Exam 4 Review: Momentum, Energy, Work, and Collisions

Momentum

Impulse

Impulse is a measure of the effect of a force acting over a period of time. It is closely related to the change in momentum of an object.

• Impulse (\( J \)): Defined as the product of force and the time interval over which it acts.

• Formula: $J = F \Delta t$ (Eqn. 9.1)

• Area under Force-Time Graph: The impulse delivered to an object is equal to the area under the force vs. time graph.

• Average Force: $F_{avg} = \frac{J}{\Delta t}$ (Eqn. 9.2)

• Example: A baseball bat striking a ball delivers an impulse, changing the ball's momentum.

Momentum and the Impulse-Momentum Theorem

Momentum is a fundamental quantity in physics, representing the product of mass and velocity. The impulse-momentum theorem relates impulse to the change in momentum.

• Momentum (\( p \)): $p = m v$ (Eqn. 9.6)

• Change in Momentum: $\Delta p = m \Delta v$ (Eqn. 9.3)

• Average Acceleration: $a_{avg} = \frac{\Delta v}{\Delta t}$ (Eqn. 9.4)

• Two-Dimensional Momentum: $\vec{p} = m \vec{v}$ (Eqn. 9.7)

• Impulse-Momentum Theorem: $J = \Delta p$ (Eqn. 9.8)

• Effect of Impulse: Impulse causes a change in momentum (Eqns. 9.9, 9.10).

• Example: A car crash involves a large impulse, resulting in a significant change in momentum.

Solving Impulse and Momentum Problems

Problem-solving tactics involve identifying the system, calculating impulse, and relating it to momentum change.

• Impulse Approximation: Approximating the area under a force-time graph to estimate impulse.

• Example: Calculating the impulse delivered by a force that acts for a short duration.

Conservation of Momentum

The law of conservation of momentum states that the total momentum of an isolated system remains constant if no external forces act.

• Action/Reaction Pairs: Newton's Third Law ensures momentum is conserved in interactions.

• Law of Conservation of Momentum: $p_{total, initial} = p_{total, final}$ (Eqns. 9.14, 9.15)

• Isolated Systems: Systems with no external forces.

• Explosions: Momentum is conserved even when objects move apart after an explosion.

• Example: Two ice skaters push off each other and move in opposite directions.

Inelastic Collisions

In inelastic collisions, objects stick together or deform, and kinetic energy is not conserved, though momentum is.

• Perfectly Inelastic Collisions: Objects stick together after collision (Fig. 9.21).

• Example: Two cars collide and move as one mass after impact.

Momentum and Collisions in Two Dimensions

Collisions can occur in two dimensions, requiring vector analysis of momentum.

• Example: Billiard balls colliding at angles; momentum is conserved in both x and y directions.

Angular Momentum

Angular momentum is the rotational analog of linear momentum, important in systems involving rotation.

• Angular Acceleration vs. Torque: $\tau = I \alpha$ (Eqn. 9.16)

• Angular Acceleration vs. Angular Speed: $\alpha = \frac{\Delta \omega}{\Delta t}$ (Eqn. 9.17)

• Definition of Angular Momentum (\( L \)): $L = I \omega$ (Eqn. 9.20)

• Conservation of Angular Momentum: $L_{total, initial} = L_{total, final}$ (Eqn. 9.22)

• Varying Angular Momentum: Changes occur due to external torques.

• Example: A figure skater spins faster by pulling in her arms.

Energy and Work

Basic Energy Model

Energy is a property of systems that enables them to do work. It exists in various forms and can be transferred or transformed.

• Forms of Energy: Kinetic, potential, thermal, etc.

• Energy Transformations: Energy can change from one form to another (e.g., potential to kinetic).

• Work-Energy Equation: $\Delta E = W$ (Eqn. 10.3)

• Law of Conservation of Energy: $E_{total, initial} = E_{total, final}$ (Eqn. 10.4)

• Isolated Systems: No energy enters or leaves the system.

Work

Work is the process of energy transfer by a force acting over a distance.

• Work by Force (Same Direction): $W = F d$ (Eqn. 10.5)

• Work by Force at an Angle: $W = F d \cos \theta$ (Eqn. 10.6)

• Forces That Do No Work: Forces perpendicular to displacement (e.g., centripetal force).

• Example: Lifting a box vertically involves work equal to the weight times the height.

Kinetic Energy

Kinetic energy is the energy of motion, both translational and rotational.

• Translational Kinetic Energy: $KE_{trans} = \frac{1}{2} m v^2$ (Eqn. 10.8)

• Rotational Kinetic Energy: $KE_{rot} = \frac{1}{2} I \omega^2$ (Eqn. 10.9)

• Work-Energy Equation: $W = \Delta KE$ (Eqn. 10.7)

• Example: A rolling wheel has both translational and rotational kinetic energy.

Potential Energy

Potential energy is stored energy due to position or configuration.

• Gravitational Potential Energy: $PE_{grav} = m g h$ (Eqn. 10.14)

• Elastic (Spring) Potential Energy: $PE_{spring} = \frac{1}{2} k x^2$ (Eqn. 10.16)

• Example: A stretched spring stores elastic potential energy.

Thermal Energy

Thermal energy arises from friction and is a form of energy dissipation.

• Thermal Energy from Friction: $E_{thermal} = f_{friction} d$ (Eqn. 10.17)

• Relation to Work-Energy: $W_{total} = \Delta KE + \Delta PE + \Delta E_{thermal}$ (Eqn. 10.19)

• Example: Sliding a block across a rough surface converts mechanical energy to thermal energy.

Conservation of Energy

Energy is conserved in isolated systems, allowing analysis of before-and-after scenarios.

• Before-After Work-Energy: $E_{initial} + W = E_{final}$ (Eqn. 10.20)

• Choosing Isolated Systems: Select boundaries so no energy enters or leaves.

• Example: A pendulum swings, converting potential energy to kinetic and back.

Energy Diagrams

Potential energy curves help visualize energy changes and equilibrium positions.

• Potential Energy Curves: Graphs showing how potential energy varies with position.

• Determining KE and Motion: Total energy minus potential energy gives kinetic energy.

• Equilibrium Positions: Stable equilibrium occurs at minima; unstable at maxima.

• Example: A ball at the bottom of a valley is in stable equilibrium.

Energy in Collisions

Collisions can be elastic or inelastic, affecting conservation of kinetic energy.

• Elastic Collisions: Both momentum and kinetic energy are conserved.

• Inelastic Collisions: Only momentum is conserved; kinetic energy is not.

• Perfectly Elastic Collisions: No energy is lost to deformation or heat.

• Example: Two billiard balls collide and bounce apart without losing kinetic energy.

Power

Power is the rate at which energy is transferred or transformed.

• Rate of Energy Transformation: $P = \frac{\Delta E}{\Delta t}$ (Eqn. 10.23)

• Power in Terms of Work: $P = \frac{W}{t}$ (Eqn. 10.24)

• Power in Terms of Force and Velocity: $P = F v$ (Eqn. 10.25)

• Example: An engine does work at a certain rate, producing power.

Comparison Table: Elastic vs. Inelastic Collisions

Type of Collision	Momentum Conserved?	Kinetic Energy Conserved?	Example
Elastic	Yes	Yes	Billiard balls colliding
Inelastic	Yes	No	Cars sticking together after collision
Perfectly Inelastic	Yes	No	Two clay balls merging

Summary of Key Equations

• $J = F \Delta t$

• $p = m v$

• $J = \Delta p$

• $\tau = I \alpha$

• $L = I \omega$

• $W = F d$

• $W = F d \cos \theta$

• $KE_{trans} = \frac{1}{2} m v^2$

• $KE_{rot} = \frac{1}{2} I \omega^2$

• $PE_{grav} = m g h$

• $PE_{spring} = \frac{1}{2} k x^2$

• $E_{thermal} = f_{friction} d$

• $P = \frac{\Delta E}{\Delta t}$

• $P = F v$

Additional info: Academic context and explanations have been expanded for clarity and completeness, including definitions, examples, and formulas for each topic.