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. 495 m
D. 990 m

Explanation

To solve the problem step by step, we will follow the outlined reasoning from the video transcript:

### Step 1: Understand the problem
We have a motorcycle that starts from rest and accelerates at a rate of \(2.2 \, \text{m/s}^2\). A siren is stationary at the starting point, and the motorcyclist hears the frequency of the siren at \(90\%\) of the frequency when stationary. We need to find the distance traveled by the motorcyclist when this happens.

**Hint:** Identify the key variables: acceleration, speed of sound, and the frequency heard by the motorcyclist.

### Step 2: Set up the frequency relationship
Let \(f_0\) be the frequency of the siren when the motorcycle is stationary. The frequency heard by the motorcyclist when moving is given by the Doppler effect formula:

\[ f' = f_0 \frac{V + V_o}{V} \]

Where:
- \(f'\) is the apparent frequency heard by the observer (motorcyclist).
- \(V\) is the speed of sound (\(330 \, \text{m/s}\)).
- \(V_o\) is the velocity of the observer (motorcyclist).

Since the motorcyclist hears the frequency at \(90\%\) of the stationary frequency:

\[ f' = 0.9 f_0 \]

Substituting this into the Doppler effect equation gives:

\[ 0.9 f_0 = f_0 \frac{V + V_o}{V} \]

**Hint:** Remember that the source is stationary, so \(V_s = 0\).

### Step 3: Simplify the equation
Cancelling \(f_0\) from both sides (assuming \(f_0 \neq 0\)):

\[ 0.9 = \frac{V + V_o}{V} \]

Rearranging gives:

\[ 0.9V = V + V_o \]

\[ V_o = 0.9V - V \]

\[ V_o = -0.1V \]

Since \(V = 330 \, \text{m/s}\):

\[ V_o = -0.1 \times 330 = -33 \, \text{m/s} \]

**Hint:** The negative sign indicates that the motorcycle is moving away from the source.

### Step 4: Relate velocity to distance
The motorcyclist starts from rest and accelerates. We can use the kinematic equation:

\[ V^2 = u^2 + 2a s \]

Where:
- \(V\) is the final velocity (\(33 \, \text{m/s}\)).
- \(u\) is the initial velocity (\(0 \, \text{m/s}\)).
- \(a\) is the acceleration (\(2.2 \, \text{m/s}^2\)).
- \(s\) is the distance traveled.

Substituting the known values:

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

\[ 1089 = 4.4s \]

**Hint:** Rearranging this equation will help you find the distance \(s\).

### Step 5: Solve for distance
Now, solve for \(s\):

\[ s = \frac{1089}{4.4} \]

Calculating this gives:

\[ s = 247.5 \, \text{m} \]

**Hint:** Ensure your calculations are correct and check the units.

### Final Answer
The distance traveled by the motorcyclist is \(247.5 \, \text{m}\).