Explain the acceleration.

(N/A) 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 $\Delta v = v_{2} - v_{1}$.
According to the definition of average acceleration:
$\text{Average acceleration} = \frac{\text{change in velocity}}{\text{time interval}}$
$\langle a \rangle = \frac{v_{2} - v_{1}}{t_{2} - t_{1}} = \frac{\Delta v}{\Delta t}$
Average acceleration is a vector quantity,and its direction is the same as the direction of the change in velocity $(\Delta v)$.
The $SI$ unit of acceleration is $m/s^{2}$.
To understand how the velocity changes at a specific instant,we define instantaneous acceleration by taking the limit $\Delta t \rightarrow 0$:
$a = \lim_{\Delta t \rightarrow 0} \frac{\Delta v}{\Delta t} = \frac{dv}{dt}$
Since velocity $v = \frac{dx}{dt}$,we can write acceleration as the second derivative of position with respect to time:
$a = \frac{d}{dt} \left( \frac{dx}{dt} \right) = \frac{d^{2}x}{dt^{2}}$
If $\frac{dv}{dt} > 0$,the acceleration is in the direction of the positive $X$-axis,and if $\frac{dv}{dt} < 0$,the acceleration is in the direction of the negative $X$-axis.