Chapter 5: Linear Momentum and Collisions

5-1 Linear Momentum and Impulse

• Objectives:
  • Compare momentum of different moving objects.
  • Compare momentum of the same object at different velocities.
  • Identify change in momentum examples.
  • Relate momentum change to force and time.
• Linear Momentum:
  • Denoted by $p$, defined as the product of mass $m$ and velocity $v$.
  • $p = mv$
  • Vector quantity, direction matches velocity.
  • SI units: $kg \cdot m/s$
• Everyday Speech vs. Physics Definition: Physics definition aligns with the intuitive use of "momentum".
• Relating Force, Time, and Momentum:
  • Newton's second law expressed as $F = \frac{\Delta p}{\Delta t}$, where $\Delta p$ is change in momentum and $\Delta t$ is time interval.
  • Rearranging: $F\Delta t = \Delta p = mv<em>f - mv</em>i$
• Impulse-Momentum Theorem:
  • $F\Delta t = \Delta p$
  • Impulse is the product of force and time interval.
  • Small force over long time can equal a large force over short time given same momentum change.
• Follow-through: Extending contact time increases momentum change.

5-2 Law of Conservation of Linear Momentum

• Objectives:
  • Describe interaction between two objects via momentum change.
  • Compare total momentum before and after interaction.
  • State the law of conservation of momentum.
  • Predict final velocities after collisions.
• Law of Conservation of Linear Momentum:
  • Total initial momentum = Total final momentum.
  • Mathematically: $p<em>{A,i} + p</em>{B,i} = p<em>{A,f} + p</em>{B,f}$
  • Total momentum remains constant in a closed system.
• General Form of Conservation Law: Total momentum of interacting objects remains constant.
• Conservation of Momentum - Pushing Away:
  • Skater example: When skaters push away from each other, their momentum is equal but opposite, so the total momentum remains zero.
  • Skater 1: $p<em>1,i=0$, Skater 2: $p</em>2,i=0$
  • Total system initial momentum: $P<em>i = p</em>1,i + p_2,i = 0$
  • $p<em>1,f + p</em>2,f = 0$
• Newton's Third Law Connects to Momentum Conservation:
  • Impulse-momentum theorem applied to two colliding bumper cars.
    • Car 1: $F<em>1\Delta t = m</em>1v<em>1f - m</em>1v_1,i$
    • Car 2: $F<em>2\Delta t = m</em>2v<em>2f - m</em>2v_2,i$
  • Newton's third law: $F<em>1 = -F</em>2$, thus $F<em>1\Delta t = -F</em>2\Delta t$
    • $\Delta p<em>1 = -\Delta p</em>2$
  • $m<em>1v</em>1f- m<em>1v</em>1,i = -(m<em>2v</em>2,f- m<em>2v</em>2,i)$

5-3 Elastic and Inelastic Collisions

• Objectives:
  • Identify different types of collisions.
  • Determine changes in kinetic energy during perfectly inelastic collisions.
  • Compare momentum and kinetic energy conservation in inelastic and elastic collisions.
  • Find final velocities in perfectly inelastic and elastic collisions.
• Types of Collisions:
  • Perfectly Inelastic: Objects stick together after collision.
  • Elastic: Objects bounce off each other, total kinetic energy conserved.
  • Inelastic: Objects bounce but total kinetic energy decreases.
• Perfectly Inelastic Collisions Analysis:
  • $m<em>1v</em>1,i + m<em>2v</em>2,i = (m<em>1 + m</em>2)v_f$
  • Kinetic energy is not constant; some converted to sound and internal energy.
• Elastic Collisions Analysis:
  • Momentum and kinetic energy are conserved.
  • $m<em>1v</em>1,i + m<em>2v</em>2,i = m<em>1v</em>1,f + m<em>2v</em>2,f$
  • ${1 \over 2}m<em>1v</em>1,i^2 + {1 \over 2}m<em>2v</em>2,i^2 = {1 \over 2}m<em>1v</em>1,f^2 + {1 \over 2}m<em>2v</em>2,f^2$
• Most collisions are neither: Colliding objects bounce but some kinetic energy is lost.