Conservation of Energy | AP Physics 1 Unit 3 Study Guide

Getting Started

Imagine a roller coaster car being pulled to the top of its first, highest hill. From that point on, it travels the entire track—through loops, dips, and turns—without another engine. The core question we will explore is: How can we predict the coaster's speed at any point on the track just by knowing its height? This chapter introduces the principle of conservation of energy, a powerful tool for analyzing motion by tracking how energy is stored and transformed within a physical system.

What You Should Be Able to Do

After completing this chapter, you will be able to:

Define a physical system and identify the kinetic and potential energies present within it.
Use the principle of conservation of mechanical energy to relate an object's speed, height, and spring compression at different points in its motion.
Explain how the choice of objects included in a system determines whether the system's energy is constant or changing.
Differentiate between situations where mechanical energy is conserved and situations where it is transformed into other forms like thermal energy.

Key Concepts & Mechanisms

This section explores the conservation of energy through the lens of Interactions & Conservation. We focus on how interactions, both within a system and with its surroundings, cause energy to be transferred or transformed, and under what conditions the total mechanical energy remains constant.

System & Preconditions

The first and most critical step in any energy problem is to define the system: a collection of objects we choose to analyze. Everything outside the system is called the surroundings. The boundary between the system and surroundings is what determines how we account for energy changes.

System Choice: The choice of system is up to you, but it dramatically affects the analysis.
- A system of a single object (e.g., a moving car) can only possess Kinetic Energy (KE), the energy of motion.
- A system of multiple interacting objects (e.g., a ball and the Earth) can possess both kinetic energy and Potential Energy (PE), which is stored energy due to the configuration or position of the objects. Potential energy is a property of the system, not of a single object.
Idealizations: For many problems, we make simplifying assumptions. The most common is to analyze an isolated system, where no energy is transferred to or from the surroundings. This often means we assume that non-conservative forces like friction and air resistance are negligible and do no work on the system.

Key Steps / Relations

Energy accounting allows us to relate the state of a system at two different points in time, an initial state (i) and a final state (f).

Identify Energy Types: First, identify the types of energy present in your chosen system. The total Mechanical Energy (ME) is the sum of the system's kinetic and potential energies.
- Kinetic Energy (KE): The energy an object has due to its motion. It is calculated as KE = ½mv², where m is mass (in kg) and v is speed (in m/s). The unit of energy is the Joule (J).
- Gravitational Potential Energy (PEg): The energy stored in an object-planet system due to the object's vertical position. It is calculated as PEg = mgh, where g is the acceleration due to gravity (~9.8 m/s²) and h is the height (in m) relative to a chosen zero-level.
- Elastic Potential Energy (PEs): The energy stored in a spring or other elastic object when it is stretched or compressed. It is calculated as PEs = ½kx², where k is the spring constant (in N/m) and x is the displacement from its equilibrium position (in m).
Define Mechanical Energy: The total mechanical energy of a system is the sum of these forms.
ME = KE + PEg + PEs
ME = ½mv² + mgh + ½kx²
Apply the General Conservation of Energy Principle: The fundamental law is that energy is never created or destroyed. Any change in a system's energy must be due to work done on the system by external forces or transformations within the system. This is expressed by the general Work-Energy equation:
E_i + W = E_f
Here, E_i and E_f are the initial and final total energies of the system, and W is the work (in J) done on the system by forces from the surroundings.
Apply Conservation of Mechanical Energy: In the idealized case where there are no non-conservative forces (like friction) doing work on the system (W_nc = 0), the total mechanical energy does not change. It is conserved.
ME_i = ME_f
KE_i + PE_i = KE_f + PE_f
This powerful equation allows you to solve for an unknown quantity (like final speed or maximum height) without analyzing the forces or acceleration involved in the process.

Outputs & Effects

Energy Transformation: When mechanical energy is conserved, it transforms between its different forms. For a falling object (in the object-Earth system), potential energy is converted into kinetic energy, causing its speed to increase. For a pendulum, energy continuously transforms from potential (at the highest points) to kinetic (at the lowest point) and back again.
Predictive Power: The primary output of this analysis is the ability to predict a final state of motion (speed, position) from an initial state, bypassing the complexities of kinematics for situations with changing acceleration.

Regulation & Limits

Domain of Validity: The simple equation ME_i = ME_f is only valid when the net work done by non-conservative forces is zero. Conservative forces, like gravity and the elastic force from an ideal spring, have potential energies associated with them and do not change the total mechanical energy. Non-conservative forces, like friction and air resistance, transform mechanical energy into thermal energy, so ME is not conserved if they are present and doing work.
The Universal Law: The law of conservation of total energy (E_i + W = E_f) is always true. The conservation of mechanical energy is a special case applicable only to idealized, frictionless systems. If friction is present, ME_f will be less than ME_i, with the difference being equal to the energy transformed into heat.

Key Models & Diagrams

The following flowchart outlines the problem-solving process for applying energy conservation.

Energy Conservation Problem-Solving Flowchart


graph TD

    A[1. Define the System] --> B{Are non-conservative forces like friction or air resistance doing work on the system?};

    B -- No --> C[Mechanical energy is conserved.<br>Use ME_i = ME_f];

    B -- Yes --> D[Mechanical energy is NOT conserved.<br>Use E_i + W_nc = E_f];

    C --> E[KE_i + PE_i = KE_f + PE_f<br>½mv_i² + mgh_i = ½mv_f² + mgh_f];

    D --> F[KE_i + PE_i + W_nc = KE_f + PE_f<br>The work W_nc is often negative, representing energy removed from the system.];

    E --> G[5. Substitute known values and solve for the unknown.];

    F --> G;


    subgraph "Step 2: Identify Forces"

    B

    end


    subgraph "Step 3: Choose Equation"

    C

    D

    end


    subgraph "Step 4: Expand Equation"

    E

    F

    end


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

    style G fill:#ccf,stroke:#333,stroke-width:2px

Key Components & Evidence

System: The object or group of objects being studied. The choice of system determines which energies are internal.
Kinetic Energy (KE):½mv². The energy of motion. Evidence: An object with speed can do work on another object upon collision. Units: Joules (J).
Gravitational Potential Energy (PEg):mgh. Energy stored in the object-Earth system due to vertical separation. Evidence: An object held at a height will gain speed (and kinetic energy) if released. Units: Joules (J).
Mechanical Energy (ME):KE + PE. The sum of the macroscopic energies of motion and position. In an isolated system with only conservative forces, ME is constant.
Work (W):Fd cos(θ). The mechanical transfer of energy into or out of a system by a force. Evidence: Pushing a box across the floor increases its kinetic energy. Units: Joules (J).
Conservative Force: A force, like gravity, for which the work done is independent of the path taken. These forces store and release potential energy, conserving the system's mechanical energy.
Non-conservative Force: A force, like friction, for which the work done depends on the path taken. These forces dissipate mechanical energy, typically as thermal energy.
Conservation of Mechanical Energy: The principle that ME_i = ME_f in a system where no work is done by non-conservative forces. Evidence: A pendulum swings to nearly the same height on each side.

Skill Snapshots

Causation

An external push or pull (positive work) causes an increase in the system's total mechanical energy.
The force of friction doing negative work causes a decrease in a system's mechanical energy, transforming it into thermal energy.
The force of gravity doing work causes the transformation of potential energy into kinetic energy (or vice-versa) within the object-Earth system.

Comparison

A conservative force like gravity allows for the reversible transformation between kinetic and potential energy, while a non-conservative force like friction causes an irreversible loss of mechanical energy.
An isolated system (e.g., a frictionless roller coaster and Earth) has a constant total mechanical energy, whereas a non-isolated system (e.g., a block being pushed across a floor) has a changing mechanical energy.
Choosing a system of only the block means gravity is an external force doing work, while choosing a system of the block and Earth means gravity is an internal force associated with a potential energy.

Change Over Time

Baseline: A skier at rest at the top of a frictionless hill. The system (skier + Earth) has maximum gravitational potential energy and zero kinetic energy.
Change 1: As the skier glides down the hill, the height h decreases, causing the system's potential energy to decrease.
Change 2: Simultaneously, the skier's speed v increases, causing the system's kinetic energy to increase by an equal amount.
Continuity: The total mechanical energy of the skier-Earth system (PEg + KE) remains constant throughout the entire descent.

Common Misconceptions & Clarifications

Misconception: Energy is "lost" in situations with friction.
Clarification: Energy is never lost or destroyed; it is transformed into other forms. Friction converts mechanical energy into thermal energy (heat), which increases the internal energy of the objects and their surroundings. Total energy is always conserved.
Misconception: An object possesses potential energy.
Clarification: Potential energy is a property of a system of interacting objects, not a single object. Gravitational potential energy belongs to the object-Earth system, and elastic potential energy belongs to the spring-object system.
Misconception: Conservation of energy means an object's speed is constant.
Clarification: Conservation of mechanical energy means the sum of kinetic and potential energy is constant. The speed changes as potential energy is converted to kinetic energy and vice versa.
Misconception: The h in mgh is always the distance from the ground.
Clarification: The height h is measured from a "zero level" that you can define arbitrarily for convenience. The important quantity is the change in height (Δh), which is independent of where you place the zero level.

One-Paragraph Summary

The principle of conservation of energy provides a powerful alternative to Newtonian dynamics for analyzing motion, allowing us to relate the properties of a system at two different points in time without considering the details of the path between them. By defining a system and accounting for its kinetic and potential energies, we can predict outcomes based on the idea that energy is transformed, not created or destroyed. The special case of conservation of mechanical energy (KE_i + PE_i = KE_f + PE_f) applies to idealized systems where non-conservative forces like friction do no work. When such forces are present, they cause a decrease in mechanical energy, which is transformed into other forms, but the total energy of the universe remains constant.

Conservation of Energy - AP Physics 1: Algebra-Based Study Guide