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.

Coulomb's Law

To quantify electric interactions we use an equation known as Coulomb's Law $$\vec{F}_{elec\space on \space 2 \space by \space 1} = \frac{1}{4\pi\epsilon_0}\frac{q_1q_2}{|\vec{r}|^2}\hat{r}$$
$\vec{r} = \vec{r}_2 - \vec{r}_1$ is the position of 2 relative to 1.
$\frac{1}{4\pi\epsilon_0}$ is a universal constant, equivalent to $9\times10^9 \frac{Nm^2}{C^2}$
The charges $q_1$ and $q_2$ are measured in units of Coulombs, abbreviated $C$.

Like gravity, it is proportional to the inverse square of the distance between the center of its objects.  The universal constant is much larger than that of the gravitational constant, meaning that the electric interaction is much stronger than gravitational interaction.

Fundamental Interactions

Gravitational
Responsible for attraction between objects that have mass.  Every object with mass exerts a gravitational force on objects within its gravitational range.  A small mass on Earth may exert a gravitational force on a couple objects in its near vicinity, while the Earth exerts a gravitational force on all the objects around Earth.

Electromagnetic
Responsible for attraction or repulsion between objects that have electric charge.  Particles with the same charge (protons/electrons) repulse each other, while opposite charges attract.  Electric forces are what keep atoms bonded together, and what cause reactions.

Strong
Responsible for interactions between objects made of quarks, such as protons and neutrons.  They are held together in the nucleus despite the repulsion between protons.

Weak
Affects all kinds of elementary interaction, but is weaker than the strong and electromagnetic interactions.  Basically is in everything, just really weak - hence the name. 

Gravitational Force

The gravitational force is between at least two objects.  It:

• Acts along a line connecting the two objects.
• Is proportional to the masses.
• Is inversely proportional to the square of the distance between the centers of the two objects.

It can be modeled in several different ways.  The approximate gravitational force is $$|\vec{F}_{grav}| = mg$$However, there is a much more accurate, albeit complicated, way of determining the forces of gravity.  To determine the force of gravity from one object upon another, you simply use the equation:$$\vec{F}_{grav \space on \space 2 \space by \space 1} = -G\frac{m_1 m_2}{|\vec{r}|^2} \hat{r}$$
$\vec{r} = \vec{r}_2 - \vec{r}_1$ extends from the center of object 1 to object 2.
$G = 6.7 \times 10^-11 \frac{N m^2}{kg^2}$ and is known as the gravitational constant ($G$).

To calculate the magnitude of gravitational field near an object's surface, use the equation:$$g = G\frac{M_E}{R_E^2}$$

Spring Force

The spring force was determined through experimentation to be accurately modeled by the equation:$$|\vec{F}_{spring}| = -k_s (|\vec{L}| - L_i)\hat{L}$$$$\vec{F} = -k_s s \hat{L}$$It's easy to determine from this equation:$$s = |\vec{L}| - L_i$$
$k_s$ is the spring constant, or stiffness, of the spring.  It dictates the stretching capabilities of the spring.
$L_i$ is the length of the relaxed spring (no compression/extension).
$\vec{L}$ extends from the point of attachment of the spring to the mass at the other end (the total, final length).
The stretch of $s$ can be negative or positive (compression or extension).

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 ($\vec{p}_f$).

Impulse

Impulse is the product of the force ($\vec{F}$) and change in time ($\Delta t$).  $$Impulse = \vec{F}\Delta t$$The impulse differs from the Momentum Principle in that it is distinct in forces.  The change in momentum ($\Delta\vec{p}$) is equal to the sum of the impulses applied, or:  $$\Delta\vec{p} = \sum\limits_{i=1}^n \vec{F}_i\Delta t$$