The displacement of a particle is moving by `x = (t - 2)^2` where `x` is in metres and `t` in second. The distance covered by the particle in first `4` seconds is.

Options

A. a. 4 m
B. b. 8 m
C. c. 12 m
D. d. 16 m

Explanation

To solve the problem of finding the distance covered by a particle with the displacement equation \( x = (t - 2)^2 \) over the first 4 seconds, we will follow these steps:

### Step 1: Understand the Displacement Equation
The displacement of the particle is given by:
\[ x = (t - 2)^2 \]
This equation describes how the position of the particle changes with time \( t \).

### Step 2: Calculate Displacement at Different Time Intervals
We will calculate the displacement \( x \) at \( t = 0, 1, 2, 3, \) and \( 4 \) seconds.

- **At \( t = 0 \):**
\[ x(0) = (0 - 2)^2 = 4 \text{ m} \]

- **At \( t = 1 \):**
\[ x(1) = (1 - 2)^2 = 1 \text{ m} \]

- **At \( t = 2 \):**
\[ x(2) = (2 - 2)^2 = 0 \text{ m} \]

- **At \( t = 3 \):**
\[ x(3) = (3 - 2)^2 = 1 \text{ m} \]

- **At \( t = 4 \):**
\[ x(4) = (4 - 2)^2 = 4 \text{ m} \]

### Step 3: Determine the Distance Covered
Now we will calculate the distance covered by the particle between these time intervals:

1. From \( t = 0 \) to \( t = 1 \):
- Initial position: \( 4 \) m
- Final position: \( 1 \) m
- Distance covered: \( |1 - 4| = 3 \) m

2. From \( t = 1 \) to \( t = 2 \):
- Initial position: \( 1 \) m
- Final position: \( 0 \) m
- Distance covered: \( |0 - 1| = 1 \) m

3. From \( t = 2 \) to \( t = 3 \):
- Initial position: \( 0 \) m
- Final position: \( 1 \) m
- Distance covered: \( |1 - 0| = 1 \) m

4. From \( t = 3 \) to \( t = 4 \):
- Initial position: \( 1 \) m
- Final position: \( 4 \) m
- Distance covered: \( |4 - 1| = 3 \) m

### Step 4: Total Distance Covered
Now, we sum all the distances covered in each interval:
\[ \text{Total Distance} = 3 + 1 + 1 + 3 = 8 \text{ m} \]

### Final Answer
The total distance covered by the particle in the first 4 seconds is:
\[ \boxed{8 \text{ m}} \]