Unit 4 Master Notes: Momentum Conservation & Collision Theory

Law of Conservation of Linear Momentum

The principle stating that the total linear momentum of a closed system remains constant if no external forces act on it.

Closed (Isolated) System

A system where the net external force is zero, and total momentum remains constant.

Open System

A system where external forces can change the total momentum.

Conservation Equation

The equation stating that total momentum before an interaction equals total momentum after: Σpinitial = Σpfinal.

Center of Mass (COM)

The point in a system of particles where the mass is evenly distributed; if external forces are zero, its velocity remains constant.

Elastic Collision

A collision where momentum and kinetic energy are both conserved; objects bounce off without permanent deformation.

Inelastic Collision

A collision where momentum is conserved but kinetic energy is not; some energy is transformed into internal energy.

Perfectly Inelastic Collision

A collision where momentum is conserved while kinetic energy is not; objects stick together after impact.

Kinetic Energy (K)

The energy that an object possesses due to its motion, which can vary in different types of collisions.

Momentum

The product of an object's mass and its velocity; a vector quantity.

Total Momentum

The vector sum of all individual momenta in a system; conserved in isolated systems.

Velocity of COM Formula

v_cm = (m1v1 + m2v2 + …)/(m1 + m2 + …); calculates the velocity of the center of mass.

Collision Types

Classified into elastic, inelastic, and perfectly inelastic based on conservation of kinetic energy.

Recoil

The backward movement of an object when another object is propelled forward, conserving momentum.

Key Characteristic of Elastic Collisions

No heat or sound generation; objects bounce perfectly.

Key Characteristic of Inelastic Collisions

Some kinetic energy is 'lost' to heat, sound, or deformation.

Key Characteristic of Perfectly Inelastic Collisions

Objects stick together and move as one mass after the collision.

Friction,

A force that opposes the motion of an object; external force affecting momentum in open systems.

Impulsive Approximation

Assumption that external forces are negligible during the very short time of impact.

Common Mistake 1

Ignoring direction; velocity must include signs based on direction in conservation equations.

Common Mistake 2

Assuming kinetic energy is conserved unless the collision is stated to be elastic.

Common Mistake 3

Confusing momentum conservation with mechanical energy conservation; momentum is always conserved.

Common Mistake 4

Forgetting that external forces affect momentum conservation; the system's boundaries must be properly defined.

Final Velocity Formula for Perfectly Inelastic Collision

m1v1i + m2v2i = (m1 + m2)*v_f.

Worked Example 1D Collision

Illustrates how to apply momentum conservation in a two-cart perfectly inelastic collision case.

Conservation of Energy

In elastic collisions, kinetic energy is conserved (Ki = Kf), while inelastic collisions lose some energy.