Momentum is "quantity of motion."

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

Units: kg - m/s

Properties:

• Vector quantity (direction = velocity direction)
• Mass × velocity
• Frame-dependent

Impulse is change in momentum.

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

Momentum formula.

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

J = ∫ F dt = Deltap:

Impulse-momentum theorem (integral form).

$\vec{J} = \int_{t_i}^{t_f} \vec{F}(t) \, dt = \Delta \vec{p} = m\vec{v}_f - m\vec{v}_i$

Physical meaning:

• Force applied over time changes momentum
• Impulse equals area under force-time graph

J = F_avg Deltat:

Average force form.

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

Where $\vec{F}_{avg} = \frac{1}{\Delta t}\int \vec{F} \, dt$

Sigmap_initial = Sigmap_final (for isolated system):

Total momentum conserved in isolated system.

$\vec{p}_{total,i} = \vec{p}_{total,f}$

$\sum_{i} m_i \vec{v}_{i,i} = \sum_{i} m_i \vec{v}_{i,f}$

When applicable:

• Net external force = 0
• Or collision time very short (internal >> external forces)

Elastic: Both momentum and kinetic energy conserved $\vec{p}_i = \vec{p}_f, \quad K_i = K_f$

Inelastic: Momentum conserved, kinetic energy NOT conserved $\vec{p}_i = \vec{p}_f, \quad K_f < K_i$

Perfectly inelastic: Objects stick together, maximum KE loss $\vec{p}_i = \vec{p}_f, \quad \vec{v}_f = \frac{\vec{p}_i}{m_1 + m_2}$

Measure of "bounciness" of collision.

$e = \frac{v_{separation}}{v_{approach}} = \frac{v_{2f} - v_{1f}}{v_{1i} - v_{2i}}$

Values:

• e = 1: perfectly elastic
• 0 < e < 1: inelastic
• e = 0: perfectly inelastic (objects stick)

Only elastic collisions conserve kinetic energy.

$K_i = K_f$

$\frac{1}{2}m_1v_{1i}^2 + \frac{1}{2}m_2v_{2i}^2 = \frac{1}{2}m_1v_{1f}^2 + \frac{1}{2}m_2v_{2f}^2$

Inelastic: $K_f < K_i$ (energy converted to heat, deformation, sound, etc.)

Center of mass moves as if total mass concentrated there.

$\vec{v}_{cm} = \frac{\vec{p}_{total}}{M} = \frac{\sum m_i \vec{v}_i}{\sum m_i}$

$\vec{a}_{cm} = \frac{\vec{F}_{net,ext}}{M}$

SigmaF_ext = Ma_cm:

Newton's Second Law for system.

$\vec{F}_{net,ext} = M\vec{a}_{cm} = \frac{d\vec{p}_{total}}{dt}$

Physical meaning:

• Only external forces affect COM motion
• Internal forces cancel by Newton's Third Law
• COM follows "particle" trajectory even for complex systems

For systems losing/gaining mass (rockets, leaking containers).

Tsiolkovsky rocket equation: $\Delta v = v_{ex} \ln\left(\frac{m_{initial}}{m_{final}}\right)$

Where $v_{ex}$ = exhaust velocity relative to rocket.

Propulsive force from mass ejection.

$F_{thrust} = \frac{dm}{dt} v_{ex}$

Where dm/dt = rate of mass change, $v_{ex}$ = exhaust velocity

Sign convention: dm/dt negative for losing mass