# Fundamentals of Physics/Linear Momentum and Collisions

## IntroductionEdit

In the previous chapters, we concerned ourselves only with isolated objects. In this chapter, we'll see what happens when two (or more) objects interact which each other, in the form of contact between the two bodies (as in a collision, or in the sudden motion of two or more objects due to a need for them to separate due to an explosion).

## Forces between two colliding objectsEdit

Collisions are typically very brief, say ${\displaystyle 1}$  ms, or ${\displaystyle 0.001}$  s. During this time, a parameter called "impulse" exists, defined as ${\displaystyle J=\Delta p}$ , which is the change in an object's momentum. How far does the bug and SUV move in this time? The bug from above will move ${\displaystyle 5}$  cm, the SUV will "move" about the diameter of an atom making up the windshield. The bug get crushed because its internal structure cannot sustain an acceleration of about 10,000g.

## MomentumEdit

This equal and opposite force idea leads to momentum, which is defined as ${\displaystyle p=mv}$  or more correctly, ${\displaystyle {\vec {p}}=m{\vec {v}}}$ . Notice ${\displaystyle {\vec {p}}}$  involves velocity directly. We also have "conservation of momentum" that says that

${\displaystyle {\vec {p}}_{before}={\vec {p}}_{after}}$ .

The "before" and "after" refer to before and after a collision. This law itself allows us to ignore the physics \emph{of the collision} and instead focus on the physics \emph{just before and just after the collision}. More correctly, the law is

${\displaystyle \Sigma {\vec {p}}_{before}=\Sigma {\vec {p}}_{after}}$ ,

indicating that the law means add all objects carrying momentum before a collision and set equal to the sum of all momenta carrying objects after the collision. Since ${\displaystyle p}$  is a vector, so you must sum the momenta of all objects in the ${\displaystyle x}$  direction, then in the ${\displaystyle y}$  direction, both before and after the collision in order the conservation law to be helpful. Lastly, there are two types of collisions, elastic and inelastic. In elastic collisions, the colliding objects bounce off of each other, while in inelastic, they all stick together creating a new "conglomerate mass" which is the sum of the individual masses that stuck together. In applying the conservation law for an inelastic collision, you typically have something like

${\displaystyle m_{1}{{\vec {v}}_{1before}}+m_{2}{{\vec {v}}_{2before}}+m_{3}{{\vec {v}}_{3before}}+...=(m_{1}+m_{2}+m_{3}+...){\vec {v}}_{after}}$ .

Notice that there's only one velocity after the collision

(${\displaystyle {\vec {v}}_{after}}$ ) because only the "big blob" is moving after they all collided and stuck together. For an elastic collisions, where two objects (1 and 2) collide along a single axis, we'll have

${\displaystyle v_{1final}={\frac {m_{1}-m_{2}}{m_{1}+m_{2}}}v_{1before}+{\frac {2m_{2}}{m_{1}+m_{2}}}v_{2before}}$

and

${\displaystyle v_{2after}={\frac {2m_{1}}{m_{1}+m_{2}}}v_{1before}+{\frac {m_{2}-m_{1}}{m_{1}+m_{2}}}v_{2before}}$ .