A motor bike is initially at rest. It then starts and further accelerates by 3 `m//s^(3)`. At the rest position or at the starting point of motion there is a loud strong and stationary electric siren. What will be the distance travelled by the bike if the biker hears the siren frequency at 80% of its value when the bike was at rest. Speed of sound = 340 m/s.

Options

A. 770.7 m
B. 467.8 m
C. 234 m
D. 500 m

Explanation

To solve the problem, we will use the Doppler effect to find the velocity of the biker when they hear the siren frequency at 80% of its original value. Then, we will use kinematic equations to find the distance traveled by the bike during its acceleration.

### Step 1: Understanding the Doppler Effect
The frequency heard by the observer (biker) is given by the formula:
\[ f = f_0 \frac{v - v_o}{v - v_s} \]
Where:
- \( f_0 \) = original frequency of the siren
- \( f \) = frequency heard by the biker
- \( v \) = speed of sound (340 m/s)
- \( v_o \) = speed of the observer (biker)
- \( v_s \) = speed of the source (siren, which is stationary, so \( v_s = 0 \))

### Step 2: Setting up the equation
Since the biker hears the frequency at 80% of its original value, we can write:
\[ f = 0.8 f_0 \]
Substituting this into the Doppler effect equation gives:
\[ 0.8 f_0 = f_0 \frac{340 - v_o}{340} \]

### Step 3: Simplifying the equation
We can cancel \( f_0 \) from both sides (assuming \( f_0 \neq 0 \)):
\[ 0.8 = \frac{340 - v_o}{340} \]

### Step 4: Cross-multiplying to solve for \( v_o \)
Cross-multiplying gives:
\[ 0.8 \times 340 = 340 - v_o \]
Calculating \( 0.8 \times 340 \):
\[ 272 = 340 - v_o \]
Rearranging gives:
\[ v_o = 340 - 272 \]
\[ v_o = 68 \, \text{m/s} \]

### Step 5: Using kinematics to find the distance
Now we know the final velocity \( v_o = 68 \, \text{m/s} \), initial velocity \( u = 0 \, \text{m/s} \), and acceleration \( a = 3 \, \text{m/s}^2 \). We can use the kinematic equation:
\[ v^2 = u^2 + 2ax \]
Substituting the known values:
\[ 68^2 = 0^2 + 2 \times 3 \times x \]
\[ 4624 = 6x \]
Now, solving for \( x \):
\[ x = \frac{4624}{6} \]
\[ x = 770.67 \, \text{m} \]

### Final Answer
The distance traveled by the bike is approximately:
\[ \boxed{770.67 \, \text{m}} \]