Unit 8: The Great Conservation Laws - Part 1: Momentum and Impulse

Unit 8: The Great Conservation Laws

Part 1: Momentum and Impulse

Unit 8 Outline

  1. Momentum and Conservation
    a. Conservation of Linear Momentum
    b. Impulse – Momentum Theorem
    c. Types of Collisions
    i. 1D
    ii. 2D
    d. Momentum Problems

  2. Center of Mass
    a. System of Particles
    b. Solid Objects (with uniform density)
    i. Triangle
    ii. Rod
    c. Motion of a System with Center of Mass

Unit 8 Learning Objectives

  • Knowledge: Understand that conservation of momentum is a tool for solving problems in collisions and locating the center of mass is essential to analyze rotation and linear motion.

  • Understanding: Recognize when conservation of momentum and conservation of energy are applicable.

  • Skills:

    1. Solve momentum and impulse problems.

    2. Center of Mass Observation:
      a. Locate the center of mass of a system consisting of two objects.
      b. State and apply the relationships between linear momentum and center-of-mass motion for a system of particles.

Extra Concept

  • Both energy and momentum are conserved due to universe symmetries.

Conservation of Momentum vs. Energy

  • Scenario: If two ice skaters push away from each other while holding hands, conservation of energy cannot determine each skater's velocity post-push.

    • Assumptions: Assume a closed system with no external forces.

    • Problem Statement: Without enough information, conservation of energy is not applicable.

  • Systems Defined:

    • System 1: Skaters as a single system

    • System 2: Each individual skater as a separate system.

  • Energy Considerations: Energy can exit the system through thermal and sound fluctuations affecting momentum calculation.

Momentum related equations:

  • For energy before and after interactions:
    K1^i + K2^i + W{nc} = K1^f + K2^f W{net} = \Delta K
    W{nc} + Wc = \Delta K

  • If work done is not equal to zero, we derive:
    \Delta K = W{nc} W{nc} = K1^f - K1^i + K2^f - K2^i
    W_{net} = \Delta K

Momentum and Conservation of Momentum

  • Newton's Third Law Applicability: The force exerted by one skater on the second is equal and opposite to the first's force.

  • Momentum Definition:

    • Linear momentum is defined as p = m v

      • Where $p$ is momentum (in kg·m/s), $m$ is mass (in kg), and $v$ is the velocity (in m/s).

Momentum Conservation Principle

  • Closed System Definition: A system with no external force is described as closed and the momentum of this system will not change, formulated as:
    \Delta p = 0
    p{Total, initial} = p{Total, final}

Impulse-Momentum Theorem

  • An impulse acting on a system produces a change in momentum represented as: J = \Delta p

    • Impulse equals the area under the curve of a force vs. time graph.

Collision Types and Conservation of Momentum

  1. Elastic Collision: Kinetic Energy conserved.

  2. Inelastic Collision: Kinetic Energy NOT conserved; two variations:

    • Objects stick together (perfectly inelastic)

    • One object breaks into multiple parts (explosion).

Collision Examples and Problem Solving

  • Collisions in 1 Dimension:

    • Set equations based on momentum conservation before and after collisions to find unknown velocities.

    • E.g.:

    • m1 v1 + m2 v2 = m1 v1' + m2 v2' (momentum equation)

  • Collisions in 2 Dimensions: Require separate calculations for each axis (X and Y) using conservation laws.

Problem Analysis (Elastic Collision Example)

  • Example Problem: A 1800-kg car strikes a 5500-kg truck.

    • Using conservation laws for momentum and kinetic energy, calculate final velocities.

Impulse Applications

  • Impulse changes based on time applied.

    • Short time = large force

    • Long time (airbags, helmets, etc.) = lower force.

Problem Solving Techniques

  1. System definition and momentum conservation equations setup.

  2. Solve key parameters based on the momentum and kinetic energy formulations.

  3. Graphical approaches to visualize impulse and ranges of force application.

Important Equations to Remember

  1. Momentum Conservation Principle:
    p{Total, initial} = p{Total, final}

  2. Impulse-Momentum Theorem:
    J = rac{F}{ riangle t} = riangle p

  3. Kinetic Energy Before and After Interactions:
    K1^i + K2^i + W{nc} = K1^f + K2^f W{net} = riangle K

  4. Work-Energy Principle:
    W{nc} + Wc = riangle K

  5. Work Done:
    riangle K = W{nc} W{nc} = K1^f - K1^i + K2^f - K2^i
    W_{net} = riangle K

  6. Momentum Equation for 1D Collisions:
    m1 v1 + m2 v2 = m1 v1' + m2 v2'

  7. Newton's Third Law:
    F{12} = -F{21} (force exerted by one skater on another is equal and opposite)