AP Physics 1 Linear Momentum: Conservation and Collisions

Linear momentum (\u2192p)

A vector measure of “quantity of motion,” defined for one object as \u2192p = m\u2192v (depends on mass and velocity direction).

Momentum as a vector

Momentum has direction because velocity has direction; you must use velocity (with sign/components), not speed.

Total momentum of a system (\u2192p_tot)

The vector sum of all object momenta in the chosen system: \u2192ptot = \u03a3\u2192pi = \u03a3 mi\u2192vi.

System (in momentum problems)

A selected collection of interacting objects analyzed together so that internal forces cancel when considering total momentum.

Conservation of linear momentum

The total momentum of a system remains constant if the net external impulse on the system is zero (or negligible).

Isolated system (momentum context)

A system experiencing zero net external impulse during the interaction, so \u2192ptot,initial = \u2192ptot,final.

External impulse (\u2192J_ext)

The impulse delivered to the system by forces from outside the system; it determines change in system momentum.

Impulse\u2013momentum relationship (system form)

\u0394\u2192psystem = \u2192Jext; if \u2192Jext = 0 then \u0394\u2192psystem = 0 and total momentum is constant.

Internal forces (in a system)

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

Newton\u2019s Third Law (collision mechanism)

Interaction forces occur in equal-and-opposite pairs, helping internal forces cancel in the total momentum of a system.

1D sign convention (momentum)

Choosing a positive direction so velocities (and momenta) are written with signs; essential for correct conservation equations.

2D momentum conservation (components)

If net external impulse is negligible, momentum is conserved separately in perpendicular directions (e.g., px and py).

Collision (physics definition)

A short-time interaction with large forces that changes velocities; analyzed primarily using momentum conservation.

Kinetic energy (K)

Energy of motion: K = \u00bdmv^2 (a scalar that may or may not be conserved in a collision).

Elastic collision

A collision in which both total momentum and total kinetic energy of the system are conserved (ideal \u201cspring back\u201d behavior).

Inelastic collision

A collision in which momentum is conserved (if isolated) but total kinetic energy is not; some K becomes heat/sound/deformation.

Perfectly inelastic collision

An inelastic collision where objects stick together and share a common final velocity; maximum kinetic energy loss consistent with momentum conservation.

Perfectly inelastic collision equation (sticking)

If two objects stick: m1v1i + m2v2i = (m1 + m2)v_f (single shared final velocity).

1D elastic collision (stationary target results)

If m1 moves at v1i and m2 starts at rest: v1f = ((m1−m2)/(m1+m2))v1i and v2f = (2m1/(m1+m2))v1i.

Recoil / explosion in 1D (push apart)

An internal interaction where total momentum is conserved; if initially at rest, total momentum stays zero.

Recoil velocity relationship (two carts)

If initially at rest and cart 1 ends with v1, then 0 = m1v1 + m2v2 \u2192 v2 = −(m1/m2)v1 (opposite direction).

2D explosion bookkeeping (fragment B components)

If initial momentum is (p0, 0) and fragment A is (pAx, pAy), then fragment B is (pBx, pBy) = (p0 − pAx, −pAy).

Common mistake: treating momentum as a scalar

Using speeds or magnitudes only (ignoring signs/components), which breaks the vector nature of momentum conservation.

Common mistake: conserving momentum for one object

Applying conservation to a single object instead of the whole interacting system; momentum conservation applies to the system total (when isolated).

Common mistake: assuming equal force means equal speed

In push-apart/recoil situations, forces are equal and opposite but speeds are not necessarily equal because accelerations differ for different masses.