Bus moving with speed v forwards a stationary wall. it produces sound of frequency `f = 420 Hz. The heard frequency of reflected sound from wall by driver is `490 Hz`. Calculate the speed v of bus. The velocity of sound in air is `330 m/s`.

Options

A. `61 Km/s`
B. `71 Km/s
C. `81 Km/s`
D. `91 Km/s`

Answer

To solve the problem, we can use the Doppler effect formula for sound. The situation involves a moving source (the bus) and a stationary observer (the wall). The steps to find the speed of the bus are as follows:

### Step 1: Understand the given values
- Frequency of sound produced by the bus (f1) = 420 Hz
- Frequency of sound heard by the driver after reflection (f2) = 490 Hz
- Velocity of sound in air (c) = 330 m/s

### Step 2: Set up the Doppler effect equations
When the bus is moving towards the wall, the frequency of the sound heard by the wall (f1) is given by:
\[ f_1 = f \left( \frac{c}{c - v} \right) \]
Where:
- \( f \) is the original frequency (420 Hz)
- \( v \) is the speed of the bus

When the sound reflects off the wall and travels back to the bus, the frequency heard by the driver (f2) is given by:
\[ f_2 = f_1 \left( \frac{c + v}{c} \right) \]

### Step 3: Substitute the known values
From the first equation:
\[ 420 = f \left( \frac{330}{330 - v} \right) \]

From the second equation:
\[ 490 = 420 \left( \frac{330 + v}{330} \right) \]

### Step 4: Solve the second equation for v
Rearranging the second equation:
\[ 490 = 420 \left( \frac{330 + v}{330} \right) \]
\[ \frac{490}{420} = \frac{330 + v}{330} \]
\[ \frac{49}{42} = \frac{330 + v}{330} \]
Cross-multiplying gives:
\[ 49 \cdot 330 = 42 \cdot (330 + v) \]

### Step 5: Expand and simplify
Expanding the equation:
\[ 16170 = 13860 + 42v \]
Rearranging gives:
\[ 16170 - 13860 = 42v \]
\[ 2310 = 42v \]
\[ v = \frac{2310}{42} \]
\[ v = 55 \text{ m/s} \]

### Step 6: Convert to km/h
To convert the speed from m/s to km/h:
\[ v_{km/h} = v_{m/s} \times \frac{18}{5} \]
\[ v_{km/h} = 55 \times \frac{18}{5} = 198 \text{ km/h} \]

### Final Answer
The speed of the bus is approximately **198 km/h**.

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