Instantaneous \[ v = \lim_{\Delta t \rightarrow 0} {{d}\over{\Delta t}} \qquad \vec{V} = \lim_{\Delta t \rightarrow 0} {{\Delta \vec{r}}\over{\Delta t}} \qquad \vec{a} = \lim_{\Delta t \rightarrow 0} {{\Delta \vec{V}}\over{\Delta t}} \] Average (note the overbar) \[ \overline{v} = {{d}\over{\Delta t}} \qquad \overline{\vec{V}} = {{\Delta \vec{r}}\over{\Delta t}} \qquad \overline{\vec{a}} = {{\Delta \vec{V}}\over{\Delta t}} \] Acceleration reflects changes in magnitude or direction of $\vec{V}$ or both

Linear motion with uniform acceleration \[ \begin{eqnarray*} v & = & v_0 + a t \\ \overline{v} & = & {\scriptstyle {1\over2}}(v + v_0) = v_0 + {\scriptstyle {1\over2}}at \\ x & = & \overline{v}t = v_0 t + {\scriptstyle {1\over2}}at^2\\ v^2 & = & v_0^2 + 2ax \end{eqnarray*} \]