Explain the acceleration.

Explanation

The time rate of change of velocity is called acceleration. Let a particle be moving in a straight line andat 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, Average acceleration `= ("change in velocity")/(“time")`
`therefore lt a gt = (v_(2) -v_(1))/(t _(2) - t _(1)) = (Delta t)/(Delta t)`
Average acceleration is a vector quantity and its direction is in the direction of change in velocity `(Deltav)`
. The unit of acceleration is `ms^(-2)`
From average acceleration we cannot know how the velocity of particle changes with time.
Taking `lim _(Delta t to 0)` in equation then we get instantaneous acceleration a at time t.
`therefore a = lim _(Delta t to 0) (Delta v )/(Delta t) = (dv)/(dt)`
Now, `v = (dx)/(dt)`
`therefore a = (dv)/(dt) = (d)/(dt) ((dx)/(dt))`
`therefore a = (d ^(2) x)/(dt ^(2)) = bar x`
In other words second derivative of position with respect to time is acceleration of a particle.
If `(dv)/(dt)` is positive, acceleration is along the positive X-axis and if it is negative the acceleration is along the negative X-axis.