A police car moving at `"30 m s"^(-1)`, chases a motorcyclist. The policeman sounds his horn at 180 Hz, while both of them move towards a stationary siren of frequency 160 HZ. He does not observe any beats then, calculate the speed `("in m s"^(-1))` of the motorcyclist round off two decimal places? [speed of sound `=330ms^(-1)`]

Options

A. `2 ms^(-1)`
B. `30 ms^(-1)`
C. `35.02 ms^(-1)`
D. `40 ms^(-1)`

Answer

To solve the problem, we need to find the speed of the motorcyclist given the frequencies involved and the speed of the police car. Here’s a step-by-step solution:

### Step 1: Understand the Problem
The police car is moving towards a stationary siren, and the policeman sounds his horn while chasing a motorcyclist. The frequencies involved are:
- Frequency of the police horn (F1) = 180 Hz
- Frequency of the stationary siren (F2) = 160 Hz
- Speed of the police car (Vs) = 30 m/s
- Speed of sound (v) = 330 m/s

### Step 2: Apparent Frequency for the Motorcyclist
Since the motorcyclist does not observe any beats, the apparent frequency of the sound from the police car must equal the frequency of the siren. The formula for apparent frequency when both the source and observer are moving is given by:

\[ f' = \frac{v + v_o}{v - v_s} \cdot f \]

Where:
- \( f' \) = apparent frequency
- \( v \) = speed of sound
- \( v_o \) = speed of the observer (motorcyclist)
- \( v_s \) = speed of the source (police car)
- \( f \) = frequency of the source (police horn)

### Step 3: Set Up the Equation
For the motorcyclist (observer) moving towards the stationary siren, we can set the apparent frequency equal to the frequency of the siren:

\[ \frac{330 + v_m}{330} \cdot 180 = 160 \]

Where \( v_m \) is the speed of the motorcyclist.

### Step 4: Solve for \( v_m \)
Cross-multiplying gives:

\[ (330 + v_m) \cdot 180 = 160 \cdot 330 \]

Expanding both sides:

\[ 59400 + 180v_m = 52800 \]

Now, isolate \( v_m \):

\[ 180v_m = 52800 - 59400 \]
\[ 180v_m = -6600 \]
\[ v_m = \frac{-6600}{180} \]
\[ v_m = -36.67 \text{ m/s} \]

Since speed cannot be negative, we take the absolute value:

\[ v_m = 36.67 \text{ m/s} \]

### Step 5: Round Off
Rounding off to two decimal places, we get:

\[ v_m = 36.67 \text{ m/s} \]

### Final Answer
The speed of the motorcyclist is **36.67 m/s**.