A car is moving towards a high cliff. The car driver sounds a horn of frequency `f`. The reflected sound heard by the driver has a frequency `2 f`. if v be the velocity of sound, then the velocity of the car, in the same velocity units, will be

Options

A. `v/sqrt2`
B. `v/3`
C. `v/4`
D. `v/2`

Answer

To solve the problem, we need to apply the concept of the Doppler effect, which describes how the frequency of a wave changes for an observer moving relative to the source of the wave.

### Step-by-step Solution:

1. **Identify the Given Information**:
- Frequency of the horn sounded by the car, \( f \).
- Frequency of the reflected sound heard by the driver, \( 2f \).
- Velocity of sound in air, \( v \).

2. **Understand the Scenario**:
- The car is moving towards a cliff, which acts as a reflector of sound.
- The car is the source of the sound, and the cliff is the observer initially.

3. **Apply the Doppler Effect for the Source**:
- When the car (source) sounds the horn, the frequency heard by the cliff (observer) can be calculated using the Doppler effect formula:
\[ f' = \frac{v}{v - v_d} f \]
where \( v_d \) is the velocity of the car.

4. **Calculate the Frequency Heard by the Cliff**:
- The frequency \( f' \) heard by the cliff is \( f' = 2f \).
- Therefore, we can write:
\[ 2f = \frac{v}{v - v_d} f \]
- Dividing both sides by \( f \) gives:
\[ 2 = \frac{v}{v - v_d} \]

5. **Rearranging the Equation**:
- Cross-multiplying gives:
\[ 2(v - v_d) = v \]
- Expanding this gives:
\[ 2v - 2v_d = v \]
- Rearranging leads to:
\[ 2v - v = 2v_d \implies v = 2v_d \]

6. **Finding the Velocity of the Car**:
- From the equation \( v = 2v_d \), we can express the velocity of the car:
\[ v_d = \frac{v}{2} \]

7. **Conclusion**:
- The velocity of the car \( v_d \) is \( \frac{v}{2} \).

### Final Answer:
The velocity of the car is \( \frac{v}{2} \).