Momentum, Impulse, and Collisions

Introduction

This chapter explores the concepts of momentum, impulse, and the conservation of momentum, which are essential for analyzing interactions such as collisions where forces are complex or unknown. These principles allow us to solve problems that cannot be addressed using Newton's second law alone.

Translational (Linear) Momentum

• Definition: The momentum p of a particle is defined as the product of its mass and velocity.

• Formula:

• Vector Nature: Momentum is a vector quantity, possessing both magnitude and direction.

• Units:

• Components: In Cartesian coordinates: , ,

• Magnitude: Momentum is large if either mass or velocity is large.

Newton's Second Law and Momentum

• Standard Form:

• Alternative Form:

• This form is valid even if the mass is changing (e.g., rocket propulsion).

• General Vector Form:

Impulse

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

• Definition:

• General Form:

• Impulse is the area under the force vs. time curve.

• Impulse-Momentum Theorem: The change in momentum of a particle during a time interval equals the impulse of the net force acting on it.

Comparing Momentum and Kinetic Energy

• Momentum Change: Depends on the time over which the net force acts.

• Kinetic Energy Change: Depends on the distance over which the net force acts.

• Formula for Kinetic Energy:

Conservation of Momentum

If no external forces act on a system, its total momentum remains constant.

• Principle:

• Internal Forces: Forces between objects within the system; these sum to zero and do not affect total system momentum.

• External Forces: Forces from outside the system; these can change the system's total momentum.

• Isolated System: No external forces act; momentum is conserved.

• Newton's Third Law: Internal forces are equal and opposite, ensuring conservation.

Mathematical Proof for Two Particles

• Action-Reaction:

• Conservation:

• Examples: Ice skaters pushing off, rifle recoil, rocket propulsion.

Momentum Conservation for a System of Particles

• System Momentum:

• Time Derivative:

• Internal Forces Cancel: By Newton's Third Law,

• Result: If , then

Collisions

A collision is an interaction where two or more objects strike each other, and if the net external force is zero, momentum is conserved.

• Types: Car crashes, billiard balls, more than two objects possible.

• Vector Nature: Momentum must be added as vectors, not scalars.

• Conservation Equation:

Types of Collisions

One-Dimensional Collisions

• General Case: Both objects may have initial velocities in either direction.

• Final Velocity (Perfectly Inelastic):

• Special Cases:

  • Equal masses:

  • Heavy body at rest:

  • Light body at rest:

Elastic Collisions

• Conservation of Momentum:

• Conservation of Kinetic Energy:

• Relative Velocity:

• Equal Masses: Velocities are exchanged after collision.

Two-Dimensional Collisions

• Momentum conservation applies to each component (x and y directions).

• Requires solving simultaneous equations for final velocities and angles.

• Example: Billiard balls scattering at known angles.

Impulse in Collisions

• Impulse on First Mass:

• Impulse on Second Mass:

• Momentum Conservation:

Center of Mass

The center of mass (CM) of a system is the mass-weighted average position, and the system behaves as if all its mass were concentrated at the CM.

• Definition: ,

• For continuous distributions:

  • Volume density:

  • Surface density:

  • Linear density:

• CM may not be located on the object itself (e.g., irregular shapes).

Motion of the Center of Mass

• Velocity of CM:

• Acceleration of CM:

• External Forces: The CM moves as if all external forces act at that point.

• Conservation: If no external forces, the velocity of the CM is constant.

Examples

• Hockey Puck: Calculate change in momentum, impulse, and average force during a bounce.

• Car Collision: Estimate average force during abrupt stop and compare to weight.

• Skaters on Ice: Use conservation of momentum to find velocities and forces during push-off.

• Car Collision (Perfectly Inelastic): Find final velocity after two cars stick together.

Summary Table: Types of Collisions

Additional info: These notes include expanded explanations, formulas, and examples to provide a comprehensive overview suitable for college-level physics students preparing for exams.