Unit 5 · Topic 03

Conservation of linear momentum

Total momentum changes only through net external impulse. Conservation requires a declared system and justification that this impulse is zero or negligible.

Unit 5ConservationOpen navigation

Concept 01

What is inside

Essential The minimum you should retain

Before conserving momentum, declare the bodies in the system and the interval studied.

UnderstandInterpret and connect

Forces between system bodies are internal; forces exerted from outside are external.

DeepenFormulation and conditions

dP/dt=F_ext and ΔP=J_ext \Delta P=J_{ext} summarize the system balance.

ExploreConnections for further study

A useful boundary can turn a complicated interaction into a simple balance.

Mathematical relation

System momentum balance

P→=Σpi,dP→dt=F→ \vec P=\sum_i\vec p_i,\ d\vec P/dt=\vec F_{ext}
Represents

Total momentum and its change from external force.

Physical interpretation

Internal forces do not change total P.

DeepenVariables, conditions, and checks

Variables

P
total momentum; usual unit: kg·m/s
F_ext
net external force; usual unit: N

Conditions of application

  • The system boundary is declared.

Dimensional check

N=d(kg·m/s)/dt.

Two bodies interact inside a boundary while an external force crosses it.system A+BABinternal forceJ_ext

The boundary separates internal forces between A and B from external impulse that changes total P.

Concept 02

Constant does not mean zero

Essential The minimum you should retain

If net external impulse is zero, P_f=P_i P_f=P_i .

UnderstandInterpret and connect

Conserved does not mean zero: a system may retain nonzero total momentum.

DeepenFormulation and conditions

Conservation applies by components, so one component may be conserved while another is not.

ExploreConnections for further study

Over a short interval, the relevant external impulse need only be negligible, not every force absent.

Mathematical relation

Momentum conservation

Jext=0⇒Pf=Pi \vec J_{ext}=0\Rightarrow\vec P_f=\vec P_i
Represents

Condition for conservation of total momentum.

Physical interpretation

The constant value need not be zero.

DeepenVariables, conditions, and checks

Variables

J_ext
net external impulse; usual unit: N·s

Conditions of application

  • J_ext=0 or negligible in the studied component.

Dimensional check

kg·m/s in both states.

Concept 03

Opposite directions

Essential The minimum you should retain

If an initially resting system separates without external impulse, final momenta sum to zero.

UnderstandInterpret and connect

The smaller mass usually acquires the larger speed to balance momentum.

DeepenFormulation and conditions

Newton's third law describes opposite internal forces; momentum balance links the states directly.

ExploreConnections for further study

Recoil, explosions, and propulsion share this internal exchange logic.

Worked example

Recoil on ideal ice

A 70 kg person throws a 5.0 kg bag at +8.0 m/s from rest.

Given
  • m_p=70 kg
  • m_b=5.0 kg
  • v_b=+8.0 m/s
Target

Calculate the person's velocity.

  1. System

    Person and bag; horizontal external impulse is negligible.

  2. Initial state

    P_i=0.

  3. Balance

    70v+(5)(8)=0.

  4. Result

    v=-40/70=-0.571 m/s.

Conclusion

The person moves at 0.571 m/s in the opposite direction.

A person and bag go from rest to motion in opposite directions.personbagpersonbagp_pp_mbefore: P=0after: p_p+p_m=0

Before, P=0; afterward, opposite person and bag momenta give the same total.

Concept 04

A controlled approximation

Essential The minimum you should retain

Exact conservation requires zero net external impulse over the interval.

UnderstandInterpret and connect

Real experiments should state which component and time scale justify the approximation.

DeepenFormulation and conditions

When J_ext is known, use P_f=P_i P_f=P_i +J_ext instead of forcing conservation.

ExploreConnections for further study

Changing the boundary can expose or hide momentum flows but does not change the physics.

Concept review

Common errors

Each warning includes a concrete way to review the reasoning, not only an incorrect-answer marker.

Believing conserved P means P=0.

P may be nonzero and constant.

Claiming internal forces prevent total-P conservation.

Total-P change depends on external impulse.