A motor cycle starts from rest and accelerates along a straight path at `2m//s^(2)`. At the starting point of the motor cycle there is a stationary electric siren. How far has the motor cycle gone when the driver hears the frequency of the siren at `94%` of its value when the motor cycle was at rest ? (Speed of sound = `330ms^(-2))`

Options

A. 49 m
B. 98m
C. 147m
D. 196m

Explanation

To solve the problem step by step, we will follow the outlined approach based on the information provided in the question.

### Step 1: Understand the motion of the motorcycle
The motorcycle starts from rest and accelerates at \(2 \, \text{m/s}^2\). The initial velocity \(u = 0\) and the acceleration \(a = 2 \, \text{m/s}^2\).

### Step 2: Determine the velocity of the motorcycle after traveling a distance \(s\)
Using the kinematic equation:
\[ v^2 = u^2 + 2as \]
Substituting the known values:
\[ v^2 = 0 + 2 \cdot 2 \cdot s \implies v^2 = 4s \]
Thus, the velocity of the motorcycle is:
\[ v = 2\sqrt{s} \]

### Step 3: Apply the Doppler effect
The frequency heard by the motorcycle driver is \(94\%\) of the original frequency when the motorcycle was at rest. According to the Doppler effect, the apparent frequency \(f'\) is given by:
\[ f' = f \frac{v - v_m}{v} \]
Where:
- \(f'\) is the apparent frequency
- \(f\) is the original frequency
- \(v\) is the speed of sound (\(330 \, \text{m/s}\))
- \(v_m\) is the speed of the motorcycle

Since \(f' = 0.94f\), we can set up the equation:
\[ 0.94f = f \frac{v - v_m}{v} \]
Dividing both sides by \(f\) (assuming \(f \neq 0\)):
\[ 0.94 = \frac{v - v_m}{v} \]
Rearranging gives:
\[ 0.94v = v - v_m \implies v_m = v - 0.94v = 0.06v \]

### Step 4: Substitute the expression for \(v_m\)
From Step 2, we have \(v_m = 2\sqrt{s}\). Now substituting this into the equation we derived from the Doppler effect:
\[ 2\sqrt{s} = 0.06v \]

### Step 5: Solve for \(s\)
Substituting \(v = 330 \, \text{m/s}\):
\[ 2\sqrt{s} = 0.06 \cdot 330 \]
Calculating the right side:
\[ 2\sqrt{s} = 19.8 \]
Dividing both sides by 2:
\[ \sqrt{s} = 9.9 \]
Now squaring both sides:
\[ s = (9.9)^2 = 98.01 \, \text{m} \]

### Conclusion
The distance \(s\) that the motorcycle has traveled when the driver hears the frequency at \(94\%\) of its original value is approximately \(98 \, \text{m}\).

### Final Answer
The motorcycle has gone approximately \(98 \, \text{meters}\).