Work, Energy, and Momentum: Review Sheet and Practice Problems

Work, Energy, and Momentum

Key Equations and Definitions

This section summarizes essential equations and concepts related to work, energy, and momentum in classical mechanics. These principles are foundational for understanding the behavior of objects in motion and the interactions of forces.

• Newton's Second Law: $F_{net} = ma$

• Friction Force: $F_f = u_k F_n$

• Work: $W = F_{net} d \cos \theta$

• Work-Energy Theorem: $W_{net} = \Delta KE = KE_f - KE_i$

• Kinetic Energy: $KE = \frac{1}{2}mv^2$

• Potential Energy (Gravitational): $PE_g = mgh$

• Potential Energy (Elastic): $PE_{elastic} = \frac{1}{2}kx^2$

• Mechanical Energy: $ME = KE + PE$

• Conservation of Energy: $ME_i = ME_f$

• Power: $P = \frac{W}{\Delta t}$

• Impulse-Momentum Theorem: $\vec{J} = F_{net} \Delta t = \Delta \vec{p}$

• Linear Momentum: $\vec{p} = m\vec{v}$

Conservation of Linear Momentum

In isolated systems, the total linear momentum remains constant unless acted upon by external forces. This principle is crucial for analyzing collisions and explosions.

• General Equation: $m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f}$

• Application: Used to solve problems involving collisions (elastic and inelastic).

Conceptual Applications

Work and Energy Transformations

Work done by forces can be positive, negative, or zero, depending on the direction of force and displacement. Energy can transform between kinetic and potential forms, especially in systems like pendulums and projectiles.

• Example: A pendulum bob swinging from point A to B converts potential energy to kinetic energy and vice versa.

• Work by Gravity: Positive when moving downward, negative when moving upward.

Mechanical Energy in Systems

Mechanical energy (sum of kinetic and potential energy) is conserved in the absence of non-conservative forces (e.g., friction). In roller coaster problems, the total mechanical energy at different points can be compared to determine speed and height relationships.

• Example: A roller coaster at different elevations has varying potential and kinetic energy, but the total mechanical energy remains constant if friction is negligible.

Projectile Motion and Energy Changes

As a projectile moves upward, its kinetic energy decreases while potential energy increases. At the peak, kinetic energy is minimized, and potential energy is maximized.

• Example: A ball launched upward loses kinetic energy and gains potential energy until it reaches its highest point.

Collisions: Elastic and Inelastic

Collisions are classified as elastic (kinetic energy conserved) or inelastic (kinetic energy not conserved, but momentum is). Perfectly inelastic collisions involve objects sticking together after impact.

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

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

• Perfectly Inelastic Collision: Objects stick together after collision.

Problem-Solving Strategies

Work and Energy Calculations

• Identify forces acting on the object.

• Calculate work using $W = Fd \cos \theta$.

• Apply the work-energy theorem to relate work to changes in kinetic energy.

Momentum and Impulse

• Use $\vec{p} = m\vec{v}$ to find momentum.

• Apply impulse-momentum theorem for force and time relationships: $\vec{J} = F_{net} \Delta t$.

• For collisions, use conservation of momentum equations to solve for final velocities.

Energy in Springs

• Elastic potential energy: $PE_{elastic} = \frac{1}{2}kx^2$

• Spring constant $k$ can be found if energy and compression/stretch are known.

Table: Comparison of Collision Types

Type of Collision	Momentum Conserved?	Kinetic Energy Conserved?	Objects Stick Together?
Elastic	Yes	Yes	No
Inelastic	Yes	No	Sometimes
Perfectly Inelastic	Yes	No	Yes

Example Problems

• Work by Multiple Forces: Calculate total work done by two forces pulling a tanker at different angles.

• Energy Change: Find minimum energy needed to change the speed of a car.

• Friction and Initial Speed: Determine initial speed of an object given friction and stopping distance.

• Spring Compression: Calculate spring constant or maximum compression using elastic potential energy.

• Impulse: Find impulse delivered to a soccer ball by a player's kick.

• Collisions: Analyze momentum and energy before and after collision of carts or objects.

Summary

• Work, energy, and momentum are interconnected concepts in physics, essential for analyzing motion and interactions.

• Conservation laws (energy and momentum) provide powerful tools for solving problems involving collisions, projectiles, and mechanical systems.

• Understanding the distinctions between elastic and inelastic collisions is crucial for predicting outcomes in real-world scenarios.

Additional info: Some context and explanations have been expanded for clarity and completeness beyond the original review sheet and questions.