Definition:Instantaneous

Definition

Instantaneous means occurring at, or associated with, a particular time instant.

Let $B$ be a body in motion.

Let $\Bbb I = \closedint t {t + h}$ be a time interval.

Let $\mathbf v_1$ and $\mathbf v_2$ be the velocity of $B$ at $t$ and $t + h$ respectively.

Let $\overline {\mathbf a}$ denote the average acceleration of $B$ over $\Bbb I$.

The instantaneous acceleration $\mathbf a$ of $B$ at time $t$ is defined as the limit of $\overline {\mathbf a}$ as $h \to 0$:

$\mathbf a := \ds \lim_{h \mathop \to 0} \dfrac {\mathbf v_2 - \mathbf v_1} h$

Let $B$ be a body in motion.

Let $\Bbb I = \closedint t {t + h}$ be a time interval.

Let $\mathbf s_1$ and $\mathbf s_2$ be the displacement of $B$ at $t$ and $t + h$ respectively.

Let $\overline {\mathbf v}$ denote the average velocity of $B$ over $\Bbb I$.

The instantaneous velocity $\mathbf v$ of $B$ at time $t$ is defined as the limit of $\overline {\mathbf v}$ as $h \to 0$:

$\mathbf v := \ds \lim_{h \mathop \to 0} \dfrac {\mathbf s_2 - \mathbf s_1} h$

Let $B$ be a body undergoing rotation.

The instantaneous center of rotation of $B$ is the point about which $B$ is rotating at a particular time instant.