Microwaves are reflected from a distant airplane approaching a stationary observer. It is found that when the reflected waves beat against the waves radiating from the source, the beat frequency is 900 Hz. Calculate the approach speed of the airplane if the wavelength of the microwave is 0.1 H. Velocity of microwave is `3 xx 10^(8)" m/s".`

Answer

To solve the problem, we need to find the approach speed of the airplane based on the given information about the beat frequency and the wavelength of the microwaves. Here’s a step-by-step solution:

### Step 1: Understand the given data
- **Beat Frequency (f_b)** = 900 Hz
- **Wavelength (λ)** = 0.1 m
- **Velocity of microwaves (v)** = \(3 \times 10^8 \, \text{m/s}\)

### Step 2: Relate the beat frequency to the speed of the airplane
The beat frequency is the difference between the frequency of the waves emitted by the source and the frequency of the waves reflected back from the airplane. The frequency of the emitted waves (f_source) can be calculated using the formula:
\[ f_{\text{source}} = \frac{v}{\lambda} \]
Substituting the values:
\[ f_{\text{source}} = \frac{3 \times 10^8 \, \text{m/s}}{0.1 \, \text{m}} = 3 \times 10^9 \, \text{Hz} \]

### Step 3: Calculate the frequency of the reflected waves
When the airplane is approaching the observer, the frequency of the reflected waves (f_reflected) can be calculated using the Doppler effect formula for a source moving towards a stationary observer:
\[ f_{\text{reflected}} = f_{\text{source}} \left(1 + \frac{v_{airplane}}{v}\right) \]
Where \(v_{airplane}\) is the speed of the airplane and \(v\) is the speed of the microwaves.

### Step 4: Set up the equation for beat frequency
The beat frequency (f_b) is given by:
\[ f_b = f_{\text{reflected}} - f_{\text{source}} \]
Substituting the expression for \(f_{\text{reflected}}\):
\[ f_b = f_{\text{source}} \left(1 + \frac{v_{airplane}}{v}\right) - f_{\text{source}} \]
This simplifies to:
\[ f_b = f_{\text{source}} \cdot \frac{v_{airplane}}{v} \]

### Step 5: Rearranging the equation to find \(v_{airplane}\)
Now, we can rearrange the equation to solve for the speed of the airplane:
\[ v_{airplane} = \frac{f_b \cdot v}{f_{\text{source}}} \]

### Step 6: Substitute the known values
Substituting the known values into the equation:
\[ v_{airplane} = \frac{900 \, \text{Hz} \cdot (3 \times 10^8 \, \text{m/s})}{3 \times 10^9 \, \text{Hz}} \]
Calculating this gives:
\[ v_{airplane} = \frac{270000000000}{3000000000} = 90 \, \text{m/s} \]

### Final Answer
The approach speed of the airplane is **90 m/s**.

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