Impulse, Collisions, and System Momentum

Driving question

What can we predict about an interaction when external impulse is limited?

Before learning: predict the collision

Two carts approach each other. One is twice as massive but moves half as fast. Before calculating, decide whether the system's momentum points left, right, or is zero. Then state what additional information you need to predict each cart's final velocity.

Hook: the interaction can be messy while the system stays predictable

A collision may bend bumpers, create sound, and turn kinetic energy into thermal energy. Yet the total momentum of a well-chosen system can remain nearly constant. Momentum lets us predict the before-to-after change without resolving every force during the brief interaction.

Momentum is a vector

For a particle of constant mass,

Momentum has units $\text{kg}\cdot\text{m/s}$. Its direction is the velocity direction. A larger object does not automatically have more momentum; mass and velocity both matter.

For a multi-object system,

Choose axes first. In one dimension, right may be positive and left negative. Never add speeds when directions differ.

The diagram is equivalent to the statement that the vector sum before equals the vector sum after when external impulse is negligible.

Impulse changes momentum

Impulse is the time accumulation of force:

For constant or average net force,

On a net-force-versus-time graph, signed area is impulse. Peak force alone does not determine momentum change.

Worked example: catching safely

A $0.150\text{ kg}$ ball moving at $+20.0\text{ m/s}$ is caught and brought to rest. Its momentum change is

If the catch lasts $0.060\text{ s}$, the average net force is

If the catcher moves their hands so the same momentum change takes longer, the average force magnitude decreases. The impulse remains $-3.00\text{ N}\cdot\text{s}$.

Momentum conservation is a system statement

For a chosen system,

If external impulse is zero or negligible during the interval,

Internal interaction forces occur in Newton's-third-law pairs. Their impulses cancel in the total system momentum, although each object can undergo a large momentum change.

Collision categories

• Elastic collision: momentum and kinetic energy are conserved for the system.
• Inelastic collision: momentum is conserved when external impulse is negligible, but kinetic energy is not.
• Perfectly inelastic collision: objects stick and share a final velocity.

Momentum conservation alone generally cannot determine two unknown final velocities. An elastic collision needs the additional kinetic-energy condition; a sticking collision supplies a shared-velocity condition.

Worked example: carts stick

A $0.40\text{ kg}$ cart moving right at $+3.0\text{ m/s}$ sticks to a $0.60\text{ kg}$ cart moving left at $-1.0\text{ m/s}$. Neglect external impulse during impact.

The positive result means the joined carts move right. Initial kinetic energy is $2.10\text{ J}$ and final kinetic energy is $0.18\text{ J}$; the difference becomes other forms of energy. Momentum conservation does not imply kinetic-energy conservation.

Explosions and recoil

If a system initially at rest separates under internal forces and external impulse is negligible,

The pieces have equal-magnitude, opposite momenta—not necessarily equal speeds. The less massive piece moves faster.

Center-of-mass connection

System momentum relates to center-of-mass velocity:

Internal collisions can radically change individual velocities while the center-of-mass motion changes only through external impulse.

During learning: model check

For each scenario, identify the system and interaction interval before choosing “momentum conserved.” Ask:

1. Which forces are external to the system?
2. Over the selected interval, is their total impulse negligible compared with object momenta?
3. Is the problem one-dimensional, or must momentum be conserved separately in $x$ and $y$?
4. What collision condition provides any additional equation?

Two-dimensional collisions

Momentum conservation is vector conservation:

Choose axes, resolve momentum vectors into components, and keep signs consistent. Kinetic energy is a scalar and does not have $x$ and $y$ components.

Misconception clinic

“Momentum is conserved for each object.” Interaction impulses usually change each object's momentum; the total system momentum is the conserved quantity.

“Kinetic energy is conserved in every collision.” It is conserved only in elastic collisions, while total energy is always accounted for in all forms.

“Equal and opposite forces mean equal accelerations.” Third-law forces are equal, but different masses can have different accelerations.

“A larger object always has more momentum.” Momentum depends on both mass and velocity.

After learning: retrieval and transfer

1. What does signed area under a net-force-time graph represent?
2. Why can internal forces change each cart's momentum without changing total system momentum?
3. Which extra condition defines a perfectly inelastic collision?
4. A firework initially at rest separates into unequal masses. Compare their momenta and speeds.
5. Describe a situation in which momentum conservation is only approximate and defend the chosen interval.

AP-style evidence routine

1. Define system, interval, axes, and initial/final states.
2. Draw a before/after momentum representation.
3. Evaluate external impulse explicitly.
4. Write vector momentum conservation symbolically.
5. Add only justified collision constraints.
6. Solve, interpret signs, check units, and compare limiting cases.

Key takeaway

Momentum conservation is not a slogan about collisions. It is a consequence of negligible external impulse on a clearly defined system over a stated interval.

Further learning and alignment

• College Board, “AP Physics 1: Algebra-Based Course,” https://apcentral.collegeboard.org/courses/ap-physics-1 (retrieved 2026-08-18). Alignment reference only; no endorsement implied.
• OpenStax, “8.1 Linear Momentum, Force, and Impulse” and “8.3 Conservation of Momentum,” https://openstax.org/books/physics/pages/8-1-linear-momentum-force-and-impulse and https://openstax.org/books/physics/pages/8-3-conservation-of-momentum (retrieved 2026-08-18). CC BY concept verification; no exercises copied.
• NASA Glenn Research Center, “Conservation of Momentum,” https://www.grc.nasa.gov/www/k-12/airplane/conmo.html (retrieved 2026-08-18). Government educational reference; no media reused.