AP Physics 1 Unit 4: Mastering Linear Momentum

Linear momentum

A measure of motion that’s hard to stop; for an object it equals mass times velocity and points in the direction of velocity.

Momentum (symbol p)

Vector quantity defined by p = mv with units kg·m/s.

Vector quantity

A physical quantity with both magnitude and direction (e.g., momentum).

Momentum direction rule

The direction of an object’s momentum is always the same as the direction of its velocity.

Momentum components

Breaking momentum into perpendicular directions: px = mvx and py = mvy.

1D sign convention (momentum)

In one dimension, direction is represented by +/− based on the chosen positive direction.

System (in momentum problems)

A chosen collection of objects analyzed together so you can track total momentum and identify internal vs. external effects.

Total system momentum

The vector sum of the momenta of all objects in the system: Ptotal = Σ pi.

Total momentum in 1D (multiple objects)

P_total = m1v1 + m2v2 + … including signs for direction.

Total momentum components in 2D

Conserve/add momentum by components: Px,total = Σ mi v{x,i} and Py,total = Σ mi v{y,i}.

Mass–speed tradeoff (momentum)

An object can have large momentum by having large mass, large speed, or both (e.g., slow truck vs. fast small car).

Net momentum interpretation

The sign/direction of total momentum tells the overall direction of the system’s motion tendency.

Impulse (symbol J)

The combined effect of net force acting over time; for constant net force J = F_net Δt.

Impulse units

N·s, which is equivalent to kg·m/s (same as momentum units).

Impulse–momentum theorem

Net impulse equals change in momentum: J = Δp.

Change in momentum (Δp)

Δp = pf − pi (a vector change; includes direction/sign).

Impulse form of Newton’s Second Law

Fnet Δt = m vf − m v_i (useful for short-time forces/collisions).

Force–time graph impulse

Impulse equals the area under the F vs. t curve (not the slope).

Average force (F_avg)

A constant force that would give the same impulse over the same time: J = F_avg Δt.

Softening a collision

Increasing stopping time Δt reduces average force for a fixed momentum change because F_avg = Δp/Δt.

Net external impulse (J_ext)

Impulse on a system due to forces from outside the system; it determines changes in total system momentum.

Momentum conservation condition (impulse form)

If Jext = 0 (or negligible), then ΔPsystem = 0 and Pf = Pi.

Internal forces

Forces objects within a system exert on each other; they can change individual momenta but not total system momentum.

External forces

Forces exerted on the system by objects outside the system; they can change total system momentum.

Newton’s Third Law (collision relevance)

Interaction forces between two objects come in equal and opposite pairs, causing internal forces to cancel in the system total.

Isolated system (AP usage)

A system with zero (or negligible) net external force/impulse during the interaction, so momentum is conserved.

Axis-by-axis momentum conservation

Momentum can be conserved in one direction even if external forces exist in another direction (e.g., horizontal conserved while vertical forces cancel).

Collision (momentum modeling)

A short, high-force interaction where external impulses are often negligible, making momentum conservation reliable.

Explosion (momentum modeling)

Objects push apart due to internal energy; if external impulse is negligible, total momentum is conserved.

Recoil

Motion opposite the ejected object in an explosion-like event, required to keep total momentum conserved.

“Starts at rest” momentum fact

If a system begins at rest, initial total momentum is zero (P_i = 0), so final total momentum must also be zero if isolated.

Opposite momenta after separation (from rest)

For two pieces: m1 v1 + m2 v2 = 0, so their momenta are equal in magnitude and opposite in direction.

Perfectly inelastic collision

A collision where objects stick together and share the same final velocity; momentum conserved (if isolated) but kinetic energy not conserved.

Shared final velocity (perfectly inelastic)

If objects stick: v_f = (m1 v1,i + m2 v2,i)/(m1 + m2).

Inelastic collision (general)

A collision in which kinetic energy decreases (converted to deformation, heat, sound), though momentum may still be conserved.

Elastic collision (1D ideal model)

A collision where both momentum and kinetic energy are conserved (assuming negligible external impulse).

1D elastic collision equations

Two conservation laws are used: momentum conservation and kinetic energy conservation to solve for two final velocities.

Standard 1D elastic collision formulas

Closed-form expressions for v1,f and v2,f derived from conserving both momentum and kinetic energy (use only when collision is stated elastic).

Equal-mass elastic collision result

In a 1D elastic collision with m1 = m2, the objects exchange velocities.

Kinetic energy (symbol K)

Energy of motion: K = (1/2)mv^2; may change even when momentum is conserved.

Momentum vs. kinetic energy distinction

Momentum conservation does not imply kinetic energy conservation; many collisions conserve momentum but not kinetic energy.

2D momentum conservation strategy

Treat momentum as a vector: choose axes, conserve Px and Py separately, then solve for velocity components/angles.

Glancing/right-angle deflection scenario

A common 2D setup where one object changes direction; solve using separate x- and y-momentum conservation equations.

Speed from velocity components

For 2D motion, speed is v = √(vx^2 + vy^2).

Center of mass (CM)

The mass-weighted balance point of a system; its motion reflects the system’s overall momentum behavior.

Center of mass position (discrete, 1D)

xcm = (Σ mi xi)/(Σ mi) (mass-weighted average position).

Center of mass position (2D)

Compute each coordinate separately: xcm = (Σ mi xi)/(Σ mi), ycm = (Σ mi yi)/(Σ mi).

Total momentum–CM velocity relationship

For total mass M, total momentum satisfies P = M v_cm.

External-force control of CM motion

Internal forces can’t change CM motion; net external force/impulse determines how the center of mass accelerates.

Two-phase momentum problems

Model a short interaction (collision/explosion) with momentum conservation, then analyze the longer motion afterward with energy/kinematics when external forces act.