Momentum & Impulse in AP Physics C: Mechanics (Unit 4 Core Ideas)

Linear momentum

Vector quantity measuring an object’s motion and how hard it is to stop/redirect; defined as (\vec p = m\vec v).

Momentum ((\vec p))

The product of mass and velocity; points in the same direction as (\vec v).

SI unit of momentum

(\text{kg}\cdot\text{m/s}) (equivalent to (\text{N}\cdot\text{s})).

System momentum (total momentum)

Vector sum of all momenta in a chosen collection of objects: (\vec P = \sumi \vec pi = \sumi mi\vec v_i).

Vector nature of momentum

Momentum has magnitude and direction; in 1D direction is handled with sign, and in 2D/3D with components.

Newton’s second law (momentum form)

Net external force equals rate of change of momentum: (\vec F_{net} = \dfrac{d\vec p}{dt}).

Newton’s second law (constant mass case)

If mass is constant, (\dfrac{d\vec p}{dt}=m\dfrac{d\vec v}{dt}=m\vec a), so (\vec F_{net}=m\vec a).

Internal forces

Forces between objects within a system; occur in equal-and-opposite pairs and cancel in the sum for the whole system.

External forces

Forces exerted on the system by objects outside the system boundary; they can change the system’s total momentum via external impulse.

External impulse ((\vec J_{ext}))

Impulse delivered by external forces to a system: (\vec J{ext}=\int \vec F{ext}\,dt); if zero, system momentum is conserved.

Impulse ((\vec J))

Measure of how much a force changes momentum over a time interval; for constant net force, (\vec J=\vec F_{net}\Delta t).

Impulse-momentum theorem

Impulse equals change in momentum: (\vec J = \Delta\vec p = \vec pf-\vec pi).

Impulse (integral definition)

For time-varying force, impulse is (\vec J = \int{ti}^{tf} \vec F{net}\,dt).

Force–time graph interpretation of impulse

Impulse equals the signed area under the (F) vs. (t) curve: (\vec J) is area (with direction/sign).

Average net force ((\vec F_{avg}))

Constant force that would produce the same impulse over (\Delta t): (\vec F_{avg}=\dfrac{\Delta\vec p}{\Delta t}).

Peak (maximum) force vs. average force

In collisions, (F{max}) can be much larger than (F{avg}); (F_{avg}) does not imply the force was constant.

Rectangular force pulse

Force-time shape with constant force; impulse is (J = F\Delta t) (with sign/direction).

Triangular force pulse

Force-time shape that rises then falls linearly; impulse magnitude is area (J=\tfrac{1}{2}F_{peak}\Delta t) (with sign/direction).

Conservation of linear momentum

If net external impulse on a system is zero, total momentum stays constant: (\vec J{ext}=\vec 0\Rightarrow \Delta\vec P=\vec 0\Rightarrow \vec Pi=\vec P_f).

Isolated system (for momentum)

A system for which external impulse is zero or negligible during the interaction, allowing momentum conservation.

Component form of momentum conservation (2D)

When external impulse is negligible, momentum is conserved separately by axis: (P{x,i}=P{x,f}) and (P{y,i}=P{y,f}).

Elastic collision

Collision in which (for an isolated system) both momentum and kinetic energy are conserved.

Inelastic collision

Collision in which (for an isolated system) momentum is conserved but kinetic energy is not conserved (some becomes heat/sound/deformation).

Perfectly inelastic collision

Inelastic collision where objects stick together; momentum conserved (if isolated), kinetic energy loss is maximum consistent with momentum conservation.

Approximate momentum conservation

Momentum can be treated as conserved when external impulses (e.g., friction) are small compared to internal collision impulses, often because collision time is very short or only certain components are considered.