Conservation of Momentum

Particle Momentum

The concept of momentum is fundamental in physics, describing the motion of a particle as the product of its mass and velocity. The impulse-momentum theorem provides a conservation law for particle motion.

• Impulse-Momentum Theorem: The change in momentum of a particle is equal to the impulse applied to it.

• Conservation Condition: When , the particle's momentum remains constant.

• Extension to Systems: Momentum is also conserved for a system of particles, not just individual particles.

Momentum Conservation for a System of Objects

In many physical situations, we consider systems composed of multiple interacting objects. If the system is isolated (no significant external forces), its total momentum is conserved.

• Isolated System: A collection of objects with negligible interaction with the environment (e.g., solar system, atom, molecule).

• Conservation Law for Systems: The total momentum of the system changes only if external forces act on it.

Internal and External Forces

Forces within a system can be classified as internal or external, depending on their origin.

• Internal Forces: Forces between objects within the system (e.g., gravitational force between two planets in the solar system).

• External Forces: Forces exerted on system objects by sources outside the system.

Example: If a system consists of objects A and B, the force A exerts on B is internal, while a force from an external object C on A or B is external.

The Momentum of a System

The total momentum of a system is the vector sum of the momenta of all its constituent objects.

• System Momentum:

• Newton's Second Law for Object :

• Newton's Second Law for the System:

Cancellation of Internal Forces

Internal forces within a system always occur in equal and opposite pairs (Newton's Third Law), so their net effect on the system's total momentum cancels out.

• Sum of Forces in the System:

• Internal forces sum to zero:

• Thus,

Conservation of Momentum

Only external forces can change the momentum of a system. If the sum of external forces is zero, the system's momentum is conserved.

• General Law:

• Isolated System: If , then and momentum is conserved.

• Note: Energy and momentum are distinct; energy can change without affecting momentum.

Example: Two Carts on a Track

Consider two carts initially at rest. An internal spring pushes them apart. Since the system is isolated, momentum is conserved.

• Initial Momentum:

• Final Momentum:

• Relation:

• Application: This principle is used in recoil problems and explosion analysis.

Momentum Conservation in Multiple Dimensions

Momentum Conservation in Three Dimensions

Momentum is a vector quantity, so conservation applies independently to each component (x, y, z).

• Component Equations:

• Each direction of motion is independent.

Collisions in Two Dimensions

In two-dimensional collisions, momentum is conserved separately in both the x and y directions.

• Before Collision:

• After Collision:

• Momentum conservation in each direction allows us to solve two-dimensional problems as two separate one-dimensional problems.

Center of Mass

Definition and Calculation

The center of mass is a special point in a system of particles that moves as if all the system's mass were concentrated there and all external forces were applied at that point.

• Mathematical Definition:

• Total Mass:

• The center of mass is the mass-weighted average position of all objects in the system.

Example: Two Objects

Given two objects with masses at and at , the center of mass is calculated as:

• Interpretation: The center of mass lies closer to the heavier object.

Center-of-Mass Velocity and System Momentum

The velocity of the center of mass is related to the total momentum of the system:

• If there are no external forces, both the total momentum and the center-of-mass velocity remain constant.

Summary Table: Key Concepts in Momentum Conservation

Take-Away Concepts

• Systems: Distinguish between internal and external forces.

• Momentum of a System:

• Newton's Second Law for a System:

• Momentum Conservation: If , then is constant.

• Component Conservation: Momentum conservation applies to each component (x, y, z) separately.

• Center of Mass:

• System Momentum and Center-of-Mass Velocity:

Example Problem

Problem: Penguin A is at rest on a sled on frictionless ice. Penguin A walks toward penguin B on shore. What happens to the center of mass of the system (penguin A + sled)?

• Answer: The center of mass remains the same distance from the shore, since there are no external horizontal forces acting on the system.