A car acceleration from rest at a constant rate `2m//s^(2)` for some time. Then, it retards at a constant rate of `4 m//s^(2)` and comes to rest. If it remains motion for 3 second, then the maximum speed attained by the car is:-

Options

A. 2m/s
B. 3m/s
C. 4m/s
D. 6m/s

Explanation

To solve the problem step-by-step, we will analyze the motion of the car during its acceleration and deceleration phases.

### Step 1: Define the variables
Let:
- \( a = 2 \, \text{m/s}^2 \) (acceleration)
- \( b = 4 \, \text{m/s}^2 \) (retardation)
- \( t_1 \) = time of acceleration
- \( t_2 \) = time of retardation
- \( v_{\text{max}} \) = maximum speed attained by the car
- Total time of motion = 3 seconds

### Step 2: Set up the time relationship
Since the total time of motion is 3 seconds, we can express the time of retardation in terms of the time of acceleration:
\[ t_1 + t_2 = 3 \quad \text{or} \quad t_2 = 3 - t_1 \]

### Step 3: Relate maximum speed to time of acceleration
The maximum speed attained during the acceleration phase can be expressed as:
\[ v_{\text{max}} = a \cdot t_1 = 2 \cdot t_1 \]

### Step 4: Relate maximum speed to time of retardation
During the deceleration phase, the maximum speed can also be expressed as:
\[ v_{\text{max}} = b \cdot t_2 = 4 \cdot t_2 \]
Substituting \( t_2 \) from Step 2:
\[ v_{\text{max}} = 4 \cdot (3 - t_1) \]

### Step 5: Set the two expressions for \( v_{\text{max}} \) equal to each other
Now we have two expressions for \( v_{\text{max}} \):
\[ 2 \cdot t_1 = 4 \cdot (3 - t_1) \]

### Step 6: Solve for \( t_1 \)
Expanding the equation:
\[ 2t_1 = 12 - 4t_1 \]
Combining like terms:
\[ 2t_1 + 4t_1 = 12 \quad \Rightarrow \quad 6t_1 = 12 \]
Dividing both sides by 6:
\[ t_1 = 2 \, \text{seconds} \]

### Step 7: Find \( t_2 \)
Now substitute \( t_1 \) back to find \( t_2 \):
\[ t_2 = 3 - t_1 = 3 - 2 = 1 \, \text{second} \]

### Step 8: Calculate \( v_{\text{max}} \)
Now we can find the maximum speed using either expression. Using \( v_{\text{max}} = 2 \cdot t_1 \):
\[ v_{\text{max}} = 2 \cdot 2 = 4 \, \text{m/s} \]

### Final Answer
The maximum speed attained by the car is:
\[ \boxed{4 \, \text{m/s}} \]