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. 98 m
C. 147 m
D. 196 m

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the problem
The motorcycle accelerates from rest at a rate of \(2 \, \text{m/s}^2\). We need to find the distance traveled by the motorcycle when the driver hears the frequency of the siren at \(94\%\) of its original value.

### Step 2: Set up the frequency equation
The apparent frequency \(f'\) heard by the observer (motorcycle driver) moving away from the stationary source (siren) can be expressed using the Doppler effect formula:
\[ f' = f_0 \left( \frac{v - v_o}{v} \right) \]
where:
- \(f_0\) is the original frequency of the siren,
- \(v\) is the speed of sound (\(330 \, \text{m/s}\)),
- \(v_o\) is the speed of the observer (motorcycle).

### Step 3: Set the equation for \(94\%\) of the original frequency
We want the apparent frequency to be \(94\%\) of the original frequency:
\[ f' = 0.94 f_0 \]
Substituting this into the frequency equation gives:
\[ 0.94 f_0 = f_0 \left( \frac{v - v_o}{v} \right) \]

### Step 4: Simplify the equation
Dividing both sides by \(f_0\) (assuming \(f_0 \neq 0\)):
\[ 0.94 = \frac{v - v_o}{v} \]
Rearranging gives:
\[ 0.94v = v - v_o \]
\[ v_o = v - 0.94v = 0.06v \]

### Step 5: Calculate the speed of the observer
Substituting \(v = 330 \, \text{m/s}\):
\[ v_o = 0.06 \times 330 = 19.8 \, \text{m/s} \]

### Step 6: Use kinematics to find the distance traveled
Using the third equation of motion:
\[ v_o = u + at \]
where \(u = 0\) (initial velocity), \(a = 2 \, \text{m/s}^2\), and \(t\) is the time taken. We can rearrange this to find \(t\):
\[ t = \frac{v_o}{a} = \frac{19.8}{2} = 9.9 \, \text{s} \]

### Step 7: Calculate the distance traveled
Using the formula for distance:
\[ s = ut + \frac{1}{2} a t^2 \]
Substituting \(u = 0\):
\[ s = \frac{1}{2} \cdot 2 \cdot (9.9)^2 \]
\[ s = 1 \cdot 98.01 = 98.01 \, \text{m} \]

### Final Result
The motorcycle has traveled approximately \(98 \, \text{m}\) when the driver hears the frequency of the siren at \(94\%\) of its original value.

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