Momentum
##### 2.4.1 Define Linear Momentum and impulse

If you get hit by something with a small mass but high velocity (a bullet) it hurts. If you get hit by something with a large mass but low velocity (slowly rolling car) it still hurts. Newton called momentum (velocity times mass) the quantity of motion, momentum is a fundamental quantity.

Newton originally wrote his second law as:

(1)
\begin{align} \vec F_{net}={\Delta \vec p \over \Delta t} \end{align}

The more common format of $F_{net}=ma$is a special case of Newton’s second law.

Let's define a few terms:

Linear Momentum – mass times linear velocity: $\vec p=m\vec {v }$

Impulse – equal to the change in momentum $\Delta \vec p = \vec F \Delta t$. Graphically the impulse is the area under the curve of a force vs. time graph.

##### 2.4.3 Derive the law of conservation of momentum for an isolated system consisting of two interacting particles.

Consider two balls, with initial momentum p1,i and p2,i approaching each other, after the collision they have final momentum, p1,f and p2,f .

Therefore we can define the change of momentum for each ball as:

(2)
\begin{align} \Delta \vec p_1 = \vec p_{1,f} - \vec p_{1,i} \end{align}
(3)
\begin{align} \Delta \vec p_{2} = \vec p_{2,f} - \vec p_{2,i} \end{align}

When they collide they each apply force to the other ball, F1 and F2, according to Newton’s 3rd law the forces must be equal and opposite.

(4)
\begin{align} \vec F_1 = - \vec F_2 \end{align}
(5)
\begin{align} {\Delta \vec P_1 \over \Delta t} = -{\Delta \vec P_2 \over \Delta t} \end{align}

Since the time the forces are applied (the time the balls are in contact) is the same we can simplify:

(6)
\begin{align} \Delta \vec P_1 = -\Delta \vec P_2 \end{align}

or

(7)
\begin{align} \vec p_{1,f} - \vec p_{1,i} = - (\vec p_{2,f} - \vec p_{2,i}) \end{align}

Grouping the initial and final momentums together:

(8)
\begin{align} \vec p_{1,i} +\vec p_{2,i} = \vec p_{1,f}+\vec p_{2,f} \end{align}

Where the left hand side is the total initial momentum and the right hand side is the final momentum.

This is the law of conservation of momentum, i.e. the amount momentum you start with is the amount you end with (direction matter too!). Momentum is ALWAYS conserved.