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/3
B. v/4
C. v/2
D. v/ 2

Explanation

To solve the problem, we will use the concept of the Doppler effect. The situation involves a car moving towards a cliff, where the car driver sounds a horn of frequency \( f \), and the reflected sound heard by the driver has a frequency \( 2f \). We need to find the velocity of the car in terms of the velocity of sound \( v \).

### Step-by-step Solution:

1. **Identify the Parameters**:
- Let the frequency of the horn be \( f \).
- The frequency of the sound heard by the driver after reflection is \( 2f \).
- Let the velocity of sound be \( v \).
- Let the velocity of the car be \( V_c \).

2. **Apply Doppler's Effect for the First Case**:
- In the first scenario, the car acts as the source of sound moving towards a stationary observer (the cliff).
- The apparent frequency \( f_1 \) heard by the cliff (observer) is given by:
\[ f_1 = f \frac{v}{v - V_c} \]
- Here, \( V_c \) is positive since the source (car) is moving towards the observer (cliff).

3. **Apply Doppler's Effect for the Second Case**:
- In the second scenario, the cliff acts as the source of sound (reflected sound) and the car is now the observer moving towards the source.
- The apparent frequency \( f' \) heard by the driver is given by:
\[ 2f = f_1 \frac{v + V_c}{v} \]
- Here, \( f_1 \) is the frequency calculated from the first case.

4. **Substitute \( f_1 \) into the Second Equation**:
- From the first case, we have:
\[ f_1 = f \frac{v}{v - V_c} \]
- Substitute this into the second equation:
\[ 2f = \left( f \frac{v}{v - V_c} \right) \frac{v + V_c}{v} \]

5. **Simplify the Equation**:
- Cancel \( f \) from both sides:
\[ 2 = \frac{v}{v - V_c} \cdot \frac{v + V_c}{v} \]
- Simplifying further:
\[ 2 = \frac{(v)(v + V_c)}{(v - V_c)(v)} \]
- This simplifies to:
\[ 2(v - V_c) = v + V_c \]

6. **Rearranging the Equation**:
- Expanding the left side:
\[ 2v - 2V_c = v + V_c \]
- Rearranging gives:
\[ 2v - v = 2V_c + V_c \]
- Thus:
\[ v = 3V_c \]

7. **Solve for \( V_c \)**:
- Rearranging gives:
\[ V_c = \frac{v}{3} \]

### Final Answer:
The velocity of the car is:
\[ V_c = \frac{v}{3} \]