Unit 5 · Topic 04

Collisions in one and two dimensions

In an isolated system, every collision conserves total momentum. Total kinetic energy is conserved only in elastic collisions.

Unit 5CollisionsOpen navigation

Concept 01

What is conserved

Essential The minimum you should retain

An elastic collision conserves P and K; an inelastic collision conserves P but not K.

UnderstandInterpret and connect

In a perfectly inelastic collision the bodies stick and share one final velocity.

DeepenFormulation and conditions

Lost K becomes deformation, sound, or internal energy; momentum is not lost.

ExploreConnections for further study

Classifying the collision determines which additional equations are available.

Three rows compare elastic, inelastic, and perfectly inelastic collisions.elastic: P and K conservedinelastic: P conserved, K decreasesstuck: shared v_f

Every isolated case conserves P; only elastic conserves K and perfectly inelastic shares v_f.

Concept 02

One final velocity

Essential The minimum you should retain

When bodies stick, their combined mass moves with v_f=(m_1v_1i+m_2v_2i)/(m_1+m_2).

UnderstandInterpret and connect

Velocity signs are essential when adding initial momentum.

DeepenFormulation and conditions

The balance conserves P, while comparing K_i and K_f generally shows a decrease.

ExploreConnections for further study

This is the limiting case of maximum kinetic-energy loss compatible with the given momentum.

Worked example

Two cars stick together

A 1200 kg car moves at +12 m/s and an 800 kg car at -5.0 m/s.

Given
  • m_1=1200 kg
  • v_1=+12 m/s
  • m_2=800 kg
  • v_2=-5.0 m/s
Target

Calculate shared final velocity.

  1. Initial momentum

    P_i=1200(12)+800(-5)=10400 kg·m/s.

  2. Final mass

    m_1+m_2=2000 kg.

  3. Calculation

    v_f=10400/2000=5.20 m/s.

  4. Interpretation

    Positive sign is the first car's direction.

Conclusion

v_f=+5.20 m/s.

Mathematical relation

Perfectly inelastic collision

vf=m1v1i+m2v2im1+m2 v_f=(m_1v_{1i}+m_2v_{2i})/(m_1+m_2)
Represents

Shared velocity after the bodies stick.

Physical interpretation

Momentum is conserved, kinetic energy generally is not.

DeepenVariables, conditions, and checks

Variables

v_f
shared final velocity; usual unit: m/s

Conditions of application

  • Isolated 1D collision; bodies stick.

Dimensional check

m/s.

Concept 03

Two laws, two unknowns

Essential The minimum you should retain

A one-dimensional elastic collision conserves momentum and kinetic energy.

UnderstandInterpret and connect

With sufficient initial data, both laws determine the two final velocities.

DeepenFormulation and conditions

The formulas for a target initially at rest depend on the mass ratio.

ExploreConnections for further study

Velocity exchange occurs only for equal masses in this special case.

Worked example

Head-on elastic collision

m_1=2.0 kg at +6.0 m/s hits m_2=4.0 kg at rest.

Given
  • m_1=2.0 kg
  • v_1i=+6.0 m/s
  • m_2=4.0 kg
  • v_2i=0
Target

Find both final velocities.

  1. Model

    Use elastic 1D formulas for a target at rest.

  2. First mass

    v_1f=((2-4)/(2+4))6=-2.0 m/s.

  3. Second mass

    v_2f=(2·2/(2+4))6=4.0 m/s.

  4. Check

    P_i=P_f=12 kg·m/s and K_i=K_f=36 J.

Conclusion

v_1f=-2.0 m/s and v_2f=+4.0 m/s.

Mathematical relation

Elastic collision in 1D

Pi=Pf,Ki=Kf P_i=P_f,\ K_i=K_f
Represents

The two conservation laws for an elastic collision.

Physical interpretation

Both equations determine final velocities.

DeepenVariables, conditions, and checks

Variables

P
total momentum; usual unit: kg·m/s
K
total kinetic energy; usual unit: J

Conditions of application

  • Isolated elastic one-dimensional collision.

Dimensional check

Each equality preserves its units.

Concept 04

Conserve components

Essential The minimum you should retain

In two dimensions, P_x and P_y are separately conserved when external impulse is negligible.

UnderstandInterpret and connect

Conserving only the magnitude |P| is not enough.

DeepenFormulation and conditions

Final vectors must close the same vector polygon as initial momentum.

ExploreConnections for further study

Momentum gives two equations; angles, elasticity, or other data may be needed to close the problem.

A vector triangle shows P_i=P_1f+P_2f.P_ip_1fp_2f

The initial vector is the sum of the two final vectors; the polygon closes in x and y.

Concept review

Common errors

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

Conserving K in every collision.

K is conserved only in elastic collisions; P may also be conserved in inelastic ones.

Saying an inelastic collision loses momentum.

An isolated system conserves P while part of K changes form.

Conserving only |P| in two dimensions.

Conserve P_x and P_y separately.

Applying velocity exchange to every collision.

It is a special elastic 1D equal-mass case.