Linear Momentum

What this unit covers

The topics below follow the published Physics C: Mech course framework for Unit 4. This unit is worth 10–20% of the exam, so budget your time against that rather than against how long the unit takes to teach.

Lessons in this unit

• Impulse & Momentum13 min · 3 objectivesDefine linear momentum p = mv and express Newton's second law as ΣF = dp/dt · Define impulse as the time integral J = ∫F dt and relate it to the change in momentum · Apply the impulse-momentum theorem, including reading impulse as area under an F-t graph
• Conservation of Momentum & Collisions14 min · 3 objectivesState conservation of momentum for an isolated system and justify it from Newton's third law · Distinguish elastic from inelastic collisions by whether kinetic energy is conserved · Solve one-dimensional collision problems, including the perfectly inelastic case
• Center of Mass14 min · 3 objectivesLocate the center of mass of a system of discrete particles · Compute the center of mass of a continuous body using x_cm = (1/M)∫x dm · Relate the motion of the center of mass to the net external force
• Variable Mass & the Rocket Equation14 min · 3 objectivesAnalyze systems whose mass changes using conservation of momentum · Relate the thrust on a rocket to its exhaust speed and burn rate · Apply the ideal rocket equation Δv = v_ex ln(m_i/m_f)

Formulas in Unit 4

Every term in Unit 4

All 16 terms we publish for Linear Momentum, with definitions. Reading them through is the fastest way to find the ones you cannot define — then drill those in cram mode until you can produce them without the prompt.

Center of mass by integration

x_cm = (1/M)∫x dm. Express dm using linear density λ dx for a rod, then integrate over its length.

Momentum and impulse as integrals

p = mv and J = ∫F dt = Δp. Impulse is the area under a force-time curve when force varies.

Conservation of momentum

Follows from Newton's third law: internal forces cancel in pairs, so total momentum changes only through external forces.

Center of mass

x_cm = Σmx/Σm for discrete masses, or (1/M)∫x dm for a continuous body.

Motion of the center of mass

Accelerates only under net external force, so it continues undisturbed through any explosion or collision.

Elastic collision in one dimension

Conserving both momentum and kinetic energy gives relative approach speed equal to relative separation speed — a shortcut avoiding the quadratic.

Perfectly inelastic collision

Objects move together at v = (m₁v₁ + m₂v₂)/(m₁ + m₂), and kinetic energy loss is maximized.

Variable-mass systems

Rocket propulsion requires ΣF = dp/dt including the dm/dt term, which is why F = ma is insufficient.

Two-dimensional collisions

Momentum conserves independently in each direction, giving two equations to solve simultaneously.

Deciding whether momentum is conserved

Ask whether an external force acts over the interval. During a brief collision, gravity and friction contribute negligible impulse.

Momentum conservation with a pivot

A pivot exerts an external force, so linear momentum is not conserved in a collision with a hinged rod — but angular momentum about the pivot is.

Ballistic pendulum sequence

Momentum conservation for the embedding collision, then energy conservation for the swing. Using energy for the collision loses the point.

Kinetic energy lost in a collision

ΔKE = KE_i − KE_f, maximized in a perfectly inelastic collision and zero in an elastic one.

Elastic collision special cases

Equal masses exchange velocities; a very heavy object striking a light one gives the light one nearly twice the heavy one's speed.

Rocket equation qualitatively

Thrust comes from expelling mass, so ΣF = dp/dt must include v(dm/dt). Final speed depends on exhaust speed and the mass ratio.

Impulse-momentum in two dimensions

Apply the theorem separately in each direction; the impulse vector points along the change in momentum, not along the velocity.

What examiners penalize here

• For a continuous body, the recipe is always the same: write dm using the density, integrate x dm for the numerator and dm for the total mass M, then divide. Never average the endpoints — that only works for a uniform object.
• The rocket equation comes from momentum conservation with changing mass, not from F = ma with constant mass. Watch the logarithm: the payoff for carrying more fuel diminishes, since Δv grows only as ln(m_i/m_f).

Practice Physics C: Mech

Our practice bank is drawn from across the whole course rather than filtered to one unit, which is closer to how the exam asks anyway — it will not tell you which unit a question is testing.

Questions about this unit

All 7 units of AP Physics C: Mechanics

1. Unit 1 · Kinematics
2. Unit 2 · Force and Translational Dynamics
3. Unit 3 · Work, Energy, and Power
4. Unit 4 · Linear Momentum
5. Unit 5 · Torque and Rotational Dynamics
6. Unit 6 · Energy and Momentum of Rotating Systems
7. Unit 7 · Oscillations