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A convertible moves toward you and then passes you; all the while, its loudspeakers are producing a sound. The speed of the car is a constant 9.00 m/s, and the speed of sound is 343 m/s. What is the ratio of the frequency you hear while the car is approaching to the frequency you hear while the car is moving away?

Short Answer

Expert verified
The frequency ratio is approximately 1.054.

Step by step solution

01

Understand the Doppler Effect Formula

The Doppler Effect relates the observed frequency of a wave to the frequency of the source, the speed of the source, the speed of the observer, and the speed of the wave. The formula for the observed frequency \( f' \) when the source is moving towards the observer is given by: \( f' = \frac{v + v_o}{v - v_s} f \), where \( v \) is the speed of sound, \( v_o \) is the speed of the observer (which is 0 because the observer is stationary), \( v_s \) is the speed of the source, and \( f \) is the frequency of the source.
02

Calculate Frequency When Approaching

When the car is approaching, the observed frequency \( f_1' \) can be calculated using the formula: \( f_1' = \frac{343 + 0}{343 - 9} f = \frac{343}{334} f \). The numerator and denominator reflect the speed of sound and the speed of the approaching car, respectively.
03

Calculate Frequency When Moving Away

When the car is moving away, the observed frequency \( f_2' \) is calculated as: \( f_2' = \frac{343 + 0}{343 + 9} f = \frac{343}{352} f \). This reflects the car moving away from the observer with speed \( v_s = 9 \text{ m/s} \).
04

Find the Ratio of Frequencies

The ratio of the frequency heard while the car is approaching \( f_1' \) to the frequency heard while the car is moving away \( f_2' \) is: \( \frac{f_1'}{f_2'} = \frac{343/334}{343/352} = \frac{352}{334} \). Simplifying this ratio gives: \( \frac{352}{334} \approx 1.054 \).

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Key Concepts

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

Frequency
Frequency is an essential part of understanding the sound and its perception. It refers to the number of cycles a wave completes in a second, measured in hertz (Hz). The frequency of a sound wave determines its pitch, which is how 鈥渉igh鈥 or 鈥渓ow鈥 we perceive a sound to be. In the context of the Doppler Effect, the frequency changes based on the relative motion between the source of the sound and the observer. When an object producing sound moves toward an observer, the sound waves become compressed, increasing the frequency and resulting in a higher-pitched sound. Conversely, as the object moves away, the sound waves become stretched, decreasing the frequency and resulting in a lower-pitched sound. This frequency change is calculated using the Doppler Effect formula, which helps determine how the frequency will appear to an observer when the source of the sound is in motion.
Sound Wave
Sound waves are mechanical waves that travel through a medium like air, water, or solids. They are created by vibrating objects, which cause the surrounding medium to vibrate as well. These waves are longitudinal, meaning that the oscillation of the medium is parallel to the direction of wave travel. Sound waves are characterized by their amplitude, frequency, and wavelength. The sound we hear is a result of these waves reaching our ears and being processed by our auditory system. The amplitude of a sound wave affects its volume, while the frequency affects its pitch. In situations where there is motion, like a moving car emitting sounds, the Doppler Effect may alter how we perceive that sound wave's frequency and, consequently, its pitch. Thus, while the original frequency of the car's speakers remains unchanged, our perception of it varies with the car's motion toward or away from us.
Wave Speed
Wave speed is the speed at which a wave travels through a medium. It is determined by the properties of the medium as well as the type of wave. The speed of sound, for instance, is around 343 m/s in air at room temperature. This speed can change with variations in the medium, such as changes in temperature or density. In the scenario with the car, the constant wave speed of sound in air is vital for calculating how the Doppler Effect changes perceived frequency. Wave speed is a crucial component of the Doppler Effect equation; it appears in both the numerator and denominator as it acts as a baseline relative to which the motion effects on frequency can be measured. Understanding wave speed helps us comprehend how fast changes in frequency, or sound pitch, propagate when either the source or observer is in motion.

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Most popular questions from this chapter

As a prank, someone drops a water-filled balloon out of a window. The balloon is released from rest at a height of 10.0 m above the ears of a man who is the target. Then, because of a guilty conscience, the prankster shouts a warning after the balloon is released. The warning will do no good, however, if shouted after the balloon reaches a certain point, even if the man could react infinitely quickly. Assuming that the air temperature is \(20^{\circ} \mathrm{C}\) and ignoring the effect of air resistance on the balloon, determine how far above the man's ears this point is.

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(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.

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