The position of a particle is given by
`x=2(t-t^(2))`
where `t` is expressed in seconds and `x` is in metre.
The total distance travelled by the paticle between `t=0` to `t=1s` is

Options

A. `0m`
B. `1m`
C. `2m`
D. `1/2m`

Answer

To find the total distance traveled by the particle between \( t = 0 \) and \( t = 1 \) second, we will follow these steps:

### Step 1: Determine the position function
The position of the particle is given by the equation:
\[ x(t) = 2(t - t^2) \]

### Step 2: Find the velocity function
To find the velocity, we differentiate the position function with respect to time \( t \):
\[ v(t) = \frac{dx}{dt} = \frac{d}{dt}[2(t - t^2)] = 2(1 - 2t) = 2 - 4t \]

### Step 3: Find when the velocity is zero
Setting the velocity function to zero to find critical points:
\[ 2 - 4t = 0 \implies 4t = 2 \implies t = \frac{1}{2} \text{ seconds} \]

### Step 4: Evaluate the position at critical points and endpoints
Now we will evaluate the position at \( t = 0 \), \( t = \frac{1}{2} \), and \( t = 1 \):
- At \( t = 0 \):
\[ x(0) = 2(0 - 0^2) = 0 \text{ m} \]
- At \( t = \frac{1}{2} \):
\[ x\left(\frac{1}{2}\right) = 2\left(\frac{1}{2} - \left(\frac{1}{2}\right)^2\right) = 2\left(\frac{1}{2} - \frac{1}{4}\right) = 2\left(\frac{1}{4}\right) = \frac{1}{2} \text{ m} \]
- At \( t = 1 \):
\[ x(1) = 2(1 - 1^2) = 2(1 - 1) = 0 \text{ m} \]

### Step 5: Calculate the total distance traveled
1. From \( t = 0 \) to \( t = \frac{1}{2} \), the particle moves from \( 0 \text{ m} \) to \( \frac{1}{2} \text{ m} \), covering a distance of:
\[ \text{Distance} = \frac{1}{2} - 0 = \frac{1}{2} \text{ m} \]
2. From \( t = \frac{1}{2} \) to \( t = 1 \), the particle returns from \( \frac{1}{2} \text{ m} \) back to \( 0 \text{ m} \), covering the same distance of:
\[ \text{Distance} = 0 - \frac{1}{2} = \frac{1}{2} \text{ m} \]

### Step 6: Add the distances
The total distance traveled by the particle is:
\[ \text{Total Distance} = \frac{1}{2} + \frac{1}{2} = 1 \text{ m} \]

Thus, the total distance traveled by the particle between \( t = 0 \) and \( t = 1 \) second is \( \boxed{1 \text{ m}} \).