The displacement x of a particle along a straight line at time t is given by : `x = a_(0) + (a_(1) t)/(2) + (a_(2))/(3) t^(2)`. The acceleration of the particle is

Options

A. `(a_(2))/(3)`
B. `(2a_(2))/(3)`
C. `(a_(1))/(2)`
D. `a_(0) + (a_(2))/(3)`

Explanation

To find the acceleration of the particle given the displacement function \( x(t) = a_0 + \frac{a_1}{2} t + \frac{a_2}{3} t^2 \), we will follow these steps:

### Step 1: Differentiate the displacement function to find velocity
The velocity \( v(t) \) is the first derivative of the displacement \( x(t) \) with respect to time \( t \):
\[ v(t) = \frac{dx}{dt} \]
Differentiating the displacement function:
\[ v(t) = \frac{d}{dt} \left( a_0 + \frac{a_1}{2} t + \frac{a_2}{3} t^2 \right) \]

### Step 2: Apply the differentiation
- The derivative of \( a_0 \) (a constant) is \( 0 \).
- The derivative of \( \frac{a_1}{2} t \) is \( \frac{a_1}{2} \).
- The derivative of \( \frac{a_2}{3} t^2 \) is \( \frac{2a_2}{3} t \) (using the power rule).

Putting it all together:
\[ v(t) = 0 + \frac{a_1}{2} + \frac{2a_2}{3} t = \frac{a_1}{2} + \frac{2a_2}{3} t \]

### Step 3: Differentiate the velocity function to find acceleration
The acceleration \( a(t) \) is the derivative of the velocity \( v(t) \) with respect to time \( t \):
\[ a(t) = \frac{dv}{dt} \]
Differentiating the velocity function:
\[ a(t) = \frac{d}{dt} \left( \frac{a_1}{2} + \frac{2a_2}{3} t \right) \]

### Step 4: Apply the differentiation
- The derivative of \( \frac{a_1}{2} \) (a constant) is \( 0 \).
- The derivative of \( \frac{2a_2}{3} t \) is \( \frac{2a_2}{3} \).

Putting it all together:
\[ a(t) = 0 + \frac{2a_2}{3} = \frac{2a_2}{3} \]

### Conclusion
The acceleration of the particle is:
\[ \boxed{\frac{2a_2}{3}} \]