Unit 5 · Topic 01

Linear momentum

Linear momentum describes motion through mass and direction. It is a vector quantity, so its components retain the sign of the chosen axis.

Unit 5Linear momentumOpen navigation

Concept 01

Mass times velocity

Essential The minimum you should retain

A particle's linear momentum is p=m v \vec p=m\vec v and points in the same direction as its velocity.

UnderstandInterpret and connect

A larger mass can have the same momentum at a lower speed; each component's sign comes from velocity.

DeepenFormulation and conditions

In Cartesian components, p_x=mv_x, p_y=mv_y, and p_z=mv_z; the unit is kg·m/s.

ExploreConnections for further study

Momentum depends on the reference frame because velocity does too.

Worked example

Momentum from velocity components

A 2.0 kg particle has v=(3.0 i-4.0 j) m/s.

Given
  • m=2.0 kg
  • v=(3.0 i-4.0 j) m/s
Target

Calculate p and its magnitude.

  1. Principle

    p=m v \vec p=m\vec v .

  2. Components

    p=(6.0 i-8.0 j) kg·m/s.

  3. Magnitude

    |p|=sqrt(6²+(-8)²)=10 kg·m/s.

  4. Interpretation

    The direction matches v.

Conclusion

p=(6.0 i-8.0 j) kg·m/s and |p|=10 kg·m/s.

Mathematical relation

Linear momentum

p→=mv→ \vec p=m\vec v
Represents

Momentum of a particle.

Physical interpretation

p has the direction and sense of v.

DeepenVariables, conditions, and checks

Variables

p
linear momentum; usual unit: kg·m/s
m
mass; usual unit: kg
v
velocity; usual unit: m/s

Conditions of application

  • Newtonian mechanics; m>0.

Dimensional check

kg·m/s.

Mathematical relation

Momentum components

px=mvx,py=mvy,pz=mvz p_x=mv_x,\ p_y=mv_y,\ p_z=mv_z
Represents

Cartesian decomposition of momentum.

Physical interpretation

Each component inherits the velocity sign.

DeepenVariables, conditions, and checks

Variables

p_x,p_y,p_z
momentum components; usual unit: kg·m/s

Conditions of application

  • Axes and signs are declared.

Dimensional check

kg·m/s.

A particle shows parallel velocity and momentum vectors.vp=mvm

Because m is positive, p and v share a direction; p length scales with mass.

Concept 02

Same momentum, different speed

Essential The minimum you should retain

For fixed nonzero momentum, the more massive object has the lower speed.

UnderstandInterpret and connect

Momentum and kinetic energy are not equivalent: for fixed p, K=p2/(2m) K=p^2/(2m) .

DeepenFormulation and conditions

The relation K=p2/(2m) K=p^2/(2m) follows by eliminating v between p=mv and K=(1/2)mv².

ExploreConnections for further study

Comparing p and K distinguishes what is conserved in interactions from what depends on mass.

For fixed p, K decreases as m increases

Kinetic energy versus mass for two momentum values.
  • p=4 kg·m/s
  • p=8 kg·m/s

The curves K=p2/(2m) K=p^2/(2m) show that equal momentum does not imply equal kinetic energy.

Concept 03

Changing a vector

Essential The minimum you should retain

Momentum can change through speed, direction, or both.

UnderstandInterpret and connect

Δp=p_f-p_i is a vector subtraction, not a subtraction of magnitudes.

DeepenFormulation and conditions

A rebound that reverses velocity can give a larger |Δp| than stopping the same object.

ExploreConnections for further study

In multiple dimensions, calculate component changes before the magnitude.

Concept 04

From one particle to a system

Essential The minimum you should retain

Total momentum is the vector sum of all particle momenta.

UnderstandInterpret and connect

Individual momenta can change while the system total remains constant.

DeepenFormulation and conditions

P=Σp_i links the microscopic description to center-of-mass motion.

ExploreConnections for further study

The system choice decides which forces are internal and external.

Concept review

Common errors

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

Treating p as mass times speed and ignoring direction.

p=m v \vec p=m\vec v is a vector; each component retains the velocity sign.

Thinking a momentum component cannot be negative.

Its sign follows from the chosen axis and velocity component.