$\vec{p} = m\vec{v}$

• Vector quantity (direction same as velocity)
• Proportional to mass and velocity
• Units: kg - m/s
• Important for collision analysis

$\vec{J} = \vec{F}_{avg}\Delta t = \Delta\vec{p}$

• Vector quantity (direction of net force)
• Change in momentum over time interval
• Units: N - s or kg - m/s
• Large force over short time can produce same impulse

$\vec{J} = \Delta\vec{p} = m\vec{v}_f - m\vec{v}_i$

• Impulse equals change in momentum
• Net force over time interval causes momentum change
• Useful for collisions (contact force often unknown)

• Impulse = area under F-t graph
• Average force: $F_{avg} = \frac{J}{\Delta t} = \frac{\Delta p}{\Delta t}$
• Larger area = larger impulse
• Important for collision analysis

• System must experience no net external force
• OR collision time very short (internal forces >> external forces)
• External forces: gravity, friction, air resistance
• Internal forces (action-reaction pairs) cancel within system

$\vec{v}_{cm} = \frac{m_1\vec{v}_1 + m_2\vec{v}_2 + ...}{m_1 + m_2 + ...}$

• Momentum of system = total mass × velocity of center of mass
• Center of mass continues at constant velocity without external forces
• Useful for analyzing system motion

• No external forces act on system
• Total momentum conserved: $\vec{p}_i = \vec{p}_f$
• Examples: collisions in space, frictionless surfaces
• Can analyze in components: $p_{ix} = p_{fx}$, $p_{iy} = p_{fy}$

• Momentum conserved: $\vec{p}_i = \vec{p}_f$
• Kinetic energy conserved: $K_i = K_f$
• Objects bounce off each other
• No energy loss
• Examples: billiard balls, atomic collisions

• Momentum conserved: $\vec{p}_i = \vec{p}_f$
• Kinetic energy NOT conserved: $K_f < K_i$
• Some kinetic energy converted to other forms (heat, sound, deformation)
• Objects may stick together or deform
• Most real collisions are inelastic

• Momentum conserved: $m_1\vec{v}_1 + m_2\vec{v}_2 = (m_1 + m_2)\vec{v}_f$
• Maximum kinetic energy loss
• Objects stick together after collision
• Move together with common final velocity
• Examples: car crashes, objects caught and moving together

• Momentum conserved in both x and y separately
• $m_1v_{1x} + m_2v_{2x} = m_1v_{1x}' + m_2v_{2x}'$
• $m_1v_{1y} + m_2v_{2y} = m_1v_{1y}' + m_2v_{2y}'$
• Elastic: also conserve kinetic energy in each component
• Inelastic: only conserve momentum, not kinetic energy

• Initially at rest: total momentum = 0
• After explosion: $m_1\vec{v}_1 + m_2\vec{v}_2 + ... = 0$
• Parts move in opposite directions
• Momentum conserved (no external forces during explosion)
• Kinetic energy gained from chemical potential energy

• Pushing apart: both gain momentum in opposite directions
• Momenta equal in magnitude, opposite in direction
• $m_1v_1 = m_2v_2$ (if only two parts)
• Lighter object gains higher velocity