Elastic and Inelastic | AP Physics C: Mech Unit 4 Study Guide

Getting Started

We consider a system of two or more objects that interact intensely over a very short time interval. This interaction, known as a collision, is governed by large internal forces that cause a rapid change in the objects' motion. The core question is: what physical quantities are conserved during this process, and how can we classify the nature of the interaction based on the conservation, or non-conservation, of the system's mechanical energy?

What You Should Be Able to Do

After studying this section, you should be able to:

Apply the integral form of Newton's second law, $\int F_{n e t} d t = Δ p$ , to justify the conservation of momentum for an isolated system during a collision.
Distinguish between elastic and inelastic collisions by evaluating the system's total kinetic energy before and after the interaction.
Model a perfectly inelastic collision as a limiting case where the colliding objects attain a common final velocity, maximizing the loss of kinetic energy.
Relate the change in a system's kinetic energy during an inelastic collision to the net work done by internal non-conservative forces, such as friction and plastic deformation.

Key Concepts & Mechanisms

The primary way to analyze and classify collisions is by comparing different physical models based on what quantities they conserve. For any collision occurring in an isolated system (one where the net external force is zero or negligible compared to the internal collision forces), total momentum is always conserved. The key distinction arises from the behavior of kinetic energy.

Feature	Model A: Elastic Collision	Model B: Inelastic Collision	Why It Matters
Momentum Conservation	Conserved. The total momentum of the system before the collision equals the total momentum after. $\sum p_{i} = \sum p_{f}$ .	Conserved. The total momentum of the system is also conserved. $\sum p_{i} = \sum p_{f}$ .	Momentum conservation is the universal principle for all collisions in isolated systems. It provides a vector equation that is always valid for relating initial and final states, regardless of the collision type.
Kinetic Energy Conservation	Conserved. The total kinetic energy of the system before the collision equals the total kinetic energy after. $\sum K_{i} = \sum K_{f}$ .	Not Conserved. The final total kinetic energy is less than the initial total kinetic energy. $\sum K_{f} < \sum K_{i}$ .	This is the defining difference. The change in kinetic energy, $Δ K = W_{in t er na l}$ , tells us about the nature of the internal forces. If $Δ K = 0$ , the forces were perfectly conservative. If $Δ K < 0$ , non-conservative forces did negative work.
Internal Forces	The net work done by internal forces is zero. The forces are purely conservative (e.g., an ideal spring-like interaction).	The internal forces are non-conservative. They do net negative work, transforming kinetic energy into other forms.	In the real world, most collisions are inelastic. The energy "loss" is a transformation into thermal energy (heating), acoustic energy (sound), and potential energy of permanent deformation.
Final State of Objects	Objects separate after the collision. Their shapes and internal states are unchanged.	Objects may separate or stick together. If they stick, the collision is perfectly inelastic.	The final state provides a clue. If objects fuse, the collision is definitively inelastic. If they separate, you must check kinetic energy to classify it.
Mathematical Constraint	Provides two conserved quantities (momentum and kinetic energy), allowing for the solution of more unknown variables.	Provides only one conserved quantity (momentum). Additional information (e.g., the final velocity of one object, or the coefficient of restitution) is needed to fully solve the system.	In one-dimensional, two-body elastic collisions, the two conservation laws provide a unique solution for the two final velocities. Inelastic collisions are less constrained.

Key Models & Diagrams

To solve any collision problem, follow this logical flowchart:


graph TD

    A[Start: Identify interacting objects as the System] --> B{Is the system isolated during the collision?};

    B -- Yes --> C[Apply Conservation of Momentum: \n Σp_i = Σp_f];

    B -- No --> D[Apply Impulse-Momentum Theorem: \n J_ext = Δp_sys];

    C --> E[Calculate Initial Kinetic Energy: \n K_i = Σ(1/2)mv_i^2];

    E --> F[Solve for final velocities using momentum conservation and any other given constraints.];

    F --> G[Calculate Final Kinetic Energy: \n K_f = Σ(1/2)mv_f^2];

    G --> H{Compare Energies: Is K_f = K_i?};

    H -- Yes --> I[Conclusion: The collision is ELASTIC];

    H -- No --> J{Do the objects have a common final velocity?};

    J -- Yes --> K[Conclusion: The collision is PERFECTLY INELASTIC];

    J -- No --> L[Conclusion: The collision is INELASTIC];


    style A fill:#f9f,stroke:#333,stroke-width:2px

    style I fill:#ccffcc,stroke:#333,stroke-width:2px

    style K fill:#ffcccc,stroke:#333,stroke-width:2px

    style L fill:#ffeecc,stroke:#333,stroke-width:2px

Key Components & Evidence

System: The collection of objects whose mutual interactions are being analyzed. For collisions, the system is chosen such that the strong, short-duration internal forces between objects are far greater than any external forces (like gravity or friction).
Linear Momentum ( $p$ ): A vector quantity defined as the product of an object's mass $m$ and its velocity $v$ , so $p = m v$ . Its SI unit is kg·m/s. The total momentum of a system is the vector sum of the individual momenta.
Conservation of Momentum: A fundamental principle stating that if the net external force on a system is zero, its total linear momentum remains constant. For a collision, this means $p_{sys t e m, ini t ia l} = p_{sys t e m, f ina l}$ .
Kinetic Energy ( $K$ ): A scalar quantity representing the energy of motion, defined as $K = \frac{1}{2} m v^{2}$ . Its SI unit is the Joule (J). Unlike momentum, the total kinetic energy of a system is the simple arithmetic sum of individual kinetic energies.
Elastic Collision: A collision in which the total kinetic energy of the system is conserved ( $Δ K_{sys t e m} = 0$ ). This is an idealization, closely approximated by interactions between billiard balls or subatomic particles.
Inelastic Collision: A collision in which the total kinetic energy of the system decreases ( $Δ K_{sys t e m} < 0$ ). The "lost" kinetic energy is converted into other forms, such as thermal energy or sound. Most macroscopic collisions are inelastic.
Perfectly Inelastic Collision: A specific type of inelastic collision where the colliding objects stick together and move with a single, common final velocity. This scenario corresponds to the maximum possible loss of kinetic energy consistent with momentum conservation.
Impulse ( $J$ ): The integral of force over a time interval, $J = \int_{t_{1}}^{t_{2}} F (t) d t$ . By Newton's second law ( $F = d p / d t$ ), impulse is equal to the change in momentum, $Δ p$ . During a collision, the equal and opposite internal forces deliver impulses of equal magnitude and opposite direction to the colliding objects.

Skill Snapshots

Causation

Driver: The net external force on a system of colliding particles is zero.
Change: The total momentum of the system remains constant throughout the interaction.
Driver: Large, non-conservative internal forces (e.g., friction, deformation) act between objects during the collision.
Change: The total kinetic energy of the system decreases as mechanical energy is converted to thermal and other forms.
Driver: The objects' material properties allow them to deform and then perfectly restore their shape, meaning internal forces are conservative.
Change: The total kinetic energy of the system is the same before and after the collision.

Comparison

Elastic vs. Inelastic Collisions: An elastic collision is an idealized model where kinetic energy is conserved, while an inelastic collision is a more realistic model for macroscopic objects where kinetic energy is transformed into other forms.
Inelastic vs. Perfectly Inelastic Collisions: A perfectly inelastic collision is the specific case of an inelastic collision where the objects fuse, resulting in the maximum possible reduction of system kinetic energy. In a general inelastic collision, objects can still rebound from each other.
Momentum vs. Kinetic Energy: Momentum is a vector quantity that is always conserved in an isolated collision. Kinetic energy is a scalar quantity that is conserved only in the special case of an elastic collision.

Change, Continuity, and Conservation

Baseline: Before the collision, the system is characterized by the initial masses and velocities of its components, defining its initial total momentum and total kinetic energy.
Change: During the brief interaction, the individual momenta and kinetic energies of the objects change dramatically due to the large internal impulsive forces.
Change: If the collision is inelastic, the total kinetic energy of the system decreases as internal forces do negative work.
Continuity: Throughout the entire process (before, during, and after), the total momentum of the isolated system remains unchanged.

Common Misconceptions & Clarifications

Misconception: If objects bounce off each other, the collision must be elastic.
Clarification: Bouncing is not a sufficient condition for an elastic collision. A rubber ball dropped on the floor bounces, but it does not return to its original height. This indicates that kinetic energy was lost during its collision with the floor, making the collision inelastic. Elasticity requires that kinetic energy is perfectly conserved.
Misconception: In an inelastic collision, momentum is lost.
Clarification: This is incorrect. In any collision within an isolated system, momentum is always conserved, regardless of whether the collision is elastic or inelastic. Energy may be lost (transformed), but momentum is not.
Misconception: A perfectly inelastic collision means the final kinetic energy is zero.
Clarification: The final kinetic energy is zero only if the total momentum of the system was zero before the collision (e.g., two objects with equal and opposite momenta collide and stick). In the more general case, the combined mass moves with a final velocity determined by momentum conservation, and thus possesses a non-zero final kinetic energy. A perfectly inelastic collision results in the maximum loss of kinetic energy, not necessarily the loss of all kinetic energy.

One-Paragraph Summary

Collisions are brief, intense interactions between objects where internal forces dominate. For any isolated system, the total vector momentum is strictly conserved, a direct consequence of Newton's third law. The classification of a collision, however, depends on the behavior of the system's total kinetic energy. In an idealized elastic collision, kinetic energy is also conserved, implying the internal forces are conservative. In a more realistic inelastic collision, kinetic energy decreases as internal non-conservative forces do work, converting mechanical energy into thermal energy, sound, or deformation. The limiting case is a perfectly inelastic collision, where objects stick together, move with a common final velocity, and experience the maximum possible loss of kinetic energy consistent with momentum conservation.

Elastic and Inelastic Collisions - AP Physics C: Mechanics Study Guide