A radar operates at wavelength 50.0 cm. If the beat frequency between the transmitted singal and the singal reflected from aircraft `(Deltav)` is 1 kHz, then velocity of the aircraft will be :

Answer

To solve the problem of determining the velocity of the aircraft based on the given parameters, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Wavelength of the radar, \( \lambda = 50.0 \, \text{cm} = 0.5 \, \text{m} \)
- Beat frequency, \( \Delta f = 1 \, \text{kHz} = 1000 \, \text{Hz} \)

2. **Use the Doppler Effect Formula:**
The beat frequency \( \Delta f \) is given by the difference between the frequency of the transmitted signal \( f \) and the frequency of the reflected signal \( f' \):
\[ \Delta f = f' - f \]

3. **Express the Reflected Frequency:**
According to the Doppler effect, the frequency of the reflected signal can be expressed as:
\[ f' = f \left(1 + \frac{2v_a}{c}\right) \]
where \( v_a \) is the velocity of the aircraft and \( c \) is the speed of light.

4. **Substituting the Frequencies:**
We can substitute \( f' \) into the beat frequency equation:
\[ \Delta f = f \left(1 + \frac{2v_a}{c}\right) - f \]
This simplifies to:
\[ \Delta f = f \cdot \frac{2v_a}{c} \]

5. **Express the Frequency in Terms of Wavelength:**
The frequency \( f \) can be related to the wavelength \( \lambda \) using the equation:
\[ f = \frac{c}{\lambda} \]
Substituting this into the beat frequency equation gives:
\[ \Delta f = \frac{c}{\lambda} \cdot \frac{2v_a}{c} \]
Simplifying this, we find:
\[ \Delta f = \frac{2v_a}{\lambda} \]

6. **Rearranging to Solve for Velocity:**
Rearranging the equation to solve for \( v_a \):
\[ v_a = \frac{\Delta f \cdot \lambda}{2} \]

7. **Substituting the Known Values:**
Now substituting the known values:
\[ v_a = \frac{1000 \, \text{Hz} \cdot 0.5 \, \text{m}}{2} \]
\[ v_a = \frac{500}{2} = 250 \, \text{m/s} \]

8. **Converting to km/h:**
To convert the velocity from meters per second to kilometers per hour:
\[ v_a = 250 \, \text{m/s} \times \frac{3600 \, \text{s}}{1000 \, \text{m}} = 900 \, \text{km/h} \]

### Final Answer:
The velocity of the aircraft is \( 250 \, \text{m/s} \) or \( 900 \, \text{km/h} \).