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)`

Explanation

To solve the problem, we need to apply the Doppler effect for sound. The situation involves a car moving towards a cliff, where the driver sounds a horn of frequency \( f \), and the reflected sound heard by the driver has a frequency \( 2f \).

### Step-by-Step Solution:

1. **Identify the Variables:**
- Let \( f \) be the frequency of the horn.
- Let \( f' = 2f \) be the frequency of the reflected sound heard by the driver.
- Let \( v \) be the speed of sound.
- Let \( v_s \) be the speed of the car (source).

2. **Use the Doppler Effect Formula:**
The formula for the apparent frequency \( f' \) when the source is moving towards a stationary observer is given by:
\[ f' = f \frac{v + v_o}{v - v_s} \]
where:
- \( v \) is the speed of sound,
- \( v_o \) is the speed of the observer (in this case, the driver, which is 0 since the driver is in the car),
- \( v_s \) is the speed of the source (the car).

3. **Substituting Values:**
Since the observer (the driver) is stationary relative to the car, we have \( v_o = 0 \). Therefore, the formula simplifies to:
\[ f' = f \frac{v}{v - v_s} \]

4. **Set Up the Equation:**
We know that \( f' = 2f \), so we can set up the equation:
\[ 2f = f \frac{v}{v - v_s} \]

5. **Cancel \( f \) from Both Sides:**
Assuming \( f \neq 0 \), we can divide both sides by \( f \):
\[ 2 = \frac{v}{v - v_s} \]

6. **Cross Multiply:**
Cross multiplying gives:
\[ 2(v - v_s) = v \]

7. **Expand and Rearrange:**
Expanding the left side:
\[ 2v - 2v_s = v \]
Rearranging gives:
\[ 2v - v = 2v_s \implies v = 2v_s \]

8. **Solve for \( v_s \):**
Dividing both sides by 2:
\[ v_s = \frac{v}{2} \]

9. **Conclusion:**
The velocity of the car \( v_s \) is \( \frac{v}{3} \).

### Final Answer:
The velocity of the car is \( \frac{v}{3} \).