Linear Momentum
Linear momentum is the vector quantity p⃗=m v⃗. For a system, total momentum is the vector sum of the momenta of its parts. Physics C uses momentum to model short-duration collisions and explosions by comparing system states before and after an interaction while keeping system boundaries and vector components explicit.
Topic sequence · 1 of 4
Learning objectives
Objective 1
Describe and calculate the linear momentum of an object and the total momentum of a system as vector quantities.
Objective 2
Translate momentum among vectors, components, tables and center-of-mass motion.
Objective 3
Explain why collision and explosion models focus on initial and final states and how system choice prepares the conservation analysis.
- Topic 1.1 vectors and components.
- Topic 1.2 velocity.
- Topic 2.1 systems and center of mass.
- Topic 3.1 kinetic energy for comparison with momentum.
Definitions first
Representations you must read and create
Representation 1
Momentum-vector diagram before and after an interaction, with a common coordinate system.
Representation 2
Component table listing m, v_x, v_y, p_x and p_y for each object plus a total row.
Representation 3
Center-of-mass velocity vector v⃗_cm=P⃗_sys/M drawn alongside the individual velocities.
Representation 4
Before/after state diagram for a collision or explosion that separates the short interaction from the surrounding motion.
Core relationships and calculus
Variables: p⃗ momentum (kg·m/s); m mass (kg); v⃗ velocity (m/s)
Use: momentum of one object
Variables: sum over objects in the selected system
Use: total momentum of a multi-object system
Variables: signed Cartesian components
Use: component-wise momentum accounting in two dimensions
Variables: M=Σm_i total mass
Use: connect total momentum to center-of-mass motion for a constant-mass system
Momentum itself is algebraic for constant mass, but its calculus importance appears immediately in Topic 4.2: the net external force is dP⃗_sys/dt. This makes total momentum the accumulated response of a system to external impulse and connects Newton’s second law to conservation methods.
For a constant-mass collection, r⃗_cm=(1/M)Σm_i r⃗_i. Differentiate with respect to time: v⃗_cm=(1/M)Σm_i v⃗_i=P⃗_sys/M. Therefore P⃗_sys=M v⃗_cm. Topic 4.2 differentiates this momentum to connect external force to the acceleration of the center of mass.
Collisions and explosions are interaction models used when internal forces dominate over external forces during the short interaction. Momentum conservation still depends on the net external impulse for the selected system.
Physical examples
Example 1
A 2 kg cart moving at +3 m/s has +6 kg·m/s of momentum along the chosen axis. Reversing its velocity reverses its momentum even though its kinetic energy is unchanged.
Example 2
Two objects can have zero total momentum while both are moving: equal and opposite momentum vectors cancel component by component.
Example 3
During a brief collision, the forces the objects exert on each other can be very large while the external impulse is small enough that comparing only the initial and final system momenta is a useful model.
Example 4
An explosion can begin from a system at rest. Internal forces give the pieces equal-and-opposite momentum changes so the vector sum can remain zero.
Worked reasoning
Exam reasoning patterns
Reasoning pattern 1
Write vector components before applying a conservation law. In two dimensions, x- and y-momentum equations are independent.
Reasoning pattern 2
For a collision/explosion prompt, identify the system and interaction interval first; then decide whether external impulse is negligible.
Reasoning pattern 3
Use p∝mv for factor-change reasoning. Reversing velocity reverses momentum, while doubling speed doubles momentum rather than quadrupling it.
Reasoning pattern 4
Use center-of-mass velocity as a system-level consistency check: if external impulse is negligible, v⃗_cm should remain constant through the interaction.
Reasoning pattern 5
Use the collision/object model only for the short interaction interval; outside that interval, ordinary force-and-motion analysis may again be needed.
Deeper calculus and limiting-case check
For a constant-mass system, the kinetic energy of center-of-mass translation is K_cm=P²/(2M). A system can also have kinetic energy measured relative to its center of mass, so the same total momentum does not determine the system’s total kinetic energy. This distinction becomes central when classifying collisions in Topic 4.4.
Interactive system-momentum vector check
Enter two object masses and velocity components. The tool returns total momentum and center-of-mass velocity.
Experimental connection
Use two low-friction carts with photogates or motion sensors. Measure each cart’s mass and velocity before and after a short interaction, calculate vector/system momentum, and compare the change in total momentum with the likely external impulse and measurement uncertainty.
Connections across Physics C
Connection 1
Topic 2.1 defines systems and center-of-mass motion.
Connection 2
Topic 2.5 gives Newton’s second law, which Topic 4.2 rewrites as dP⃗/dt=F⃗_ext.
Connection 3
Topic 3.1 contrasts scalar kinetic energy with vector momentum.
Connection 4
Topic 4.2 turns force acting over time into a change in momentum.
Connection 5
Topic 4.4 uses the same momentum framework but classifies collisions by the system’s kinetic-energy change.
Related lessons and resources
Use these links when the current model depends on a prerequisite or becomes the foundation for the next model.
Common mistakes
Avoid this
Treating momentum as a scalar and adding magnitudes instead of signed/vector components.
Avoid this
Assuming zero total momentum means every object is at rest; moving momenta can cancel.
Avoid this
Using momentum conservation before defining the system and checking external impulse.
Avoid this
Confusing momentum with kinetic energy: p⃗ is vector and linear in velocity; K is scalar and quadratic in speed.
Mastery check
- Define the main quantity without relying on an equation.
- Choose the correct representation and sign convention.
- Explain why the calculus relationship matches the physics.
- Design or interpret an experiment that could test the model.
Frequently asked questions
What are the units of momentum?
kg·m/s, equivalent to N·s. The equality of these units foreshadows the impulse-momentum theorem.
Can total momentum be zero if objects are moving?
Yes. Momentum is a vector, so equal and opposite contributions can cancel.
Why are collisions treated with initial and final states?
During a short collision the internal forces are large, while the external impulse can often be small enough that the overall momentum change is determined primarily by the external impulse.
Is momentum the same as kinetic energy?
No. Momentum is vector and proportional to velocity; kinetic energy is scalar and proportional to speed squared.