Instantaneous Acceleration

When an object is moving with variable acceleration, then the object possesses different accelerations at different instants. The acceleration of the object at a given instant of time or at a given point of motion, is called its instantaneous acceleration.

Consider an object moving with non-uniform acceleration along a straight line. Let $ \displaystyle \overrightarrow{v}$ and $ \displaystyle \overrightarrow{v}+\Delta \overrightarrow{v}$ be the velocities of the object at instants of time $ \displaystyle t$ and $ \displaystyle t+\Delta t$, there $ \displaystyle \Delta t$ is very small time interval. The acceleration of the object at an instant $ \displaystyle t$ i.e., instantaneous acceleration is nearly equal to average acceleration in small time interval $ \displaystyle \Delta t$, because in this time-interval the acceleration of the object can be considered to be uniform, even though the object has non-uniform acceleration.

Therefore, instantaneous acceleration ‘a’ will be approximately equal to average acceleration in small time $ \displaystyle t+\Delta t$ and in given by

$ \displaystyle \overrightarrow{a}=\dfrac{{\Delta \overrightarrow{v}}}{{\Delta t}}$

The sense of approximation in the above expression can be removed by making the time interval $ \displaystyle \Delta t$ tends to zero. Thus, instantaneous acceleration

$ \displaystyle \overrightarrow{a}=\underset{{\Delta t\to 0}}{\mathop{{\lim }}}\,\left( {\dfrac{{\Delta \overrightarrow{v}}}{{\Delta t}}} \right)$

Hence, instantaneous acceleration of an object at a given instant is defined as the limiting value of the average acceleration in a small time interval around the given instant, when the time interval tends to zero.

Therefore, instantaneous acceleration,

$ \displaystyle \overrightarrow{a}=\underset{{\Delta t\to 0}}{\mathop{{\lim }}}\,\left( {\dfrac{{\Delta \overrightarrow{v}}}{{\Delta t}}} \right)=\dfrac{{dv}}{{dt}}$

$ \displaystyle \begin{array}{l}\text{As, }\overrightarrow{v}=\dfrac{{d\overrightarrow{x}}}{{dt}},\text{ therefore,}\\\overrightarrow{a}=\dfrac{d}{{dt}}\left( {\dfrac{{d\overrightarrow{x}}}{{dt}}} \right)=\dfrac{{{{d}^{2}}x}}{{d{{t}^{2}}}}\end{array}$

Thus, instantaneous acceleration of an object is also equal to the second derivative of the position of the object at the given instant.