The displacement of particle is given by $x=a_0+\frac{a_1 t}{2}-\frac{a_2 t^2}{3}$ What is its acceleration?

Key Idea : Acceleration is the rate of change of velocity and velocity is the rate of change of displacement. The displacement equation is given by $x=a_0+\frac{a_1 t}{2}-\frac{a_2 t^2}{3}$ Velocity $=$ rate of change of displacement ie, $\quad v=\frac{d x}{d t}$ $\begin{aligned} & =\frac{d}{d t}\left(a_0+\frac{a_1 t}{2}-\frac{a_2 t^2}{3}\right) \\ & =0+\frac{a_1}{2}-\frac{2 a_2 t}{3} \\ & =\frac{a_1}{2}-\frac{2 a_2 t}{3}\end{aligned}$ Acceleration $=$ rate of change of velocity $i e$, $\begin{aligned} a & =\frac{d v}{d t} \\ & =\frac{d}{d t}\left(\frac{a_1}{2}-\frac{2 a_2}{3} t\right) \\ & =0-\frac{2 a_2}{3} \\ & =-\frac{2 a_2}{3}\end{aligned}$