Problem 47 - University Physics with Modern Physics

Question

A car alarm is emitting sound waves of frequency 520 \(\mathrm{Hz}\) You are on a motorcycle, traveling directly away from the car. How fast must you be traveling if you detect a frequency of 490 \(\mathrm{Hz}\) ?

Step-by-Step Solution

Answer

You must travel at approximately 21.0 m/s away from the car.

Step 1: Understanding the Doppler Effect

The problem involves the Doppler Effect, which is the change in frequency (or wavelength) of a wave in relation to an observer moving relative to the source of the wave. To solve this problem, we'll use the formula for the Doppler Effect for sound. Since the observer is moving away from the source, the observed frequency is lower than the source frequency.

Step 2: The Formula for the Doppler Effect

The Doppler Effect formula for sound when the observer is moving away from the stationary source is given by: \[ f' = \frac{f}{1 + \frac{v_o}{v}} \] where \(f'\) is the observed frequency (490 Hz), \(f\) is the source frequency (520 Hz), \(v_o\) is the speed of the observer (the motorcycle), and \(v\) is the speed of sound in air (approximately 343 m/s at room temperature).

Step 3: Rearranging the Formula

To find the speed of the motorcycle \(v_o\), we need to rearrange the formula for \(v_o\): \[ 1 + \frac{v_o}{v} = \frac{f}{f'} \] Then it simplifies to: \[ \frac{v_o}{v} = \frac{f}{f'} - 1 \] \[ v_o = v \left( \frac{f}{f'} - 1 \right) \]

Step 4: Substitute Values and Solve

Now substitute the known values into the equation: \[ v_o = 343 \left( \frac{520}{490} - 1 \right) \] \[ v_o = 343 \left( 1.0612 - 1 \right) \] \[ v_o \approx 343 \times 0.0612 \] \[ v_o \approx 21.0 \text{ m/s} \]

Step 5: Conclusion

The speed of the motorcycle must be approximately 21.0 m/s (or about 75.6 km/h) traveling away from the car to detect a frequency of 490 Hz.

Key Concepts

Sound Waves

Sound waves are vibrations that travel through a medium, like air or water, and can be heard when they reach a person's or animal's ear. These waves are characterized by their frequency, which is measured in hertz (Hz). The frequency of a sound wave determines its pitch. Higher frequencies correspond to higher pitches, while lower frequencies correspond to lower pitches.
Sound waves are longitudinal waves, meaning that the air particles vibrate parallel to the direction of wave propagation. This characteristic is different from transverse waves, like light waves or waves on a string, where particles vibrate perpendicular to the wave direction.
Understanding sound waves' properties is key to grasping phenomena like the Doppler Effect, where the observed frequency changes due to relative motion between the source and the observer.

Frequency Change

The concept of frequency change is central to understanding the Doppler Effect. When either the source or the observer of a sound wave is moving, the frequency of the wave seems to change. This change is due to the relative motion between the observer and the source.
When the observer moves away from the sound source, the observed frequency decreases, making the sound seem lower in pitch. Conversely, if the observer moves toward the source, the frequency increases, and the sound seems higher.

• This phenomenon occurs because as the observer moves away, the waves are spaced further apart, leading to a lower frequency.
• If the observer moves closer, the waves are compressed, resulting in a higher frequency.

The exercise illustrates this by comparing the source frequency (520 Hz) with the observed lower frequency (490 Hz) as the observer moves away from the source.

Observer Motion

The motion of the observer dramatically influences the perceived sound frequency due to the Doppler Effect. If the observer of the sound wave is moving, this directly impacts how often wavefronts reach the observer.
In this case, the observer is traveling away from the sound source (the car's alarm), causing fewer wavefronts to hit them per second, which reduces the perceived frequency. This change allows us to calculate the speed at which the observer (e.g., someone on a motorcycle) is moving.

• If moving away, the frequency decreases.
• If moving toward, the frequency increases.

Understanding this principle is why we can calculate the observer's speed precisely using the Doppler Effect formula.

Speed Calculation

Calculating the speed of the observer involves using the Doppler Effect formula for sound, especially when the observer is moving relative to the source.
The formula for an observer moving away from a source is given by: \[ f' = \frac{f}{1 + \frac{v_o}{v}} \]where:

• \( f' \) is the observed frequency.
• \( f \) is the source frequency.
• \( v_o \) is the observer's speed.
• \( v \) is the speed of sound in the medium, (343 m/s at room temperature in air).

By rearranging this formula, we find the speed of the observer \( v_o \):\[ v_o = v \left( \frac{f}{f'} - 1 \right) \]Substituting the known values, you can determine how fast the motorcycle needs to move to experience the reduced frequency of 490 Hz. This results in \( v_o \approx 21.0 \text{ m/s} \).
Understanding and applying the formula correctly is crucial for solving such problems related to the Doppler Effect.