The acceleration is nothing but the rate of change of velocity. Suppose the velocity of a particle at time \(t_{1}\) is \(\overrightarrow{v_{1}}\) and at time \(t_{2}\) it is \(\overrightarrow{v_{2}}\). The change in velocity produced in time interval \(t_{1}\) to \(t_{2}\) is \(\overrightarrow{v_{2}}-\overrightarrow{v_{1}}\). We define the average acceleration \(\vec{a}_{a v}\) as the change in velocity divided by the time interval. Thus,
\( \vec{a}_{a v}=\frac{\overrightarrow{v_{2}}-\overrightarrow{v_{1}}}{t_{2}-t_{1}}\).

Instantaneous acceleration of a particle at time \(t\) is defined as
\(
\vec{a}=\lim _{\Delta t \rightarrow 0} \frac{\Delta \vec{v}}{\Delta t}=\frac{d \vec{v}}{d t}
\)
where \(\Delta \vec{v}\) is the change in velocity between the time \(t\) and \(t+\Delta t\). At time \(t\) the velocity is \(\vec{v}\) and at time \(t+\Delta t\) it becomes \(\vec{v}+\Delta \vec{v} \cdot \frac{\Delta \vec{v}}{\Delta t}\) is the average acceleration of the particle in the interval \(\Delta t\). As \(\Delta t\) approaches zero, this average acceleration becomes the instantaneous acceleration. The dimension of acceleration is \(\mathrm{LT}^{-2}\) and its SI unit is metre/second \({ }^{2}\) (\(\mathrm{m} / \mathrm{s}^{2}\)).

Example: Find the average acceleration for the velocity-time graph shown in Fig. below for different time intervals \(0 \mathrm{~s}-10 \mathrm{~s}, 10 \mathrm{~s}-18 \mathrm{~s}\), and \(18 s-20 s\).

Solution:

\(
\begin{aligned}
& 0 \mathrm{~s}-10 \mathrm{~s} \quad \vec{a_{a v}}=\frac{(24-0) \mathrm{ms}^{-1}}{(10-0) \mathrm{s}}=2.4 \mathrm{~m} \mathrm{~s}^{-2} \\
& 10 \mathrm{~s}-18 \mathrm{~s} \quad \vec{a_{a v}}=\frac{(24-24) \mathrm{m} \mathrm{s}^{-1}}{(18-10) \mathrm{s}}=0 \mathrm{~m} \mathrm{~s}^{-2} \\
& 18 \mathrm{~s}-20 \mathrm{~s} \quad \vec{a_{a v}}=\frac{(0-24) \mathrm{ms}^{-1}}{(20-18) \mathrm{s}}=-12 \mathrm{~m} \mathrm{~s}^{-2}
\end{aligned}
\)

Position time graph for motion with positive, negative and zero acceleration

Acceleration, therefore, may result from a change in speed (magnitude), a change in direction or changes in both. Like velocity, acceleration can also be positive, negative or zero. Position-time graphs for motion with positive, negative and zero acceleration are shown in Figs. 3.9 (a), (b) and (c), respectively. Note that the graph curves upward for positive acceleration; downward for negative acceleration and it is a straight line for zero acceleration.

Velocity time graph for motions with constant acceleration

Let’s look at Fig 3.3 to describe this behavior.

Fig. 3.10 shows the velocity-time graph for motion with constant acceleration for the following cases :

Case-a: An object is moving in a positive direction with a positive acceleration, for example, the motion of the car in Fig. 3.3 between \(t=0 \mathrm{~s}\) and \(t=10 \mathrm{~s}\).
Case-b: An object is moving in positive direction with a negative acceleration, for example, motion of the car in Fig 3.3 between \(t=18 \mathrm{~s}\) and \(20 \mathrm{~s}\).
Case-c: An object is moving in negative direction with a negative acceleration, for example the motion of a car moving from \(\mathrm{O}\) in Fig. 3.1 in negative \(x\)-direction with increasing speed.
Case-d: An object is moving in positive direction till time \(t_1\), and then turns back with the same negative acceleration, for example, the motion of a car from point \(O\) to point Q in Fig. 3.1 till time \(t_1\) with decreasing speed and turning back and moving with the same negative acceleration.

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