Linear Momentum

Definition and Properties

Linear momentum is a fundamental concept in physics that describes the quantity of motion an object possesses. It is a vector quantity, meaning it has both magnitude and direction, and is defined for a particle or object as the product of its mass and velocity.

• Definition: The linear momentum p of a particle is given by:

• Direction: The direction of momentum is the same as the direction of velocity.

• Units: The SI unit of momentum is kg·m/s.

• Vector Components: For three dimensions:

Example: A 2 kg ball moving at 3 m/s east has a momentum of 6 kg·m/s east.

Newton and Momentum

Newton's Second Law and Momentum

Newton's Second Law connects force and momentum, providing a way to analyze how forces change an object's motion.

• Constant Mass: For a particle with constant mass, Newton's Second Law can be written as:

• General Form: For cases where mass changes (e.g., rockets):

• Interpretation: The net force on a particle equals the time rate of change of its momentum.

Example: A rocket burning fuel loses mass, so the general form of Newton's Second Law is required.

Conservation of Linear Momentum

Isolated Systems and Conservation Law

In an isolated system (no external forces), the total linear momentum remains constant during interactions.

• Conservation Principle:

• For Two Particles:

• Component Form: Conservation applies to each component (x, y, z):

• Generalization: Applies to any number of particles in an isolated system.

Example: Two ice skaters push off each other and move in opposite directions; their total momentum before and after is the same.

Conservation of Momentum (Archer Example)

This example illustrates conservation of momentum when an archer releases an arrow while standing on a frictionless surface.

• Before Release: (both at rest)

• After Release:

• Solving for Archer's Velocity:

Example: If the arrow moves forward, the archer moves backward to conserve momentum.

Impulse and Momentum

Impulse-Momentum Theorem

Impulse quantifies the effect of a force acting over a time interval, resulting in a change in momentum.

• Impulse:

• Impulse-Momentum Theorem:

• For Constant Force:

• Units: Impulse has units of kg·m/s (same as momentum).

Example: A baseball bat striking a ball applies a large force over a short time, changing the ball's momentum.

Time-Averaged Force

The time-averaged force is the constant force that would produce the same impulse as a time-varying force over a given interval.

• Formula:

• Graphical Interpretation: The area under the force-time curve equals the impulse.

Example: The shaded area under a force vs. time graph represents the impulse delivered to an object.

Impulse Approximation

In many collisions, one force dominates for a short time, allowing us to approximate the change in momentum using only the impulsive force.

• Application: Only the dominant force is considered during the collision.

• Momenta: and represent momenta immediately before and after the collision.

Example: A golf ball struck by a club experiences a large impulsive force compared to other forces.

Collisions

Characteristics of Collisions

A collision is an event where two particles come close and interact via forces, possibly involving physical contact. The interaction time is typically short, and the forces involved are much greater than any external forces.

• Direct Contact: Collisions may result from direct contact (e.g., billiard balls).

• No Contact: Collisions can also occur without physical contact (e.g., charged particles interacting via electromagnetic forces).

• Momentum Conservation: Momentum is conserved in all collisions within an isolated system.

Example: Two protons deflecting each other via electric repulsion.

Types of Collisions

Collisions are classified based on whether kinetic energy is conserved.

• Elastic Collision: Both momentum and kinetic energy are conserved. - Occur on a microscopic level (e.g., gas molecules).

• Inelastic Collision: Momentum is conserved, but kinetic energy is not. - Some energy is lost to deformation, sound, heat, etc.

• Perfectly Inelastic Collision: Objects stick together after collision; maximum kinetic energy is lost consistent with momentum conservation.

Example: Two cars crash and stick together (perfectly inelastic); billiard balls bounce off each other (elastic).

Summary Table: Types of Collisions

Practice and Conceptual Questions

Clicker Questions (Conceptual Understanding)

• How does the momentum of a system change when objects interact in the absence of external forces?

• What is the effect of applying equal forces to carts of different masses for the same time interval?

• How does the kinetic energy of lighter and heavier carts compare after being pushed with equal force for equal time?

• In which scenario does a ball experience the largest change in momentum: stopping, reversing direction, or being pushed to a higher speed?

Example: If two carts of mass m and 2m are pushed with equal force for equal time, the lighter cart will have twice the momentum of the heavier cart.

Additional info: These notes expand on the brief points and examples provided in the slides, adding definitions, formulas, and academic context for clarity and completeness.