/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 42 Trooper \(B\) is chasing speeder... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 18: Problem 42

Trooper \(B\) is chasing speeder \(A\) along a straight stretch of road. Both are moving at a speed of \(160 \mathrm{~km} / \mathrm{h}\). Trooper \(B\), failing to catch up, sounds his siren again. Take the speed of sound in air to be \(343 \mathrm{~m} / \mathrm{s}\) and the frequency of the source to be \(500 \mathrm{~Hz}\). What is the Doppler shift in the frequency heard by speeder \(A\) ?

Short Answer

Expert verified
The Doppler shift is 0 Hz.

Step by step solution

01

- Convert the speeds to m/s

The speeds of Trooper B and Speeder A are given in km/h. Convert these speeds to m/s. Conversion factor: \[ 1 \text{ km/h} = \frac{1000}{3600} \text{ m/s} = \frac{5}{18} \text{ m/s} \]Thus, \[ 160 \text{ km/h} = 160 \times \frac{5}{18} \text{ m/s} = \frac{800}{9} \text{ m/s} \approx 44.44 \text{ m/s} \]
02

- Understanding the Doppler Effect

In this scenario, both the source (Trooper B) and the observer (Speeder A) are moving at the same speed. Therefore, the relative velocity between the source and the observer is zero because \[ v_s = v_o = v = 44.44 \text{ m/s} \]
03

- Applying the Doppler Effect Formula

For the Doppler Effect where both the source and observer are moving along the same direction with the same velocity: \[ f' = f \times \frac{v + v_o}{v + v_s} \]Given that both the observer (Speeder A) and the source (Trooper B) are moving at the same speed, the formula simplifies as follows: \[ f' = f \times \frac{v + v}{v + v} = f \times \frac{343 + 44.44}{343 + 44.44} = f \times 1 = 500 \text{ Hz} \]
04

- Calculate the Doppler Shift

The Doppler shift in frequency is the difference between the observed frequency and the source frequency:\[ \text{Doppler Shift} = f' - f = 500 \text{ Hz} - 500 \text{ Hz} = 0 \text{ Hz} \]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

speed conversion
When dealing with problems in physics, we often encounter different units for speed and need to convert them to maintain consistency. In this exercise, the speeds are given in kilometers per hour (km/h) but need to be converted to meters per second (m/s) for ease of calculation. The conversion factor is simple:
  • 1 km/h is equal to 1000 meters per 3600 seconds.
  • This simplifies to 1 km/h = 5/18 m/s.
To convert a speed of 160 km/h to m/s, multiply by the conversion factor:
160 km/h 脳 5/18 m/s 鈮� 44.44 m/s.
Converting units is crucial, as it allows us to apply formulas correctly and ensures all terms in an equation have compatible units.
relative velocity
Relative velocity refers to the velocity of an object in relation to another object. In this problem, both Trooper B and Speeder A are moving at the same speed of 44.44 m/s. Because their speeds are identical and they are moving in the same direction, the relative velocity between them is zero.
  • Relative velocity: the speed of one object as observed from another moving object.
  • If two objects move in the same direction with the same speed, their relative velocity is zero.
This means that from the perspective of either the trooper or the speeder, the other appears to be stationary. This is a key concept for understanding why there is no Doppler shift in their situation.
sound frequency shift
The Doppler Effect explains how the frequency of a wave changes for an observer moving relative to the source of the wave. However, in this case, both observer and source are moving together at the same speed in the same direction. Because relative velocity is zero, there is no change in the frequency of sound waves reaching Speeder A from Trooper B. The observed frequency of the siren is the same as the emitted frequency.
Mathematically, the Doppler Effect formula for sound frequency shift is:
\[ f' = f \times \frac{v + v_o}{v + v_s} \]Where:
  • \( f \) is the source frequency (500 Hz).
  • \( v \) is the speed of sound in air (343 m/s).
  • \( v_o \) is the speed of the observer (Speeder A).
  • \( v_s \) is the speed of the source (Trooper B).
Given \( v_o = v_s = 44.44 \) m/s, the formula becomes:
\[ f' = f \times \frac{343 + 44.44}{343 + 44.44} = f \times 1 = 500 \text{ Hz} \]Thus, the Doppler shift is zero.
Understanding how and why the frequency does not change helps solidify the concept of relative motion and frequency shifts in the Doppler Effect.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Pipe \(A\), which is \(1.2 \mathrm{~m}\) long and open at both ends, oscillates at its third lowest harmonic frequency. It is filled with air for which the speed of sound is \(343 \mathrm{~m} / \mathrm{s}\). Pipe \(B\), which is closed at one end, oscillates at its second lowest harmonic frequency. These frequencies of pipes \(A\) and \(B\) happen to match. (a) If an \(x\) axis extends along the interior of pipe \(A\), with \(x=0\) at one end, where along the axis are the displacement nodes? (b) How long is pipe \(B ?\) (c) What is the lowest harmonic frequency of pipe \(A\) ?

A girl is sitting near the open window of a train that is moving at a velocity of \(10.00 \mathrm{~m} / \mathrm{s}\) to the east. The girl's uncle stands near the tracks and watches the train move away. The locomotive whistle emits sound at frequency \(500.0 \mathrm{~Hz}\). The air is still. (a) What frequency does the uncle hear? (b) What frequency does the girl hear? A wind begins to blow from the east at \(10.00 \mathrm{~m} / \mathrm{s}\). (c) What frequency does the uncle now hear? (d) What frequency does the girl now hear?

Two point sources of sound waves of identical wavelength \(\lambda\) and amplitude are separated by distance \(D=\) \(2.0 \lambda\). The sources are in phase. (a) How many points of maximum signal (that is, maximum constructive interference) lie along a large circle around the sources? (b) How many points of minimum signal (destructive interference) lie around the circle?

Two loudspeakers are located \(3.55 \mathrm{~m}\) apart on an outdoor stage. A listener is \(18.3 \mathrm{~m}\) from one and \(19.5 \mathrm{~m}\) from the other. During the sound check, a signal generator drives the two speakers in phase with the same amplitude and frequency. The transmitted frequency is swept through the audible range \((20 \mathrm{~Hz}\) to \(20 \mathrm{kHz})\). (a) What are the three lowest frequencies at which the listener will hear a minimum signal because of destructive interference? (b) What are the three lowest frequencies at which the listener will hear a maximum signal?

A French submarine and a U.S. submarine move toward each other during maneuvers in motionless water in the North Atlantic (Fig. 18-33). The French sub moves at \(50.0 \mathrm{~km} / \mathrm{h}\), and the U.S. sub at \(70.0 \mathrm{~km} / \mathrm{h}\). The French sub sends out a sonar signal (sound wave in water) at \(1000 \mathrm{~Hz}\). Sonar waves travel at \(5470 \mathrm{~km} / \mathrm{h}\). (a) What is the signal's frequency as detected by the U.S. sub? (b) What frequency is detected by the French sub in the signal reflected back to it by the U.S. sub?
See all solutions

Recommended explanations on Physics Textbooks

Kinematics Physics

Read Explanation

Modern Physics

Read Explanation

Waves Physics

Read Explanation

Force

Read Explanation

Radiation

Read Explanation

Turning Points in Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.