A motorcycle starts from rest and accelerates along a straight line at `2.2(m)/(s^2)`. The speed of sound is `330(m)/(s)`. A siren at the starting point remains stationary. When the driver hears the frequency of the siren at `90%` of when motorcycle is stationary, the distance travelled by the motorcyclist is

Options

A. 123.75 m
B. 247.5 m
C. 495m
D. 990m

Explanation

To solve the problem step by step, we will use the Doppler effect and the equations of motion.

### Step 1: Understand the Doppler Effect
The frequency heard by the observer (motorcyclist) is affected by their motion relative to the source (siren). The formula for the observed frequency \( n' \) is given by:

\[ n' = n \frac{v + u_0}{v - u_s} \]

Where:
- \( n' \) = observed frequency
- \( n \) = emitted frequency
- \( v \) = speed of sound
- \( u_0 \) = speed of the observer (motorcyclist)
- \( u_s \) = speed of the source (siren)

In this case, the siren is stationary, so \( u_s = 0 \).

### Step 2: Set Up the Equation
We know that the motorcyclist hears the frequency at 90% of the stationary frequency:

\[ n' = 0.9n \]

Substituting this into the Doppler effect equation gives:

\[ 0.9n = n \frac{v + u_0}{v} \]

### Step 3: Simplify the Equation
Dividing both sides by \( n \) (assuming \( n \neq 0 \)):

\[ 0.9 = \frac{v + u_0}{v} \]

Rearranging gives:

\[ 0.9v = v + u_0 \]

Thus,

\[ u_0 = 0.9v - v = -0.1v \]

Since the motorcycle is moving away from the siren, we take the positive value:

\[ u_0 = 0.1v \]

### Step 4: Substitute the Speed of Sound
Given that the speed of sound \( v = 330 \, m/s \):

\[ u_0 = 0.1 \times 330 = 33 \, m/s \]

### Step 5: Use the Equation of Motion
Now we can use the equation of motion to find the distance \( s \) traveled by the motorcycle. The equation is:

\[ v^2 = u^2 + 2as \]

Where:
- \( v \) = final velocity
- \( u \) = initial velocity (0, since it starts from rest)
- \( a \) = acceleration (given as \( 2.2 \, m/s^2 \))

Substituting the known values:

\[ (33)^2 = 0 + 2 \times 2.2 \times s \]

### Step 6: Solve for \( s \)
Calculating \( 33^2 \):

\[ 1089 = 4.4s \]

Now, solving for \( s \):

\[ s = \frac{1089}{4.4} \approx 247.5 \, m \]

### Final Answer
The distance traveled by the motorcyclist is approximately **247.5 meters**.

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