3.4 Conservation of Energy

AP Physics 1 3.4 Conservation of Energy Summary

Conservation of energy in AP Physics 1 means the total energy of a system stays constant unless energy is transferred across the system boundary by work. In Topic 3.4, you track mechanical energy (kinetic plus potential) and decide whether it stays constant based on how you define your system and whether nonconservative forces like friction act inside it.

Why This Matters for the AP Physics 1 Exam

Energy conservation is one of the most reusable tools in the course. Instead of tracking forces and acceleration moment by moment, you can compare a starting state and an ending state and connect them through energy. That makes it powerful for both multiple-choice reasoning and free-response problems where you set up and solve equations.

This topic also trains the exam skills you need across the course: defining a system, building energy bar charts and equations, and justifying claims with evidence. Energy ideas show up again in momentum, rotation, oscillations, and fluids, so getting comfortable here pays off later. The free-response section rewards clear, organized reasoning that cites physical principles, and energy problems are a natural place to practice that kind of writing.

Key Takeaways

• A system of just one object can only have kinetic energy. Potential energy requires a system of interacting objects, like object-Earth or block-spring.
• Mechanical energy is kinetic plus potential: $E_{mech} = KE + PE$.
• If the work done on a chosen system is zero and no nonconservative forces act inside it, total mechanical energy stays constant.
• If nonzero work is done on the system, energy transfers across the boundary and the system's total energy changes.
• Your choice of system boundary decides whether a force is an internal interaction or an outside energy transfer.
• Nonconservative forces like friction and air resistance can dissipate mechanical energy as thermal energy or sound.

System Energy Types

Potential energy is not assigned to a single object by itself. It belongs to a system of interacting objects, such as an object-Earth system or a block-spring system. A system made of only one object can have kinetic energy, but not gravitational or elastic potential energy.

Single Object Kinetic Energy

Kinetic energy is the energy of motion and is the only form of energy a single-object system can have.

Kinetic energy depends on mass and velocity, with velocity having the bigger effect because it is squared:

• The formula is $KE = \frac{1}{2}mv^2$
• Kinetic energy is always positive (or zero when the object is at rest)
• Doubling velocity multiplies kinetic energy by four
• Doubling mass only doubles kinetic energy

Systems with only kinetic energy include:

• An asteroid moving through empty space
• A hockey puck gliding across nearly frictionless ice after it leaves the stick
• A cart moving at constant height on a horizontal track when the system is just the cart

Interacting Objects Energy Types

When objects interact through conservative forces or change shape reversibly, the system can store energy as both kinetic and potential.

Conservative forces let energy convert between kinetic and potential forms without loss. They have two key features:

• The work done depends only on the initial and final positions, not the path
• The work done around any closed path is zero

The main types of potential energy in AP Physics 1:

• Gravitational potential energy near Earth's surface: $\Delta U_g = mg\Delta y$
• Elastic potential energy in an ideal spring: $U_s = \frac{1}{2}k(\Delta x)^2$

Systems with both kinetic and potential energy:

• A pendulum swinging back and forth (gravitational potential ↔ kinetic)
• A mass oscillating on a spring (elastic potential ↔ kinetic)
• A roller coaster moving along its track (gravitational potential ↔ kinetic)

Conservation of Mechanical Energy

Mechanical Energy Components

Mechanical energy is the sum of a system's kinetic and potential energies: $E_{mech} = KE + PE$

In a typical mechanical system:

• Kinetic energy is the energy of motion
• Potential energy is stored energy that depends on position or configuration
• The total mechanical energy stays constant when only conservative forces act inside the system

A roller coaster is a clean way to picture this. At the top of a hill it has maximum potential energy and minimum kinetic energy. As it descends, potential energy drops and kinetic energy rises. At the bottom the relationship flips, with maximum kinetic and minimum potential energy. 🎢

Energy Changes and Transfers

Energy follows strict accounting. Any change in one type of energy must be balanced by a change in another type inside the system or by a transfer between the system and its surroundings.

When energy changes form within a system:

• The decrease in one form equals the increase in another
• When an object falls, the lost potential energy equals the gained kinetic energy
• The total stays constant, just spread differently among the forms

In this topic the most important transfer mechanism is work done on or by the system. You should also recognize that nonconservative forces can turn mechanical energy into thermal energy or sound.

Constant Energy Systems

By choosing the system boundary carefully, you can set up situations where total energy stays constant even while things change inside.

A system with no energy entering or leaving keeps a constant total energy. In these systems:

• All energy changes happen internally between forms
• The sum of all energy forms stays the same at every moment
• Energy can keep converting between forms, but the total is preserved

Approximately constant-energy systems include:

• An ideal pendulum (ignoring air resistance and friction)
• A planet orbiting the sun (ignoring other influences)
• A ball bouncing on a perfectly elastic surface (theoretical)

System Energy Changes

When a system's total energy changes, that change matches the energy transferred across the system boundary.

Energy transfer across boundaries follows these ideas:

• Energy entering the system increases its total energy
• Energy leaving decreases its total energy
• The net change equals the sum of all transfers in and out

For example, consider a person pushing a box across a rough floor with the system defined as the box alone:

• The person does positive work on the box, adding energy
• Friction does negative work on the box, removing energy
• The change in the box's kinetic energy equals the net work done on it

System Selection and Energy Changes

Whether a system's energy changes depends on how you define the system. If the system is only a falling ball, gravity does work on it, so the ball's energy changes. If the system is ball + Earth, gravity becomes an internal interaction, and the system's mechanical energy can stay constant when air resistance is negligible. Choosing the boundary decides whether an interaction is internal or an energy transfer with the environment.

Energy Conservation in Interactions

All interactions obey energy conservation no matter how you draw your boundaries.

The universal principle means:

• The total energy of the universe stays constant
• Energy may change form or transfer between systems, but none is lost
• Everything can be accounted for if you track all forms and transfers

In collisions, energy may look like it "disappears" but it converts to:

• Thermal energy (increased molecular motion)
• Sound energy (pressure waves)
• Deformation energy (permanent shape changes)

Zero Work and Constant Energy

When no external work is done on a system and there are no internal nonconservative interactions, mechanical energy stays constant.

In that case:

• No energy enters or leaves the system
• Energy still converts between kinetic and potential forms internally
• The sum of kinetic and potential energy stays the same

This applies to:

• An object in free fall (ignoring air resistance)
• A planet orbiting the sun (ignoring other interactions)
• A pendulum swinging with negligible friction
• A mass oscillating on an ideal spring

Nonzero Work and Energy Transfer

When work is done on a system, energy transfers between the system and its environment, changing the system's total energy.

Work and energy transfer are directly linked:

• Positive work increases the system's total energy
• Negative work decreases the system's total energy
• The energy transferred equals the work done

Examples:

• Pushing a box across a floor transfers energy into the box-floor system
• A car's brakes do negative work, transferring energy out of the car-road system
• Friction converts mechanical energy into thermal energy, effectively moving usable energy out of the system

How to Use This on the AP Physics 1 Exam

Problem Solving

Most energy problems come down to comparing a starting state with an ending state. A reliable approach:

• Pick your system and state your zero level for potential energy. The zero is your choice, so use it to simplify the math.
• Decide whether any nonconservative force (friction, air resistance) acts inside the system.
• If no nonzero work is done on the system and no nonconservative forces act inside, set $KE_i + PE_i = KE_f + PE_f$.
• If friction acts, account for the dissipated energy. The mechanical energy lost equals the friction force times the path length.

Free Response

Energy questions often ask you to justify a claim, not just plug in numbers. When you explain:

• Name the principle you are using (conservation of mechanical energy) and the condition that makes it valid (no nonzero external work, no internal nonconservative forces).
• Connect each equation back to its physical meaning, including units.
• Keep your reasoning organized and sequential so a reader can follow each step.

Common Trap

Watch how the system is defined. Gravity is an internal interaction only if Earth is part of the system. If your system is just the ball, gravity does work and mechanical energy of that system is not constant on its own.

Common Misconceptions

• "A single object can have potential energy." Potential energy belongs to a system of interacting objects, not to one object alone. You need at least two interacting objects, like object-Earth or block-spring.
• "Energy disappears when an object slows down from friction." The mechanical energy is converted to thermal energy and sound, so total energy is still conserved.
• "Conservation of mechanical energy always works." It only holds when no nonzero work is done on the system and no nonconservative forces act inside it. With friction or air resistance inside the system, mechanical energy is not constant.
• "Whether energy is conserved is a fixed fact about the situation." It depends on how you define the system. The same falling ball can have changing or constant energy depending on whether Earth is included.
• "Potential energy must be zero at the ground." The zero level is your choice. Picking a convenient reference point can simplify a problem without changing the physics.

Practice Problem 1: Gravitational Potential Energy

Solution

Calculate the work done against gravity, which equals the change in gravitational potential energy.

Step 1: Identify the relevant information.

• Mass (m) = 5.0 kg
• Height change (h) = 3.0 m
• Gravitational acceleration (g) = 9.8 m/s²

Step 2: Calculate the work done against gravity. Work = Force × Distance = Weight × Height = mg × h Work = 5.0 kg × 9.8 m/s² × 3.0 m = 147 J

Step 3: Determine the gravitational potential energy at the new height. With the ground as the reference point (h = 0), the potential energy at height h is: PE = mgh = 5.0 kg × 9.8 m/s² × 3.0 m = 147 J

So 147 joules of work is done against gravity, and the object has 147 joules of gravitational potential energy relative to the ground.

Practice Problem 2: Conservation of Mechanical Energy

Solution

Use conservation of mechanical energy, since the ramp is frictionless (no nonconservative forces).

Step 1: Identify the initial and final conditions.

• Initial height (hi) = 10.0 m
• Initial velocity (vi) = 0 m/s (released from rest)
• Final height (hf) = 0 m (bottom of ramp)
• Final velocity (vf) = ? (what we're solving for)
• Mass (m) = 2.0 kg

Step 2: Apply conservation of mechanical energy. Initial mechanical energy = Final mechanical energy Initial PE + Initial KE = Final PE + Final KE mgh₁ + ½mv₁² = mgh₂ + ½mv₂²

Step 3: Substitute the known values. (2.0 kg)(9.8 m/s²)(10.0 m) + ½(2.0 kg)(0 m/s)² = (2.0 kg)(9.8 m/s²)(0 m) + ½(2.0 kg)(vf)² 196 J + 0 J = 0 J + (1.0 kg)(vf)² 196 J = (1.0 kg)(vf)²

Step 4: Solve for the final velocity. vf² = 196 m²/s² vf = √196 m/s = 14.0 m/s

So the ball's speed at the bottom of the ramp is 14.0 m/s.

Practice Problem 3: Elastic Potential Energy

Solution

This problem converts elastic potential energy into kinetic energy.

Step 1: Calculate the elastic potential energy stored in the compressed spring. Elastic PE = ½kx² Elastic PE = ½(250 N/m)(0.15 m)² = ½(250 N/m)(0.0225 m²) = 2.81 J

Step 2: Apply conservation of energy. When the spring is fully released, all the elastic potential energy converts to kinetic energy of the block. Elastic PE = Kinetic Energy 2.81 J = ½mv²

Step 3: Solve for the maximum speed. 2.81 J = ½(0.50 kg)v² 2.81 J = 0.25 kg·v² v² = 2.81 \text{ J} / 0.25 \text{ kg} = 11.24 \text{ m}^2/\text{s}^2 v = √11.24 m/s = 3.35 m/s

So the maximum speed the block will achieve is 3.35 m/s.

Related AP Physics 1 Guides

• 3.2 Work
• 3.3 Potential Energy
• 3.5 Power
• 3.1 Translational Kinetic Energy