Lecture on Momentum and Impulse

Definitions and Basics

Momentum

• Definition: Product of an object's mass and velocity ($P = m \times v$).
• Units:
  • Kilogram meters per second (kg·m/s)
  • Newton seconds (N·s)

Impulse

• Definition: Change in momentum ($\Delta P$).
• Units: Same as momentum (kg·m/s or N·s).
• Equation: $F \times \Delta t = m \times \Delta v$

Equation Manipulations

• Rearranging momentum equation:
  • For mass: $m = \frac{P}{v}$
  • For velocity: $v = \frac{P}{m}$
• Impulse Equation: $\Delta P = F \times \Delta t = m \times \Delta v$
• Always remember to use $\Delta v$ for changes in velocity, not just $v$.

Vector Nature of Momentum and Impulse

• Momentum is a vector; it has direction.
• Use the sign (positive or negative) to indicate direction.
• Be cautious of x.y and y-components in calculations.

Concepts and Principles

Physics Principles Involving Momentum and Impulse

• Uniform Motion: Constant velocity.
• Accelerated Motion: Changing velocity.
• Conservation of Momentum: Total momentum remains constant in isolated systems.
• Impulse-Momentum Theorem: Impulse causes a change in momentum.

Predetermined vs. Non-Predetermined Impulse

• Predetermined: Known initial and final momenta.
  • E.g., stopping a moving car.
• Non-Predetermined: Unknown final momentum.
  • E.g., hitting a golf ball.
• Strategies:
  • For predetermined: Spread impulse over longer time to reduce force (e.g., airbags).
  • For non-predetermined: Increasing contact time or force increases impulse.

Problem-Solving Strategies

Multiple Choice and Calculation Problems

1. Identify Known Variables: Determine given and unknown variables.
2. Use Appropriate Equations: Apply impulse or momentum equations depending on given data.
3. Unit Consistency: Ensure all units are in standard forms (kgs, meters, seconds).
4. Vector Components: Always resolve vectors into x and y-components if dealing with angles.
5. Check Signs: Ensure correct positive or negative signs are used for directions.

Force-Time Graphs

• Key Concept: Area under force-time graph equals impulse.
• Conversions: Convert units appropriately (e.g., kilonewtons to newtons, milliseconds to seconds).
• Total Impulse: Sum positive and negative areas if graph crosses x-axis.

Conservation of Momentum in One and Two Dimensions

• One Dimension: $P_{initial} = P_{final}$ for single-axis motion.
• Two Dimensions: Break into x and y-components and apply $P_{initial} = P_{final}$ for both.

Examples and Sample Problems

Simple Momentum Problem

• Given: Mass and velocity of objects.
• Find: Final momentum after collision or impulse after a force is applied.

Force-Time Graphs Problem

• Given: Force-time graph with specific values.
• Calculate: Area under graph for impulse, then use impulse to find changes in velocity/mass.

Two-Dimensional Collision/Explosion

• Given: Multiple objects with initial and final velocities at angles.
• Calculate: Resolve into x and y-components; apply conservation of momentum in both dimensions.

Elastic vs. Inelastic Collisions

• Inelastic: Objects stick together post-collision; kinetic energy not conserved.
• Elastic: Objects bounce apart; kinetic energy is conserved.

Common Errors and Tips

• Initial and Final Values: Ensure clarity between initial and final values of variables.
• Direction Signs: Always check and correct vector directions.
• Funny Angles: Resolve angles into components immediately to simplify calculations.
• Consistency in Units: Always double-check unit consistency before solving.

Summary

• Momentum and impulse are fundamental concepts linked through the impulse-momentum theorem.
• Conservation of momentum is a key principle in collisions and explosions, both in one and two dimensions.
• Proper handling of vector components and unit consistency is critical in solving problems accurately.