Problem 84 - Physics

Question

A bird is flying directly toward a stationary bird-watcher and emits a frequency of 1250 Hz. The bird-watcher, however, hears a frequency of \(1290 \mathrm{~Hz}\). What is the speed of the bird, expressed as a percentage of the speed of sound?

Step-by-Step Solution

Answer

The speed of the bird is 3.2% of the speed of sound.

Step 1: Understand the Doppler Effect Formula

The Doppler Effect formula for sound frequency is given by \( f' = \frac{f}{1 - \frac{v}{v_s}} \), where \( f' \) is the observed frequency (1290 Hz), \( f \) is the source frequency (1250 Hz), \( v \) is the speed of the source (the bird), and \( v_s \) is the speed of sound in air.

Step 2: Substitute Given Values into the Formula

Insert the given frequencies into the formula: \( 1290 = \frac{1250}{1 - \frac{v}{v_s}} \). We need to solve for \( \frac{v}{v_s} \), which represents the speed of the bird as a percentage of the speed of sound.

Step 3: Solve the Equation for \( \frac{v}{v_s} \)

First, rearrange the equation: \( 1 - \frac{v}{v_s} = \frac{1250}{1290} \). Then solve for \( \frac{v}{v_s} \): \( \frac{v}{v_s} = 1 - \frac{1250}{1290} \).

Step 4: Calculate the Numerical Value

Calculate \( \frac{1250}{1290} \) to get approximately \( 0.968 \). Then, \( \frac{v}{v_s} = 1 - 0.968 = 0.032 \). Thus, the speed of the bird is 3.2% of the speed of sound.

Key Concepts

Frequency

Frequency is a fundamental concept when discussing sound waves and the Doppler Effect. It represents the number of sound wave cycles that pass a particular point each second. The unit of frequency is Hertz (Hz). In the context of our bird-watcher problem, frequency is crucial because the changing frequency of sound helps us understand how objects are moving relative to each other.
When a source of sound, like our flying bird, moves toward a stationary observer, the sound waves get compressed. This

• Increases the number of waves reaching the observer.
• Results in a higher observed frequency.
• Explains why the bird-watcher hears a frequency of 1290 Hz instead of the emitted 1250 Hz by the bird.

The Doppler Effect formula helps us calculate the change in frequency when the source or observer is moving. Understanding frequency's role allows us to predict and analyze such changes accurately.

Sound Waves

Sound waves are mechanical waves that travel through a medium such as air, water, or solids. They are created by vibrating objects and are characterized by compressions and rarefactions in the medium they travel through. In our scenario, the bird emits sound waves as it flies, which are detected by the bird-watcher on the ground.
Sound waves carry energy and information. Here's how they work in the context of the Doppler Effect:

• Sound is perceived due to vibrating molecules set in motion by the emitter, in this situation, the bird.
• The frequency of these waves is altered when the source or observer is moving, leading to changes in the pitch.
• The bird moving toward the observer causes the sound waves to reach him more frequently, explaining why he perceives a higher frequency than emitted.

Understanding sound waves' properties helps deepen our comprehension of phenomena such as the Doppler Effect and its impact on perceived frequencies.

Speed of Sound

The speed of sound is the rate at which sound waves travel through a medium. It varies based on factors such as temperature, medium, and atmospheric pressure. In air at sea level and typical room temperature, sound travels at approximately 343 meters per second.
The Doppler Effect equation uses the speed of sound to calculate how the frequency of waves changes when the source is in motion.

• In our exercis, the bird's speed is compared to the speed of sound.
• The calculated value tells us how fast the bird is flying relative to this constant speed.
• Knowing the speed of sound helps us determine how factors such as temperature or medium can affect sound waves.

By understanding the speed of sound, we can calculate how fast an object must move to cause significant frequency changes, which is key to solving Doppler Effect problems.

Motion

Motion is all about the change in position of an object over time. In the context of the Doppler Effect, the movement of the sound source or the observer introduces apparent changes in the frequency of the sound.

• The bird is moving towards the stationary observer, causing a shift in the observed frequency.
• This motion is directly linked to the change from the emitted frequency to the one the observer perceives.
• Understanding motion allows us to use the Doppler Effect equations effectively and determine the relative speed.

The Doppler Effect demonstrates how motion can alter perception, turning motion-based changes into quantifiable data, like determining the bird's speed relative to the speed of sound.