Collisions & Momentum: Conservation Principles and Applications

Collisions & Momentum

Introduction

This study guide covers the fundamental principles of collisions and momentum, focusing on the conservation laws, types of collisions, and their analysis in one and two dimensions. These concepts are essential in classical mechanics and are widely applicable in physics and engineering.

Types of Collisions

Classification of Collisions

Collisions are categorized based on how kinetic energy is conserved and how objects interact post-collision.

• Elastic Collision: Objects bounce apart, and there is no loss of kinetic energy. $\Delta KE = 0$

• Inelastic Collision: Objects bounce apart, but some kinetic energy is lost. $\Delta KE \neq 0$

• Completely Inelastic Collision: Objects stick together or explode, and kinetic energy is not conserved. $\Delta KE \neq 0$

Examples and Applications

• Elastic: Billiard balls bouncing off each other, a ball bouncing off concrete and returning to its original height.

• Inelastic: Balls collide and separate, losing a portion of their combined kinetic energy; balls deform and move off in separate directions.

• Completely Inelastic: Objects collide, attach, and move together; a child jumps onto a sled and both slide together; a ball sticks to another ball and they move together.

Conservation Laws in Collisions

Conservation of Momentum

Momentum is conserved in all collisions, provided no external forces act on the system.

• Momentum Conservation Equation: $\vec{p}_{\text{initial}} = \vec{p}_{\text{final}}$ $m_1 v_1 + m_2 v_2 = m_1 v_1' + m_2 v_2'$

• Impulse-Momentum Theorem: $\vec{F}_{\text{avg}} \Delta t = \Delta \vec{p}$

Conservation of Energy

Kinetic energy is only conserved in elastic collisions. In inelastic collisions, some energy is transformed into other forms (e.g., heat, deformation).

• Work-Energy Principle: $W_{NC} = \Delta KE + \Delta PE$

Solving Collision Problems

General Approach

1. Identify the system and whether it is open or closed (i.e., whether external forces act).

2. Apply conservation of momentum and, if applicable, conservation of kinetic energy.

3. Set up equations for the initial and final states.

4. Solve for unknowns (e.g., final velocities).

Elastic Collision Equations (1D)

• For two objects with masses $m_1$ and $m_2$, and initial velocities $v_1$ and $v_2$:

• Final velocity of object 1: $v_1' = \frac{m_1 - m_2}{m_1 + m_2} v_1 + \frac{2m_2}{m_1 + m_2} v_2$

• Final velocity of object 2: $v_2' = \frac{2m_1}{m_1 + m_2} v_1 + \frac{m_2 - m_1}{m_1 + m_2} v_2$

Special Cases

• If $m_1 \gg m_2$, $v_1' \approx v_1$, $v_2' \approx 2v_1$

• If $m_1 \approx m_2$, $v_1' \approx v_2$, $v_2' \approx v_1$

Worked Examples

Example 1: Ball and Car Elastic Collision

A child tosses a ball at $18\,\text{m/s}$ towards a car moving at $14\,\text{m/s}$. Find the ball's speed after it rebounds elastically.

• Use: $v_1' = \frac{m_1 - m_2}{m_1 + m_2} v_1 + \frac{2m_2}{m_1 + m_2} v_2$

Example 2: Child and Tire Swing (Conservation of Energy)

A $30\,\text{kg}$ child runs at $3\,\text{m/s}$ and grabs a $10\,\text{kg}$ tire swing. How high does the system reach if the swing hangs from a $2\,\text{m}$ chain?

• Conservation of momentum for initial velocity: $v' = \frac{m_1 v_1}{m_1 + m_2}$

• Conservation of energy for height: $\frac{1}{2} m v^2 = mgh$ $h = \frac{v^2}{2g}$

Example 3: Bowling Ball and Pin (2D Collision)

A $5.5\,\text{kg}$ bowling ball moving at $9.0\,\text{m/s}$ collides with a $0.825\,\text{kg}$ pin, which is scattered at $85^\circ$ with a speed of $15\,\text{m/s}$. Find the final velocity of the bowling ball.

• Apply conservation of momentum in both $x$ and $y$ directions:

• $m_1 v_{1x} + m_2 v_{2x} = m_1 v_{1x}' + m_2 v_{2x}'$

• $m_1 v_{1y} + m_2 v_{2y} = m_1 v_{1y}' + m_2 v_{2y}'$

Summary Table: Collision Types

Type	Objects After Collision	Kinetic Energy Change
Elastic	Bounce apart	$\Delta KE = 0$
Inelastic	Bounce apart	$\Delta KE \neq 0$
Completely Inelastic	Stick together	$\Delta KE \neq 0$

Problem-Solving Flowcharts

Energy Conservation Flowchart

• Identify system and whether it is open or closed.

• Include energy gains/losses as $W_{NC}$ or $W_{ext}$ if open.

• Compare total energy at different snapshots.

Momentum Conservation Flowchart

• Identify system and whether it is open or closed.

• Momentum is conserved if no external forces act.

• Classify interaction: inelastic, elastic, or explosion.

Key Terms

• Momentum ($p$): Product of mass and velocity, $p = mv$.

• Kinetic Energy (KE): Energy due to motion, $KE = \frac{1}{2}mv^2$.

• Impulse: Change in momentum due to a force over time.

• System: The set of objects considered for analysis.

Additional info:

• Some context and equations were inferred from standard physics curriculum and the provided handwritten notes.