Conservation of Linear Momentum
Momentum is conserved in every interaction when the complete interacting system plus surroundings are considered. For a selected subsystem, its total momentum stays constant only when the net external impulse is zero or negligible. Internal forces transfer momentum between system parts in equal-and-opposite amounts, so Physics C conservation problems begin by defining the system boundary and interaction interval.
Topic sequence · 3 of 4
Learning objectives
Objective 1
Use external impulse and system selection to determine whether total linear momentum is constant.
Objective 2
Apply conservation component-wise to one- and two-dimensional collisions and explosions and connect it to center-of-mass velocity.
Objective 3
Explain momentum transfer with Newton’s third law and test conservation with experimental data and uncertainty.
- Topic 4.1 system momentum and center-of-mass links.
- Topic 4.2 external impulse and ΔP⃗=J⃗_ext.
- Topic 2.3 Newton’s third law.
- Topic 2.1 system selection and center of mass.
Definitions first
Representations you must read and create
Representation 1
Before/after vector diagram with one common coordinate system and total-momentum vector shown in both states.
Representation 2
Component conservation table with P_x and P_y totals before and after.
Representation 3
System-boundary diagram labeling internal interaction forces and any external forces that can contribute impulse.
Representation 4
Center-of-mass trajectory/velocity representation showing constant v⃗_cm for zero net external force.
Core relationships and calculus
Variables: system momentum change and net external impulse
Use: general momentum balance
Variables: vector equality
Use: conservation of linear momentum
Variables: constant masses
Use: before/after momentum equation for a closed interaction model
Variables: M total mass
Use: center-of-mass velocity
Variables: when J_x=J_y=0
Use: component-wise conservation in two dimensions
For a selected system, integrate F⃗_ext=dP⃗_sys/dt across the interaction: J⃗_ext=ΔP⃗_sys. If J⃗_ext=0, then ΔP⃗_sys=0 and total momentum is constant. Newton’s third law explains how internal impulses cancel pairwise, while the center-of-mass relation P⃗_sys=Mv⃗_cm shows that v⃗_cm remains constant when net external force is zero and total mass is constant.
For objects inside the system, pairwise internal forces satisfy F⃗_{ij}=−F⃗_{ji}. Summing d p⃗_i/dt over all objects cancels the internal pairs, leaving dP⃗_sys/dt=F⃗_ext. Integrating gives ΔP⃗_sys=J⃗_ext. Therefore if J⃗_ext=0, P⃗_sys is constant. With constant total mass, P⃗_sys=Mv⃗_cm, so the center-of-mass velocity is also constant.
The current Mechanics framework expects quantitative collision/interaction analysis in one or two dimensions. Three-dimensional collisions may be analyzed qualitatively, so the lesson emphasizes component-wise 1D/2D equations rather than full 3D algebra.
Physical examples
Example 1
Two carts collide on a nearly frictionless track. If horizontal external impulse during the short interaction is negligible, the total horizontal momentum before and after is the same even if kinetic energy changes.
Example 2
A person steps from a canoe. Person and canoe exert internal forces on each other, so their momenta change in opposite directions while the total horizontal momentum of the person-canoe system remains approximately constant.
Example 3
An object explodes from rest into two pieces. If external impulse is negligible, their momentum vectors must sum to zero, though their speeds need not be equal when their masses differ.
Example 4
In a two-dimensional interaction, conserving the x component does not automatically conserve the y component unless the external impulse in y is also zero or negligible.
Worked reasoning
Exam reasoning patterns
Reasoning pattern 1
State the system and justify why external impulse is zero or negligible before writing P⃗_i=P⃗_f.
Reasoning pattern 2
In two dimensions, write separate x and y equations. Vector conservation gives two scalar equations in a plane.
Reasoning pattern 3
Do not use kinetic-energy conservation unless the collision is separately established as elastic; momentum conservation is more general.
Reasoning pattern 4
Use v⃗_cm as a check: for a constant-mass isolated system, a changed center-of-mass velocity signals an external impulse or an algebra/sign error.
Reasoning pattern 5
Distinguish global conservation from subsystem conservation: a selected system can gain or lose momentum through external impulse even though total momentum of the larger closed interaction remains conserved.
Deeper calculus and limiting-case check
Momentum conservation is a consequence of translational symmetry in more advanced mechanics, but the AP-level derivation through Newton’s third law and external impulse is sufficient here. The center-of-mass frame is especially useful in collisions because the total momentum is zero in that frame, simplifying many before/after vector relationships.
Model-selection workflow: decide what the problem is before choosing an equation
For Conservation of Linear Momentum, the fastest reliable route is not equation hunting. Start by identifying the system and time interval, encode it with the momentum/impulse representation, choose the impulse-momentum or conservation equation, and then test the external-impulse assumption. This sequence prevents a mathematically correct manipulation from being attached to the wrong physical model.
Define the physical object or system
Name what is being modeled, what is outside the system, the coordinate or orientation convention, and the interval or geometry over which the statement applies. Treat momentum conservation and external impulse as physical quantities first and symbols second.
Choose a representation
Build Before/after vector diagram with one common coordinate system and total-momentum vector shown in both states.. If the prompt supplies Component conservation table with P_x and P_y totals before and after., translate it into the same sign, axis, orientation, or reference convention before combining information.
Write the governing relationship
A useful starting relationship is P⃗_f−P⃗_i = J⃗_ext. State what the variables mean and why this relation—not merely a familiar formula—matches the physical situation.
Check conditions and limits
The current Mechanics framework expects quantitative collision/interaction analysis in one or two dimensions. Three-dimensional collisions may be analyzed qualitatively, so the lesson emphasizes component-wise 1D/2D equations rather than full 3D algebra. After solving, test units, sign or direction, zero/large-parameter limits, and whether the answer remains consistent with the original representation.
Representation → equation → calculus → physics chain
| Stage | What to write or inspect | What proves the step is physically valid |
|---|---|---|
| Representation | Before/after vector diagram with one common coordinate system and total-momentum vector shown in both states. | Every arrow, sign, axis, node, surface, path, or reference has an explicit meaning. The diagram must contain enough information to determine the relevant components or constraints. |
| Primary model | P⃗_f−P⃗_i = J⃗_ext | general momentum balance. Preserve the variables and units long enough to see dependence before inserting numerical values. |
| Second relationship | P⃗_i = P⃗_f if J⃗_ext=0 | conservation of linear momentum. Combine relationships only after confirming they describe the same system, interval, frame, or geometry. |
| Calculus operation | For a selected system, integrate F⃗_ext=dP⃗_sys/dt across the interaction: J⃗_ext=ΔP⃗_sys. If J⃗_ext=0, then ΔP⃗_sys=0 and total momentum is constant. Newton’s third law explains how internal impulses cancel pairwise, while the center-of-mass relation P⃗_sys=Mv⃗_cm shows that v⃗_cm remains constant when net external force is zero and total mass is constant. | State what the derivative, integral, differential equation, or limiting process represents physically; do not treat calculus notation as an algebraic decoration. |
| Physical check | Return to the original words and representation. | The final sign, direction, units, endpoint behavior, and functional dependence must all agree. A correct-looking expression that violates one of these checks needs repair. |
Advanced synthesis: how to handle an unfamiliar version of this topic
A 2.0 kg puck moving at (4.0 î+1.0 ĵ) m/s interacts with a 1.0 kg puck initially at rest. Afterward the 2.0 kg puck moves at (1.0 î+2.0 ĵ) m/s. With negligible external impulse, P⃗_i=(8 î+2 ĵ) kg·m/s and puck 1 ends with (2 î+4 ĵ). Therefore puck 2 must carry (6 î−2 ĵ) kg·m/s, giving v⃗_{2,f}=(6 î−2 ĵ) m/s. Solve x and y components separately. To deepen this example, change one assumption or parameter, predict the effect before recomputing, and then use the symbolic dependence to verify the prediction.
FRQ-quality response architecture for Topic 4.3
Mathematical routines
Start from the governing law or definition, keep the derivation symbolic, identify the condition that permits any simplification, and finish with units plus a limiting-case check. For this topic, the relationship P⃗_f−P⃗_i = J⃗_ext should appear only when its physical conditions have been established.
Translation between representations
Move deliberately between Before/after vector diagram with one common coordinate system and total-momentum vector shown in both states. and Component conservation table with P_x and P_y totals before and after.. Explain what a slope, area, direction, curvature, intercept, node value, field orientation, or other feature means instead of merely redrawing it.
Experimental design and analysis
Turn the topic into a measurable relationship. State the independent and dependent variables, important controls, repeated/ranged trials, and the graph or transformed graph whose slope/intercept would test the model or extract a physical parameter.
Qualitative/quantitative translation
Make the comparison or direction claim first, justify it through the model, derive the symbolic dependence independently, and then reconcile the derivation with the original qualitative claim.
Do not let an equation stand in for an explanation. A complete response names what changes, identifies the governing physical relationship, follows the causal chain, and states the requested outcome explicitly.
Experiment and data-analysis upgrade
Use two carts on a low-friction track for one-dimensional trials and air-table pucks or video analysis for two-dimensional trials. Measure masses and velocity vectors before/after, calculate component-wise total momentum, estimate external impulse, and compare residual momentum differences with experimental uncertainty.
Operationalize the variables
Say exactly what instrument or observable supplies each measured quantity. Distinguish a directly measured variable from a quantity calculated later from those measurements.
Predict the graph before collecting data
Use the governing relationship to decide what should be plotted. If the raw relationship is nonlinear, choose a transformation that produces a line whose slope or intercept has a defined physical meaning.
Control and repeat for a reason
Hold variables fixed when they would change the predicted relationship. Repeat trials to estimate scatter; vary the independent quantity over a useful range to test functional dependence. These solve different experimental problems.
Evaluate the model, not only the number
Compare the best-fit result with the theoretical prediction, inspect residual or endpoint behavior when appropriate, and identify whether a discrepancy is random scatter, instrument resolution, or a systematic violation of an ideal-model assumption.
Error-repair matrix
| Failure pattern | Repair move | Self-check |
|---|---|---|
| Saying “momentum is conserved” without naming the system and interaction interval. | Return to the momentum/impulse representation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Assuming momentum conservation requires kinetic-energy conservation; inelastic collisions can conserve momentum while losing mechanical kinetic energy. | Return to the impulse-momentum or conservation equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Canceling internal forces across different objects in individual-object equations instead of only in the total-system momentum balance. | Return to the momentum/impulse representation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Applying a one-dimensional sign equation to a two-dimensional interaction without separately conserving components. | Return to the impulse-momentum or conservation equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Ignoring external impulse merely because the interaction is called a collision; its negligibility must be justified by the model. | Return to the momentum/impulse representation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
Spiral review: make this topic transferable
A topic is exam-ready only when it can be recognized inside a mixed problem. Use the connections below as transfer tests rather than as optional reading.
Transfer 1
Topic 2.3 explains equal-and-opposite internal force pairs. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 2
Topic 4.2 gives ΔP⃗=J⃗_ext, so zero external impulse produces conservation. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 3
Topic 3.1 helps distinguish momentum conservation from kinetic-energy conservation. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Formula-condition retrieval
From memory, write P⃗_f−P⃗_i = J⃗_ext and one condition that must be true before using it. Then state one situation where a more general derivative, integral, conservation law, or field/force model is required.
Topic 4.3 mastery ladder
Interactive 2D momentum-conservation solver
Two objects interact with negligible external impulse. Enter initial velocities and object 1 final velocity; solve object 2 final velocity component by component.
Experimental connection
Use two carts on a low-friction track for one-dimensional trials and air-table pucks or video analysis for two-dimensional trials. Measure masses and velocity vectors before/after, calculate component-wise total momentum, estimate external impulse, and compare residual momentum differences with experimental uncertainty.
Connections across Physics C
Connection 1
Topic 2.3 explains equal-and-opposite internal force pairs.
Connection 2
Topic 4.2 gives ΔP⃗=J⃗_ext, so zero external impulse produces conservation.
Connection 3
Topic 3.1 helps distinguish momentum conservation from kinetic-energy conservation.
Connection 4
Topic 4.4 classifies elastic/inelastic collisions by kinetic-energy behavior while retaining momentum conservation for an isolated system.
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
Saying “momentum is conserved” without naming the system and interaction interval.
Avoid this
Assuming momentum conservation requires kinetic-energy conservation; inelastic collisions can conserve momentum while losing mechanical kinetic energy.
Avoid this
Canceling internal forces across different objects in individual-object equations instead of only in the total-system momentum balance.
Avoid this
Applying a one-dimensional sign equation to a two-dimensional interaction without separately conserving components.
Avoid this
Ignoring external impulse merely because the interaction is called a collision; its negligibility must be justified by the model.
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
When is momentum conserved?
For the selected system over the chosen interval, total momentum is constant when the net external impulse is zero or negligible.
Is momentum conserved in an inelastic collision?
Yes, if the external impulse on the selected system is negligible. Kinetic energy need not be conserved.
Why do internal forces not change total momentum?
They occur in equal-and-opposite pairs, so their impulses redistribute momentum among system parts but sum to zero for the whole system.
Can momentum conservation be used in two dimensions?
Yes. Conserve each component separately when the corresponding external impulse is zero or negligible.