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:$$\vec{p} = \gamma\times m\vec{v}$$$$\vec{p}_{sys} = M_{total}\vec{v}_{CM}$$$$\vec{p}_{sys} = \sum\limits_{i=0}^n m_i \vec{v}_i$$

Conservation of Momentum

The concept of conservation of momentum is simple to understand by looking at the Momentum Principle.  The Momentum Principle predicts how much the momentum of a system will change based on external forces, or forces in the surroundings.  Momentum gained by the system is transferred from the surroundings, so we say the momentum is conserved.$$\Delta \vec{p}_{sys} = -\Delta \vec{p}_{surr}$$$$\Delta \vec{p}_{sys} + \Delta \vec{p}_{surr} = \vec{0}$$Basically, all force is equal and opposite.

Momentum Update Formula

The Momentum Update Formula is simply a rearrangement of the Momentum Principle:$$\Delta\vec{p} = \vec{p}_f - \vec{p}_i = \vec{F}_{net}\Delta t$$$$\vec{p}_f = \vec{p}_i + \vec{F}_{net} \Delta t$$If you know the initial momentum ($\vec{p}_i$) of an object and the net force ($\vec{F}_{net}$) over a period of time ($\Delta t$) short enough to make it approximately constant, you can predict the final momentum ($\vec{p}_f$).

Momentum Principle

The Momentum Principle is Newton's Second Law - a fundamental principle of physics.  It is used to predict the behavior of objects, restating Newton's First Law in a form that is measurable and causal.  $$\Delta\vec{p} = \vec{F}_{net}\Delta t$$This equation simply states that:  The change in momentum ($\Delta\vec{p}$) is equal to the net force ($\vec{F}_{net}$) times the elapsed time ($\Delta t$).

This is only effective if the net force is nearly constant.  So, either the net force must be constant, or the measurement used in increments of time that are small enough to model nearly constant force.

For instance, if we had a net force that changed at $t=3$ and $t=7$, respectively, we could make the measurement from $t=0$, $\Delta\vec{p} = \vec{F}_{net}\times3$, and then again from $t=3$ as $\Delta\vec{p} = \vec{F}_{net}\times4$ to conclude at $t=7$.

System vs. Surroundings

In physics, a system is a collection of objects that interacts with objects making up the surroundings.  The system is observed and recorded, while the surroundings are what creates change in the system.  The momentum of a system can only be changed by interactions with its surroundings.

Momentum

In real life, it seems obvious that stopping a baseball traveling at high speed is much easier than, say, a refrigerator hurdling towards you at high speed.  The mass, or weight, of the refrigerator is much greater than the baseball!  Everyone knows that it becomes increasingly difficult to move or change something as it becomes heavier, but how are we to describe this phenomenon using physics?

In order to represent the combination of an object's mass ($m$) and velocity ($\vec{v}$), a vector quantity called momentum, $\vec{p}$, is defined.  Notice that mass is a scalar and velocity a vector.  In our visible, daily lives, this equation exists simply as $\vec{p} = m \times \vec{v}$.  However there is another important factor that we must account for.

Certain experiments have shown that as particles travel closer and closer to the speed of light ($c = 3 \times 10^8$ m/s), the amount of interaction required in order to create an increase in velocity becomes increasingly large.  This is Einstein's "relativistic" definition of momentum, defined below.

In order to model the disparity between low speed velocity and velocity that approaches the speed of light, we introduce the scalar proportionality factor, gamma $\gamma$, which is equal to:  $$\gamma = \frac{1}{\sqrt{1-(\frac{|\vec{v}|}{c})^2}}$$
Using this proportionality factor, we can now write the correct equation to model momentum at all speeds:  $$\vec{p} = \frac{1}{\sqrt{1-(\frac{|\vec{v}|}{c})^2}} m \vec{v}$$ $$\vec{p} = \gamma \times m \vec{v}$$

Generally, gamma is approximately exactly 1 at low speeds, and is only effective at very high speeds.  We can simplify our momentum equations in low speed situations for the sake of ease down to:  $$\vec{p} = 1 \times m \vec{v} = m \vec{v}$$

Momentum is the integral of the net force ($\vec{F}_{net}$)$$\vec{p} = \int \vec{F}_{net}$$Using the definition of acceleration and its relation to velocity ($\vec{a} = \frac{\delta \vec{v}}{\delta t}$), we can define the approximate rate of change of momentum as$$\frac{\delta \vec{p}}{\delta t} = m\vec{a}$$$$\vec{F}_{net} = (\frac{\delta \vec{p}}{\delta t} = m\frac{\delta \vec{v}}{\delta t}) = m\vec{a}$$