Average and Instantaneous Acceleration

If the velocity of a particle remains constant as time passes, we say that it is moving with uniform velocity. If the velocity changes with time, it is said to be accelerated.

The acceleration is the rate of change of velocity.

Velocity is a vector quantity hence a change in its magnitude or direction or both will change the velocity, which means if even direction of velocity of an object is changing then its accelerating.

Suppose that velocity of a particle at time t1 is v1 vector and at time t2 is v2 vector. The change produced in time interval t1 to t2 is v2 vector – v1 vector.

We define the Average Acceleration aav vector as change in velocity divided by time interval.

\begin{equation} \vec{a_{av}} = \frac {\vec{v_2} – \vec{v_1}} {t_{2} – t_{1}} \end{equation}

The average acceleration depends only on the velocities at time t1 and t2. How the velocity changed in between t1 and t2 is not important in defining the average acceleration.

On the other hand, Instantaneous Acceleration of a particle at time t is defined as

\begin{equation} \vec{a}=\lim _{\Delta t \rightarrow 0} \frac{\Delta \vec{v}}{\Delta t}=\frac{d \vec{v}}{d t} \end{equation}

Where Δv vector is change in velocity between time t and t + Δt.

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