces a distinct and powerful sound, characterized not just by its volume but also by its frequency, which is measured in Hertz (Hz). In our exercise, the stationary jet engine produces a sound with a frequency of 495 Hz. This frequency can vary, depending on whether the listener is stationary or in motion relative to the engine. The noise from jet engines can seem intimidating due to its high energy output and rapid vibration of air particles, which contributes to the perception of loudness and frequency. As jets move at significant portions of the speed of sound, analyzing such scenarios helps us understand how jet noise varies for different observers. These shifts in sound frequency are explained by the Doppler Effect, which provides a framework for understanding how high-speed motion affects what we hear.

Sound Waves

Sound waves are mechanical waves that travel through a medium, namely air in this context. Sound wave propagation involves compressions and rarefactions that travel outward from the source, such as a jet engine. These waves have frequencies and wavelengths, and the speed at which they move through a medium is the speed of sound. When dealing with sound and its perception, it is crucial to appreciate that motion relative to the sound source or observer changes the observed sound properties. Higher frequencies can occur when the source and observer move towards each other, making the sound waves compress. Conversely, lower frequencies are perceived when they move apart. Understanding these changes offers a comprehensive view of how physical movements and the nature of sound waves contribute to phenomena like the Doppler Effect.

Speed of Sound

The speed of sound is a fundamental concept when discussing sound waves. It is defined as the speed at which sound travels through a medium. In air at room temperature, this speed is approximately 343 meters per second (m/s). This speed is influenced by factors such as temperature and pressure of the medium. In exercises dealing with high-speed objects, like jets moving at a significant fraction of the speed of sound (denoted as \(0.825c\) in this problem), calculating the effects on frequency becomes crucial. Through understanding and calculating how speeds close to that of sound affect perception, learners grasp how powerful velocities can warp sound experience due to compression and rarefaction of sound waves. The Doppler Effect equations incorporate this speed to allow us to predict observed frequency changes due to motion. Grasping these basics about sound's speed enables students to interpret real-life scenarios where extreme speeds affect sound perception, as commonly experienced with high-speed jets.