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 Hz. What is the speed of the bird, expressed as a percentage of the speed of sound?

Options

A. 0.02
B. 0.0097
C. 0.031
D. 0.014

Answer

To solve the problem, we will use the Doppler effect formula for sound. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Doppler Effect
The Doppler effect describes the change in frequency (or wavelength) of a wave in relation to an observer moving relative to the source of the wave. In this case, the bird (source) is moving toward the stationary bird-watcher (observer).

### Step 2: Identify Given Values
- Frequency of the source (Fs) = 1250 Hz
- Frequency heard by the observer (Fo) = 1290 Hz
- Speed of sound in air (V) = approximately 343 m/s (at 20°C, this value can vary slightly with temperature)

### Step 3: Write the Doppler Effect Formula
For a stationary observer and a moving source approaching the observer, the formula for the observed frequency (Fo) is given by:

\[ Fo = \frac{V}{V - Vs} \cdot Fs \]

Where:
- \(Vs\) = speed of the source (bird)
- \(V\) = speed of sound

### Step 4: Rearrange the Formula to Solve for Vs
Rearranging the formula to isolate \(Vs\):

\[ Fo (V - Vs) = V \cdot Fs \]

Expanding gives:

\[ Fo \cdot V - Fo \cdot Vs = V \cdot Fs \]

Now, isolate \(Vs\):

\[ Fo \cdot Vs = Fo \cdot V - V \cdot Fs \]

\[ Vs = \frac{Fo \cdot V - V \cdot Fs}{Fo} \]

### Step 5: Substitute Values and Calculate Vs
Substituting the known values into the equation:

\[ Vs = \frac{1290 \cdot 343 - 343 \cdot 1250}{1290} \]

Calculating the numerator:

\[ Vs = \frac{442470 - 433750}{1290} \]

\[ Vs = \frac{9720}{1290} \approx 7.53 \text{ m/s} \]

### Step 6: Calculate the Speed as a Percentage of the Speed of Sound
To express \(Vs\) as a percentage of the speed of sound:

\[ \text{Percentage} = \left(\frac{Vs}{V}\right) \cdot 100 \]

Substituting the values:

\[ \text{Percentage} = \left(\frac{7.53}{343}\right) \cdot 100 \approx 2.19\% \]

### Final Answer
The speed of the bird expressed as a percentage of the speed of sound is approximately **2.19%**.

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