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Chapter 16: Problem 81

A car is parked 20.0 m directly south of a railroad crossing. A train is approaching the crossing from the west, headed directly east at a speed of 55.0 m/s. The train sounds a short blast of its 289-Hz horn when it reaches a point 20.0 m west of the crossing. What frequency does the car鈥檚 driver hear when the horn blast reaches the car? The speed of sound in air is 343 m/s. (Hint: Assume that only the component of the train鈥檚 velocity that is directed toward the car affects the frequency heard by the driver.)

Short Answer

Expert verified
The driver hears a frequency of 307.4 Hz.

Step by step solution

01

Understand the Doppler Effect

The Doppler Effect describes the change in frequency of a wave (such as sound) for an observer moving relative to the source of the wave. When the source of the wave is moving closer to the observer, the observed frequency increases.
02

Determine the Relative Velocity Component

Since the train is moving east and the car is stationary south of the crossing, only the component of the train's velocity that is directed towards the car affects the frequency heard. Use trigonometry to find this component: when the train is 20 m west of the crossing and the car is 20 m south, the distance from the car to the train is \( d = \sqrt{20^2 + 20^2} = 20\sqrt{2} \). The relevant component of velocity is \( v_{component} = v_{train} \times \cos(45^{\circ}) = 55 \times \frac{1}{\sqrt{2}} \).
03

Apply the Doppler Effect Formula

The formula for the frequency heard by the observer is \( f' = f \times \frac{v_{sound}}{v_{sound} - v_{component}} \), where \( f \) is the original frequency, \( v_{sound} \) is the speed of sound, and \( v_{component} \) is the relevant component of the train's velocity. Substitute the values: \( f' = 289 \times \frac{343}{343 - 55 \times \frac{1}{\sqrt{2}}} \).
04

Calculate the Observed Frequency

Perform the calculation for the adjusted frequency: \( f' = 289 \times \frac{343}{343 - 38.9} \approx 307.4 \).
05

Conclusion

The driver of the car hears a frequency of approximately 307.4 Hz when the train sounds its horn.

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Key Concepts

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

Wave Frequency Change
Wave frequency change is an outcome of the Doppler Effect. This concept explains why we hear sounds differently when they move towards or away from us. For example, imagine you're listening to an ambulance passing by. As it approaches, the siren sounds higher in pitch, and as it moves away, the pitch lowers.

This alteration in frequency happens because the waves are being compressed or stretched depending on the motion.
  • If the source of the sound is moving towards you, the sound waves get compressed, making the frequency appear higher.
  • Conversely, if the source moves away, the waves are stretched, leading to a lower frequency.

In the context of our example with the train and car, the frequency the driver hears changes due to the train's motion as it moves closer to the stationary vehicle.
Relative Motion
Understanding relative motion is key to grasping the Doppler Effect. When we talk about relative motion, we are considering how fast two objects are moving in relation to one another.

In the case of the parked car and the approaching train:
  • The car is stationary in terms of this scenario, so we only need to consider the train's movement.
  • The relative motion here influences how the sound from the train's horn is perceived by the driver in the car.

Because the train is moving towards the car, the distance between them is decreasing, which in turn affects the frequency of the sound waves that reach the car. This is why understanding relative motion is essential to predicting how the frequency changes.
Sound Waves
Sound waves are vibrations that travel through the air or any other medium. When a sound source moves, like a train sounding its horn, these waves compress or elongate, based on the direction of the movement.

In our scenario, as the train approaches the crossing, it generates sound waves moving outward from its horn.
  • These waves are perceived as a distinct pitch or frequency.
  • The medium, which in this case is air, allows these waves to travel from the source to the observer 鈥� the driver in our scenario.

It's the movement of these sound waves that the Doppler Effect explains to show how the observed frequency shifts for a stationary observer as the source moves closer or further away.
Velocity Components
When objects move, their velocity can be split into components, especially when they move at an angle. In problems involving sound and the Doppler Effect, only the component of velocity that moves towards or away from the observer matters.

For our train problem:
  • We calculate how much of the train's speed is actually directed toward the car.'s location.
  • This involves breaking down the train's velocity into components using trigonometry.

In this exercise, the train's approach introduces a critical angle, 45 degrees relative to the car. Calculating the effective component towards the car involves using the cosine of this angle, which reveals how much of its velocity affects the perceived sound frequency.

Understanding these components is essential for applying the Doppler Effect formula accurately to predict the frequency change the driver will hear.

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Most popular questions from this chapter

(a) A uniform rope of mass m and length L is hanging straight down from the ceiling. A small-amplitude transverse wave is sent up the rope from the bottom end. Derive an expression that gives the speed v of the wave on the rope in terms of the distance y above the bottom end of the rope and the magnitude g of the acceleration due to gravity. (b) Use the expression that you have derived to calculate the speeds at distances of 0.50 m and 2.0 m above the bottom end of the rope.

Two submarines are under water and approaching each other head-on. Sub A has a speed of 12 m/s and sub B has a speed of 8 m/s. Sub A sends out a 1550-Hz sonar wave that travels at a speed of 1522 m/s. (a) What is the frequency detected by sub B? (b) Part of the sonar wave is reflected from sub B and returns to sub A. What frequency does sub A detect for this reflected wave?

Two sources of sound are located on the x axis, and each emits power uniformly in all directions. There are no reflections. One source is positioned at the origin and the other at x 123 m. The source at the origin emits four times as much power as the other source. Where on the x axis are the two sounds equal in intensity? Note that there are two answers.

When one person shouts at a football game, the sound intensity level at the center of the field is 60.0 dB. When all the people shout together, the intensity level increases to 109 dB. Assuming that each person generates the same sound intensity at the center of the field, how many people are at the game?

A microphone is attached to a spring that is suspended from the ceiling, as the drawing indicates. Directly below on the floor is a stationary 440-Hz source of sound. The microphone vibrates up and down in simple harmonic motion with a period of 2.0 s. The difference between the maximum and minimum sound frequencies detected by the microphone is 2.1 Hz. Ignoring any reflections of sound in the room and using 343 m/s for the speed of sound, determine the amplitude of the simple harmonic motion.
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