Elastic and Inelastic | AP Physics 1 Unit 4 Study Guide

Getting Started

When objects collide, from billiard balls striking each other to cars crashing, they exert forces and exchange momentum and energy. We will analyze these brief but powerful interactions by defining a system that includes all colliding objects. The core question is: while the total momentum of an isolated system is always conserved during a collision, what happens to its total energy of motion?

What You Should Be able to Do

After working through this section, you should be able to:

Define elastic, inelastic, and perfectly inelastic collisions in terms of a system's kinetic energy.
Determine whether a described collision is elastic or inelastic by comparing the system's kinetic energy before and after the event.
Explain why total momentum is conserved in all collisions within an isolated system, while kinetic energy may not be.
Identify a perfectly inelastic collision from the fact that the objects stick together and move with a common final velocity.

Key Concepts & Mechanisms

Our analysis of collisions is centered on the principles of Interactions & Conservation. We treat a collision as an interaction between objects within a defined system and use conservation laws to understand the outcome.

System & Preconditions

To analyze a collision, we first define the system as the collection of all objects involved in the interaction (e.g., two colliding carts). We make a critical idealization: the system is isolated. This means there are no significant external net forces acting on the system during the brief moment of collision (e.g., we ignore friction). This precondition is what allows us to apply the law of conservation of momentum.

Key Steps / Relations

Conserve Momentum (Always): The foundational principle for all collisions in an isolated system is the conservation of momentum. Momentum ( $p$ ) is the product of an object's mass ( $m$ , in kilograms, kg) and velocity ( $v$ , in meters per second, m/s), so $p = m v$ . It is a vector quantity measured in kg·m/s. For any system of objects, the total momentum before the collision is equal to the total momentum after the collision.
$Σ p_{initial} = Σ p_{final}$
For a two-object collision, this is written as:
$m_{1} v_{1 i} + m_{2} v_{2 i} = m_{1} v_{1 f} + m_{2} v_{2 f}$
Account for Kinetic Energy: Next, we examine the system's kinetic energy ( $K$ ), which is the energy of motion. Kinetic energy is a scalar quantity calculated as $K = \frac{1}{2} m v^{2}$ and is measured in Joules (J). Unlike momentum, the total kinetic energy of the system is not always conserved during a collision.
The Energy Test: To classify a collision, we compare the total kinetic energy of the system before the interaction ( $Σ K_{initial}$ ) to the total kinetic energy after the interaction ( $Σ K_{final}$ ).
$Σ K_{initial} = \frac{1}{2} m_{1} v_{1 i}^{2} + \frac{1}{2} m_{2} v_{2 i}^{2}$
$Σ K_{final} = \frac{1}{2} m_{1} v_{1 f}^{2} + \frac{1}{2} m_{2} v_{2 f}^{2}$
Classify the Collision: Based on the energy test, we can categorize the interaction.
- Elastic Collision: If $Σ K_{initial} = Σ K_{final}$ , the collision is elastic. Kinetic energy is conserved. The objects bounce off each other without any loss of motional energy.
- Inelastic Collision: If $Σ K_{initial} > Σ K_{final}$ , the collision is inelastic. Kinetic energy is not conserved; some of it is transformed into other forms, such as thermal energy (making the objects warmer) and sound energy.
- Perfectly Inelastic Collision: This is a special case of an inelastic collision where the objects stick together after impact and move with a single, common final velocity ( $v_{f}$ ). This type of collision results in the maximum possible loss of kinetic energy for the system.

Outputs & Effects

What Changes: In any collision, the velocities of the individual objects almost always change. In an inelastic collision, the total kinetic energy of the system decreases.
What Remains Constant: For any collision in an isolated system, the total mass and the total momentum are always conserved. In the special case of an elastic collision, the total kinetic energy is also conserved.

Regulation & Limits

The principles of momentum conservation are strictly valid only for isolated systems. In the real world, external forces like friction are always present. However, during the very short duration of a collision, the internal forces the objects exert on each other are often vastly larger than the external forces. For this reason, we can approximate the system as isolated for the moment of impact and confidently apply momentum conservation. The conservation of kinetic energy in elastic collisions is an idealization; most macroscopic collisions are inelastic to some degree.

Key Models & Diagrams

The classification of a collision depends entirely on how the system's kinetic energy behaves, while momentum conservation serves as the backdrop for all interactions.

Collision Type	Key Conservation Rules	Predicted Observables & Outcome
Elastic	1. Momentum is conserved: $Σ p_{i} = Σ p_{f}$ 2. Kinetic energy is conserved: $Σ K_{i} = Σ K_{f}$	Objects bounce off each other perfectly. There is no energy lost to deformation, sound, or heat. This is an idealization, closely approximated by colliding billiard balls or subatomic particles.
Inelastic	1. Momentum is conserved: $Σ p_{i} = Σ p_{f}$ 2. Kinetic energy is NOT conserved: $Σ K_{i} > Σ K_{f}$	Objects may bounce or separate, but the collision produces sound, heat, or permanent deformation. A dropped basketball that doesn't return to its original height is a classic example.
Perfectly Inelastic	1. Momentum is conserved: $Σ p_{i} = Σ p_{f}$ 2. Max KE is lost: $Σ K_{i} > Σ K_{f}$ 3. Objects stick together: $v_{1 f} = v_{2 f} = v_{f}$	Objects collide and move as a single mass afterward. Examples include two lumps of clay colliding, or a railroad car coupling with another.

Key Components & Evidence

System: The defined set of interacting objects. Its boundaries separate it from the external environment.
Momentum ( $p$ ): A vector quantity ( $m v$ ) representing an object's "quantity of motion," measured in kg·m/s. The total momentum of an isolated system is always conserved.
Kinetic Energy ( $K$ ): A scalar quantity ( $\frac{1}{2} m v^{2}$ ) representing an object's energy of motion, measured in Joules (J). It is the key quantity used to classify collisions.
Conservation of Momentum: A fundamental law stating that the total momentum of an isolated system remains constant before, during, and after an interaction.
Elastic Collision: An idealized interaction where the system's total kinetic energy is conserved.
Inelastic Collision: A common, real-world interaction where the system's total kinetic energy decreases, being converted into other energy forms.
Perfectly Inelastic Collision: A specific type of inelastic collision where the colliding objects stick together, resulting in the largest possible decrease in kinetic energy.
Internal Forces: The forces that objects within the system exert on each other during the collision. These forces are responsible for changing the individual momenta of the objects but not the total momentum of the system.
Transformation of Energy: The process in an inelastic collision where kinetic energy is converted into non-mechanical forms like thermal energy (heat) and sound.

Skill Snapshots

Causation

The strong internal forces during a collision cause the velocity of each object to change, but they cannot change the total momentum of the isolated system.
The permanent deformation of materials or the generation of sound waves during a collision causes a decrease in the system's total kinetic energy, defining the collision as inelastic.
In a perfectly inelastic collision, the coupling mechanism that makes the objects stick together causes them to share a single final velocity.

Comparison

An elastic collision is an ideal model where kinetic energy is conserved, whereas a real-world inelastic collision always involves some transformation of kinetic energy into heat or sound.
In both an elastic and a perfectly inelastic collision, the total momentum of an isolated system is conserved, but only in the elastic case is kinetic energy also conserved.
A perfectly inelastic collision results in the maximum possible loss of kinetic energy, while a general inelastic collision (where objects bounce apart) involves a smaller, non-zero loss of kinetic energy.

Change Over Time

Baseline: Before the collision, the system is defined by the initial masses, velocities, and a total initial kinetic energy ( $Σ K_{i}$ ).
Change 1: During the brief interaction, the objects' velocities change rapidly, and in an inelastic collision, some kinetic energy is irreversibly transformed into thermal and sound energy.
Change 2: After the collision, the system has a new set of final velocities and a final total kinetic energy ( $Σ K_{f}$ ) that is either equal to (elastic) or less than (inelastic) the baseline value.
Continuity: Throughout the entire process—before, during, and after the collision—the total momentum of the isolated system remains constant.

Common Misconceptions & Clarifications

Misconception: "Inelastic" means the objects must stick together.
- Clarification: Sticking together defines a perfectly inelastic collision, which is a specific subtype. A collision is inelastic any time kinetic energy is lost. A ball that deforms and makes a sound when it bounces off a wall is an example of an inelastic collision where the objects separate.
Misconception: Momentum is lost during an inelastic collision.
- Clarification: This is incorrect. Momentum is conserved in all collisions within an isolated system, whether elastic or inelastic. It is kinetic energy that may be lost (i.e., transformed into other forms). Do not confuse the conservation of momentum with the conservation of kinetic energy.
Misconception: Energy is not conserved in an inelastic collision.
- Clarification: Be precise with language. Kinetic energy is not conserved in an inelastic collision. However, the total energy of the system is always conserved. The "lost" kinetic energy is not destroyed; it is transformed into an equivalent amount of other energy forms, such as heat and sound, which are harder to measure.

One-Paragraph Summary

Collisions are interactions classified by their effect on a system's kinetic energy. The single most important principle is that for any isolated system, total momentum is always conserved, regardless of the type of collision. The distinction arises when we analyze kinetic energy: in a perfectly elastic collision, kinetic energy is also conserved. In an inelastic collision, which describes most real-world events, some kinetic energy is transformed into other forms like heat and sound, so the system's total kinetic energy decreases. The extreme case is a perfectly inelastic collision, where objects stick together and move as one, resulting in the maximum possible loss of kinetic energy. Therefore, by comparing the total kinetic energy before and after an interaction, we can definitively classify the nature of the collision.

Elastic and Inelastic Collisions - AP Physics 1: Algebra-Based Study Guide