Elastic and Inelastic Collisions: Physics Study Notes

Elastic and Inelastic Collisions

Introduction

Collisions are a fundamental topic in physics, especially in the study of momentum and energy. Collisions can be classified as elastic or inelastic based on whether kinetic energy is conserved during the event. Understanding these types is crucial for analyzing physical interactions in both microscopic and macroscopic systems.

Definitions and Key Concepts

• Elastic Collision: A collision in which the total kinetic energy of the system is conserved. Typically occurs at the atomic or molecular level.

• Inelastic Collision: A collision in which the total kinetic energy is not conserved. Common in everyday, macroscopic events.

• Completely Inelastic Collision: A special case where colliding objects stick together after impact.

• Momentum Conservation: Momentum is always conserved in both elastic and inelastic collisions.

Comparison of Elastic and Inelastic Collisions

Type of Collision	Kinetic Energy Conservation	Typical Occurrence	Momentum Conservation	Objects Stick Together?
Elastic	Yes	Microscopic (atoms, molecules)	Yes	No
Inelastic	No	Macroscopic (everyday objects)	Yes	Sometimes (completely inelastic)

Energy Considerations in Collisions

• Elastic Collisions:

  • Total kinetic energy before and after the collision is equal.

  • Conservation of kinetic energy equation: $E_{k,\text{initial}} = E_{k,\text{final}}$

  • Example equation for two objects: $\frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2 = \frac{1}{2} m_1 v_1'^2 + \frac{1}{2} m_2 v_2'^2$

• Inelastic Collisions:

  • Kinetic energy is not conserved; some is transformed into other forms (heat, sound, deformation).

  • Conservation of kinetic energy does not apply: $E_{k,\text{initial}} \neq E_{k,\text{final}}$

  • Energy loss depends on the nature of the collision (e.g., objects sticking together, breaking, or producing sound/light).

Momentum Conservation in Collisions

Regardless of the type of collision, the law of conservation of momentum always applies:

• General equation: $m_1 v_1 + m_2 v_2 = m_1 v_1' + m_2 v_2'$

• For completely inelastic collisions (objects stick together): $(m_1 + m_2) v' = m_1 v_1 + m_2 v_2$

Determining Collision Type

To determine whether a collision is elastic or inelastic:

1. Calculate the kinetic energy of each object before the collision and sum for total initial kinetic energy.

2. Calculate the kinetic energy of each object after the collision and sum for total final kinetic energy.

3. Compare the totals:

   • If $E_{k,\text{initial}} = E_{k,\text{final}}$, the collision is elastic.

   • If $E_{k,\text{initial}} > E_{k,\text{final}}$, the collision is inelastic.

Example 1: 2D Collision Analysis

Given: Only the red ball is moving before the collision.

• Initial kinetic energy: $E_k = \frac{1}{2} m v^2 = \frac{1}{2} \times 1.20 \times (17.1)^2 = 175 \text{ J}$

• After collision, both balls move: $E_k = \frac{1}{2} m_1 v_1'^2 + \frac{1}{2} m_2 v_2'^2 = \frac{1}{2} \times 1.20 \times (13.5)^2 + \frac{1}{2} \times 2.31 \times (3.66)^2 = 125 \text{ J}$

• Since $175 \text{ J} \neq 125 \text{ J}$, the collision is inelastic.

Example 2: Ballistic Pendulum

The ballistic pendulum is a device used to measure the speed of a bullet by analyzing its collision with a suspended block. The bullet embeds in the block, and together they swing upward.

• Given: Bullet mass $m_b = 0.0200 \text{ kg}$, pendulum mass $m_p = 5.7500 \text{ kg}$, maximum height $h = 0.386 \text{ m}$.

• After collision, use conservation of energy to find velocity:

• Gravitational potential energy: $E_p = m g h$

• Kinetic energy just after collision: $E_k = \frac{1}{2} m v^2$

• Set $E_k = E_p$:

• $\frac{1}{2} v^2 = g h$ $v = \sqrt{2 g h}$ $v = \sqrt{2 \times 9.81 \times 0.386} = 2.75 \text{ m/s}$

• Use conservation of momentum to find bullet velocity before impact:

• $m_b v_b = (m_p + m_b) v'$ $0.0200 v_b = 2.75 \times (5.7500 + 0.0200)$ $0.0200 v_b = 2.75 \times 5.7700 = 15.9$ $v_b = \frac{15.9}{0.0200} = 794 \text{ m/s}$

Example 3: Checking Elasticity of Ballistic Pendulum Collision

• Initial kinetic energy (bullet only): $E_k = \frac{1}{2} m v^2 = \frac{1}{2} \times 0.0200 \times (794)^2 = 6304.36 \text{ J} \approx 6.30 \times 10^3 \text{ J}$

• Final kinetic energy (bullet + block together): $E_k = \frac{1}{2} m v^2 = \frac{1}{2} \times 5.7700 \times (2.75)^2 = 21.8 \text{ J}$

• Only about 0.346% of the initial kinetic energy remains after the collision, confirming it is inelastic.

Summary Table: Elastic vs. Inelastic Collisions

Collision Type	Kinetic Energy Conserved?	Momentum Conserved?	Objects Stick Together?	Typical Example
Elastic	Yes	Yes	No	Atomic/molecular collisions
Inelastic	No	Yes	Sometimes	Car crash, ballistic pendulum

Key Takeaways

• Elastic collisions conserve both momentum and kinetic energy; inelastic collisions conserve only momentum.

• Most macroscopic collisions are inelastic, with some energy lost to heat, sound, or deformation.

• To determine collision type, compare total kinetic energy before and after the event.

• Ballistic pendulum is a classic example of a completely inelastic collision.

Additional info: The notes clarify that elastic collisions are rare in everyday life and typically occur at the microscopic level, while inelastic collisions are common in macroscopic systems. The ballistic pendulum example demonstrates practical application of momentum and energy conservation principles.