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

Answer

To solve the problem of finding the distance traveled by a particle moving along the x-axis given its position function \( x(t) = 2 + 8t - 4t^2 \), we will follow these steps:

### Step 1: Find the velocity function
The velocity \( v(t) \) is the derivative of the position function \( x(t) \) with respect to time \( t \).

\[ v(t) = \frac{dx}{dt} = \frac{d}{dt}(2 + 8t - 4t^2) = 0 + 8 - 8t = 8 - 8t \]

### Step 2: Determine the time intervals when the particle changes direction
To find when the particle changes direction, we need to find when the velocity is zero.

\[ 8 - 8t = 0 \implies t = 1 \text{ second} \]

### Step 3: Calculate the position at \( t = 0 \), \( t = 1 \), and \( t = 2 \)
Now we will calculate the position of the particle at these times.

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

- At \( t = 1 \):
\[ x(1) = 2 + 8(1) - 4(1)^2 = 2 + 8 - 4 = 6 \]

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

### Step 4: Calculate the distance traveled
The distance traveled by the particle from \( t = 0 \) to \( t = 2 \) can be calculated by finding the total distance covered in the intervals \( [0, 1] \) and \( [1, 2] \).

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

2. From \( t = 1 \) to \( t = 2 \):
\[ \text{Distance} = x(2) - x(1) = 2 - 6 = -4 \text{ meters} \]
Since distance cannot be negative, we take the absolute value:
\[ \text{Distance} = 4 \text{ meters} \]

### Step 5: Total distance traveled
Now, we add the distances from both intervals:
\[ \text{Total Distance} = 4 + 4 = 8 \text{ meters} \]

Thus, the distance traveled by the particle from \( t = 0 \) to \( t = 2 \) is **8 meters**.

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

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