A car travels at a speed of 'a' towards a high wall. The driver sounds a horn of frequency 'n'. If V is the velocity of sound in air, frequency of reflected sound heard by the driver is

Options

A. `n(V+a)/(V-a)`
B. `n(V-a)/(V+a)`
C. `n(V+a)/(V)`
D. `n(V-a)/(V)`

Answer

To solve the problem of finding the frequency of the reflected sound heard by the driver in the car, we can break down the solution into clear steps.

### Step-by-Step Solution:

1. **Identify the Given Variables:**
- Speed of the car (source) = \( a \)
- Frequency of the horn (source frequency) = \( n \)
- Velocity of sound in air = \( V \)

2. **Determine the Frequency Heard by the Wall (Observer):**
- The car is moving towards a stationary wall.
- The formula for the frequency heard by a stationary observer when the source is moving towards it is given by:
\[ F_1 = F_0 \frac{V}{V - V_s} \]
- Here, \( F_0 = n \) (the frequency of the horn), and \( V_s = a \) (the speed of the car).
- Substituting these values into the formula:
\[ F_1 = n \frac{V}{V - a} \]

3. **Determine the Frequency of the Reflected Sound Heard by the Driver:**
- Now, the wall acts as a source of sound emitting frequency \( F_1 \), and the driver in the car is the observer moving towards this source.
- The formula for the frequency heard by a moving observer when the source is stationary is:
\[ F_2 = F_1 \frac{V + V_o}{V} \]
- Here, \( V_o = a \) (the speed of the driver towards the wall).
- Substituting for \( F_1 \):
\[ F_2 = \left(n \frac{V}{V - a}\right) \frac{V + a}{V} \]

4. **Simplify the Expression:**
- Simplifying \( F_2 \):
\[ F_2 = n \frac{V (V + a)}{V(V - a)} = n \frac{V + a}{V - a} \]

5. **Final Result:**
- The frequency of the reflected sound heard by the driver is:
\[ F_2 = n \frac{V + a}{V - a} \]