Instantaneous Velocity

Instantaneous velocity measures the precise velocity of the ball at a given position.  We can approximate the instantaneous velocity by using the average velocity ($\vec{v}_{avg}$) from one position to another ($\Delta \vec{r}$) over a time interval ($\Delta t$), but it will be unlikely to be accurate.  In order to increase the accuracy of the measurement of the instantaneous velocity, we must make the time interval and position interval increasingly small.  The smaller the position and time interval, the more precise a value we will obtain for the instantaneous velocity:
$$\vec{v} = \lim_{\Delta t \to 0}\frac{\Delta \vec{r}}{\Delta t}$$

Otherwise known in calculus as

$$\vec{v} = \frac{\delta \vec{r}}{\delta t}$$

 As the values become smaller, they become more immediate, or "instantaneous."  The more complicated explanation of this equation:

$$\vec{v} = \frac{\delta\vec{r}}{\delta t} = \frac{\delta}{\delta t}<x, y, z> = <\frac{\delta x}{\delta t}, \frac{\delta y}{\delta t}, \frac{\delta z}{\delta t}> = <v_x, v_y, v_z>$$

This gives us the definition of velocity as being the "time rate of change of position:  $\vec{v} = \frac{\delta \vec{r}}{\delta t}$."