Position of a particle moving along x-axis is given by `x=2+8t-4t^(2)`. The distance travelled by the particle from `t=0` to `t=2` is:-

Options

A. 0
B. 8
C. 12
D. 16

Explanation

To find the distance travelled by the particle from \( t = 0 \) to \( t = 2 \) given the position function \( x(t) = 2 + 8t - 4t^2 \), we will follow these steps:

### Step 1: Calculate the position at \( t = 0 \) and \( t = 2 \)

1. **At \( t = 0 \)**:
\[ x(0) = 2 + 8(0) - 4(0)^2 = 2 \]

2. **At \( t = 2 \)**:
\[ x(2) = 2 + 8(2) - 4(2)^2 = 2 + 16 - 16 = 2 \]

### Step 2: Determine the velocity function

The velocity \( v(t) \) is the derivative of the position function \( x(t) \):
\[ v(t) = \frac{dx}{dt} = \frac{d}{dt}(2 + 8t - 4t^2) = 8 - 8t \]

### Step 3: Find when the velocity is zero

Set the velocity function to zero to find critical points:
\[ 8 - 8t = 0 \implies t = 1 \]

### Step 4: Evaluate the position at the critical point

Calculate the position at \( t = 1 \):
\[ x(1) = 2 + 8(1) - 4(1)^2 = 2 + 8 - 4 = 6 \]

### Step 5: Analyze the motion

- From \( t = 0 \) to \( t = 1 \), the particle moves from \( x(0) = 2 \) to \( x(1) = 6 \).
- From \( t = 1 \) to \( t = 2 \), the particle moves from \( x(1) = 6 \) back to \( x(2) = 2 \).

### Step 6: Calculate the total distance travelled

1. **Distance from \( t = 0 \) to \( t = 1 \)**:
\[ \text{Distance} = x(1) - x(0) = 6 - 2 = 4 \]

2. **Distance from \( t = 1 \) to \( t = 2 \)**:
\[ \text{Distance} = x(2) - x(1) = 2 - 6 = 4 \]

### Step 7: Total distance travelled

The total distance travelled by the particle from \( t = 0 \) to \( t = 2 \) is:
\[ \text{Total Distance} = 4 + 4 = 8 \text{ meters} \]

### Final Answer
The distance travelled by the particle from \( t = 0 \) to \( t = 2 \) is **8 meters**.

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