The time rate of change of velocity is called acceleration.

Let a particle be moving in a straight line and at time $t_{1}$ and $t_{2}$ its velocities are $v_{1}$ and $v_{2}$ respectively. Thus, the change in velocity of the particle in time interval $\Delta t=t_{2}-t_{1}$ is $v_{2}-v_{1}$. According to definition of average acceleration,

$\text { Average acceleration }=\frac{\text { change in velocity }}{\text { time }}$

$\therefore\langle a\rangle=\frac{v_{2}-v_{1}}{t_{2}-t_{1}}=\frac{\Delta v}{\Delta t}$

Average acceleration is a vector quantity and its direction is in the direction of change in velocity $(\Delta v)$.

Taking $\lim _{\Delta t \rightarrow 0}$ in equation then we get instantaneous acceleration $a$ at time $t$.