Physics C: Mechanics · Unit 4: Conservation of linear momentum and collisions · 14 min read · Updated 2026-05-11
Conservation of linear momentum and collisions — AP Physics C: Mechanics
AP Physics C: Mechanics · Unit 4: Conservation of linear momentum and collisions · 14 min read
1. Core Principle: Conservation of Linear Momentum★★☆☆☆⏱ 4 min
Conservation of linear momentum is a fundamental principle governing interactions between objects in isolated systems, and is the primary tool for analyzing collision events. This topic accounts for 14-18% of the total AP Physics C Mechanics exam weight, appearing in both multiple choice and free response, often combined with energy or kinematics concepts.
The formal condition for momentum conservation comes from Newton's second law for systems:
\frac{d\vec{P}_{total}}{dt} = \vec{F}_{net,ext}
If $\vec{F}_{net,ext} = 0$, then total momentum is constant, because internal forces cancel out in equal and opposite pairs per Newton's third law. A key result is that the velocity of the center of mass of an isolated system is always constant.
Exam tip: Always explicitly define your system before writing a momentum conservation equation—this helps you catch unaccounted net external forces and earns you reasoning points on FRQs.
2. One-Dimensional Collisions★★★☆☆⏱ 4 min
Collisions in one dimension (all motion along a single line) are classified by whether kinetic energy is conserved during the interaction:
**Perfectly inelastic**: Objects stick together after collision and move with a common final velocity. Kinetic energy is lost to heat, deformation, or sound.
**Inelastic**: Kinetic energy is lost, but objects remain separate after collision.
**Elastic**: Both momentum and kinetic energy are conserved (an idealization, common for billiard balls or atomic collisions).
For perfectly inelastic collisions, the momentum conservation equation is:
m_1 v_{1i} + m_2 v_{2i} = (m_1 + m_2) v_f
For 1D elastic collisions, two conservation equations can be rearranged to a convenient relative velocity relation that avoids solving a quadratic equation, saving time on the exam:
v_{1i} - v_{2i} = v_{2f} - v_{1f}
Exam tip: Memorize the relative velocity relation for 1D elastic collisions—it saves 2-3 minutes of algebra and avoids quadratic solution errors.
3. Two-Dimensional Collisions★★★★☆⏱ 4 min
For collisions in two dimensions, momentum is a vector, so it is conserved separately in the $x$ and $y$ directions. For elastic 2D collisions, we add a third equation for conservation of kinetic energy to solve for unknowns. The most common AP exam problem is a glancing collision between a moving mass and a stationary target.
Exam tip: Always split momentum into $x$ and $y$ components for 2D collisions—never add momentum magnitudes directly, as momentum is a vector quantity.
4. AP Style Worked Practice Problems★★★☆☆⏱ 6 min
Common Pitfalls
Why: Students confuse velocity (vector) with speed (scalar), so they add all velocities as positive, leading to wrong total momentum.
Why: Students incorrectly assume all collisions conserve KE after learning elastic collision rules.
Why: Students overapply the 'short interaction time' approximation to all problems.
Why: Students mix up perfectly inelastic (sticking) and elastic (separate) collisions.
Why: Students confuse average velocity with center of mass velocity.