Center of Mass

The center of mass is the center of a system of masses, located at a point that is central in relation to the masses of all of the objects within the system.  To describe the position of the center of mass, we use the equation$$\vec{r}_{CM} = \frac{\sum\limits_{i=0}^n m_i \vec{r}_i}{\sum\limits_{i=0}^n m_i}$$$$\vec{r}_{CM} = \frac{\sum\limits_{i=0}^n m_i \vec{r}_i}{M_{total}}$$

The velocity of the center of mass is the same thing, except finding velocity instead of position.$$\vec{v}_{CM} = \frac{\sum\limits_{i=0}^n m_i \vec{v}_i}{M_{total}}$$

If the velocity is very small compared to the speed of light, $\gamma = 1$ we can say that the total velocities and masses of the system are equal based off of the equation for momentum.  This helps us find the momentum of the center of mass:$$\vec{p} = \gamma\times m\vec{v}$$$$\vec{p}_{sys} = M_{total}\vec{v}_{CM}$$$$\vec{p}_{sys} = \sum\limits_{i=0}^n m_i \vec{v}_i$$

Position Update Formula

The position update formula simply states that given the initial position ($\vec{r}_i$) of an object, we can predict the final position ($\vec{r}_f$) using the average velocity ($\vec{v}_{avg}$) and change in time ($\Delta t$).

$$\Delta \vec{r} = \vec{v}_{avg} \Delta t$$
$$\vec{r}_f - \vec{r}_i = \vec{v}_{avg}(t_f - t_i)$$

Or just simply

$$\vec{r}_f = \vec{r}_i + \vec{v}_{avg} \Delta t$$

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:  $\vec{v} = \frac{\delta \vec{r}}{\delta t}$."