Unit 4 Linear Momentum — AP Physics C: Mechanics Practice Test

Impulse, Momentum, and Center of Mass via Calculus

Unit 4 of AP Physics C: Mechanics extends the momentum framework into calculus territory. The impulse-momentum theorem is derived through integration of Newton's second law, and the center of mass of continuous mass distributions is computed via integral calculus — skills that appear repeatedly in AP FRQs.

Core Concepts in Unit 4

Impulse as the Integral of Force

Impulse is defined as J = ∫F dt over a time interval. When force varies with time — as in a collision or a thrust that changes — the impulse cannot be found by simple multiplication. Students must evaluate the definite integral of F(t) over the collision duration. The impulse-momentum theorem then connects this integral directly to the change in momentum: J = Δp.

Conservation of Linear Momentum

When the net external force on a system is zero, the total linear momentum is conserved. This principle applies to both discrete multi-body systems and, via integration, to continuous mass distributions. Understanding the conditions under which momentum is conserved — and recognising when external forces violate those conditions — is critical for AP exam FRQs.

Center of Mass via Integration

For a continuous mass distribution, the position of the center of mass is given by x_cm = (1/M) ∫x dm, where the integral is taken over the entire mass distribution. Setting up this integral requires expressing dm in terms of a spatial coordinate using the appropriate linear, area, or volume mass density. This calculation appears frequently in AP Physics C: Mechanics FRQs involving rods, plates, and non-uniform density objects.

Elastic and Inelastic Collisions

Collision problems in Unit 4 require applying momentum conservation and, where applicable, kinetic energy conservation (elastic collisions). AP Physics C: Mechanics problems often combine collision analysis with subsequent motion analysis using dynamics or energy methods from earlier units.

Key AP Skills for Unit 4

• Evaluating impulse integrals for time-varying forces.
• Setting up and solving center of mass integrals for one-dimensional and two-dimensional mass distributions.
• Applying momentum conservation correctly in both elastic and inelastic collision scenarios.
• Distinguishing when momentum is conserved versus when it is not based on external force analysis.
• Computing the velocity of the center of mass of a system before and after a collision.

AP FRQ Patterns in Unit 4

Multi-Part Problems Combining Units

A common FRQ structure presents a collision (Unit 4) followed by a dynamics or energy analysis (Units 2-3). Recognising the boundary between the collision phase and the post-collision phase — and switching between momentum conservation and energy/force methods accordingly — is an advanced AP Physics C: Mechanics skill.

1. Always check whether a collision is elastic or inelastic before deciding which conservation laws apply.
2. When setting up a center of mass integral, clearly define your coordinate system and the expression for dm before integrating.
3. Express your final center of mass result as a fraction of the total length or dimension as a self-consistency check.