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 acceleration of the particle is

Options

A. 0
B. `4m//s^(2)`
C. `-4m//s^(2)`
D. None of these

Answer

To find the acceleration of the particle given the position function \( x = 2(t - t^2) \), we will follow these steps:

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

### Step 2: Simplify the position function
We can simplify the position function:
\[ x = 2t - 2t^2 \]

### Step 3: Differentiate the position function to find velocity
To find the velocity \( v \), we differentiate the position \( x \) with respect to time \( t \):
\[ v = \frac{dx}{dt} = \frac{d}{dt}(2t - 2t^2) \]
Using the power rule of differentiation:
\[ v = 2 \cdot \frac{d}{dt}(t) - 2 \cdot \frac{d}{dt}(t^2) = 2 \cdot 1 - 2 \cdot 2t = 2 - 4t \]

### Step 4: Differentiate the velocity function to find acceleration
Now, we differentiate the velocity \( v \) with respect to time \( t \) to find the acceleration \( a \):
\[ a = \frac{dv}{dt} = \frac{d}{dt}(2 - 4t) \]
Again, applying the differentiation rules:
\[ a = 0 - 4 = -4 \]

### Step 5: State the final result for acceleration
Thus, the acceleration of the particle is:
\[ a = -4 \, \text{m/s}^2 \]