The driver of a car approaching a vertical wall notices that the frequency of the horn of his car changes from 400 Hz to 450 Hz after being reflected from the wall. Assuming speed of sound to be `340(m)/(s)`, the speed of approach of car towards the wall is

Options

A. `10(m)/(s)`
B. `20(m)/(s)`
C. `30(m)/(s)`
D. `40(m)/(s)`

Explanation

To solve the problem of finding the speed of approach of the car towards the wall, we can use the Doppler effect formula for sound waves. Here’s a step-by-step solution:

### Step 1: Understand the Doppler Effect Formula
When a source of sound (the car horn) is moving towards a stationary observer (the wall), the frequency observed after reflection can be calculated using the formula:
\[ f' = f \cdot \frac{V + V_s}{V - V_s} \]
where:
- \( f' \) = observed frequency after reflection (450 Hz)
- \( f \) = emitted frequency (400 Hz)
- \( V \) = speed of sound (340 m/s)
- \( V_s \) = speed of the source (car) towards the wall (to be determined)

### Step 2: Substitute Known Values
We can substitute the known values into the formula:
\[ 450 = 400 \cdot \frac{340 + V_s}{340 - V_s} \]

### Step 3: Simplify the Equation
To simplify, we first divide both sides by 400:
\[ \frac{450}{400} = \frac{340 + V_s}{340 - V_s} \]
This simplifies to:
\[ \frac{9}{8} = \frac{340 + V_s}{340 - V_s} \]

### Step 4: Cross-Multiply
Now we cross-multiply to eliminate the fraction:
\[ 9(340 - V_s) = 8(340 + V_s) \]

### Step 5: Expand Both Sides
Expanding both sides gives us:
\[ 3060 - 9V_s = 2720 + 8V_s \]

### Step 6: Combine Like Terms
Now, we can combine like terms:
\[ 3060 - 2720 = 9V_s + 8V_s \]
\[ 340 = 17V_s \]

### Step 7: Solve for \( V_s \)
Now, we can solve for \( V_s \):
\[ V_s = \frac{340}{17} = 20 \text{ m/s} \]

### Conclusion
The speed of approach of the car towards the wall is \( 20 \text{ m/s} \).