Conservation of Energy
Energy is conserved in every interaction, but mechanical energy K+U is constant only under the right system choice and interaction conditions. Physics C therefore treats the system boundary as part of the solution: include conservative interacting objects when you want potential energy inside the system, then represent any remaining energy transfer or nonconservative work exactly once.
Topic sequence · 4 of 5
Learning objectives
Objective 1
Describe what energy forms are present for different system choices and distinguish total-energy conservation from mechanical-energy conservation.
Objective 2
Use K+U energy accounting with external transfer and nonconservative dissipation while keeping the chosen system boundary consistent.
Objective 3
Translate among equations, energy bar charts, physical system diagrams, and experimental data to justify whether mechanical energy is conserved.
- Topic 3.1 translational kinetic energy.
- Topic 3.2 work and the work–energy theorem.
- Topic 3.3 potential energy and conservative interactions.
- Topic 2.7 friction and Topic 2.9 resistive forces as nonconservative interactions.
Definitions first
Representations you must read and create
Representation 1
Energy bar charts comparing K, U, and nonmechanical/dissipated energy at initial and final states.
Representation 2
System-boundary diagrams showing which interacting objects are inside the system and which forces transfer energy across the boundary.
Representation 3
K(x), U(x), and total mechanical-energy line on the same position axis, where the gap E_mech−U gives kinetic energy.
Representation 4
Before/after state table listing K, each relevant U term, external transfer, and mechanical dissipation in joules.
Core relationships and calculus
Variables: K translational kinetic energy (J); U total mechanical potential energy (J)
Use: mechanical-energy inventory for a selected system
Variables: initial/final energies in joules
Use: mechanical-energy conservation when no energy is transferred across the boundary and no nonconservative interaction reduces mechanical energy
Variables: total system-energy change and net transfer across boundary (J)
Use: general energy bookkeeping for a selected system
Variables: E_mech=K+U; W_other is work not already represented by the conservative potential-energy terms included in U
Use: canonical mechanical-energy bookkeeping after the system boundary is fixed
Variables: all terms in joules
Use: state-to-state energy equation when other work is represented explicitly
Combine the work–energy theorem with potential energy. Write ΔK=W_c+W_other and use W_c=−ΔU. Then ΔK+ΔU=W_other, so Δ(K+U)=W_other. Which terms belong in W_other depends on the chosen system boundary. If there is no energy transfer across the boundary and no nonconservative interaction reducing mechanical energy, Δ(K+U)=0. Enlarging the system can move terms from “external work” into internal energy changes without violating total-energy conservation.
Start with the object-model work–energy theorem ΔK=W_net. Separate the work into conservative and other contributions: ΔK=W_c+W_other. By definition W_c=−ΔU, so ΔK=−ΔU+W_other and therefore Δ(K+U)=W_other. If the chosen system includes the conservative interacting objects and W_other=0, then K+U is constant. If nonconservative interactions convert mechanical energy to thermal or sound, K+U decreases even though total energy of a sufficiently inclusive system remains conserved.
The current Mechanics framework expects students to know that mechanical energy can be dissipated as thermal energy or sound by nonconservative forces. The unit does not require a full thermodynamic model; keep the distinction between total energy conservation and mechanical-energy accounting clear.
Physical examples
Example 1
A cart–spring system on a frictionless track converts spring potential energy into kinetic energy while K+U stays constant; choosing only the cart as the system instead makes the spring interaction appear as external work.
Example 2
A block sliding on a rough floor can lose mechanical energy while total energy is still conserved. If the block and floor are inside the system, the missing mechanical energy appears mainly as thermal energy within that larger system.
Example 3
A satellite–planet system can exchange gravitational potential and kinetic energy with no external work, so mechanical energy remains constant in the ideal two-body model.
Example 4
An external motor can increase a selected system’s energy by doing work across the boundary; conservation of energy does not mean the selected system’s energy must remain constant.
Worked reasoning
Exam reasoning patterns
Reasoning pattern 1
Write the system at the top of the solution. The same physical situation can have different-looking energy equations under different valid system choices.
Reasoning pattern 2
Use energy when the question asks for a speed at a position and the path details are irrelevant or can be represented by a known nonconservative work term.
Reasoning pattern 3
If friction is present, decide whether its effect is external work or internal mechanical-to-thermal conversion based on the system boundary; do not include it twice.
Reasoning pattern 4
Check whether a calculated kinetic energy is nonnegative. If your ledger predicts K<0, the assumed final state is not physically reachable under the model.
Reasoning pattern 5
After defining the system, group work not represented by included conservative potentials into one W_other term. Then use Δ(K+U)=W_other; do not count the same frictional interaction twice.
Deeper calculus and limiting-case check
For motion in a one-dimensional conservative potential, a constant total mechanical energy E gives K(x)=E−U(x). Classically allowed positions satisfy E≥U(x), and turning points occur where E=U. This viewpoint lets a single U(x) graph predict speed, turning points, and equilibrium without solving the time-dependent equation of motion.
Model-selection workflow: decide what the problem is before choosing an equation
For Conservation of Energy, the fastest reliable route is not equation hunting. Start by identifying the system boundary, encode it with the energy/work representation, choose the work-energy or conservation equation, and then test the conservative-force and transfer assumption. This sequence prevents a mathematically correct manipulation from being attached to the wrong physical model.
Define the physical object or system
Name what is being modeled, what is outside the system, the coordinate or orientation convention, and the interval or geometry over which the statement applies. Treat system and mechanical energy as physical quantities first and symbols second.
Choose a representation
Build Energy bar charts comparing K, U, and nonmechanical/dissipated energy at initial and final states.. If the prompt supplies System-boundary diagrams showing which interacting objects are inside the system and which forces transfer energy across the boundary., translate it into the same sign, axis, orientation, or reference convention before combining information.
Write the governing relationship
A useful starting relationship is E_mech = K + U. State what the variables mean and why this relation—not merely a familiar formula—matches the physical situation.
Check conditions and limits
The current Mechanics framework expects students to know that mechanical energy can be dissipated as thermal energy or sound by nonconservative forces. The unit does not require a full thermodynamic model; keep the distinction between total energy conservation and mechanical-energy accounting clear. After solving, test units, sign or direction, zero/large-parameter limits, and whether the answer remains consistent with the original representation.
Representation → equation → calculus → physics chain
| Stage | What to write or inspect | What proves the step is physically valid |
|---|---|---|
| Representation | Energy bar charts comparing K, U, and nonmechanical/dissipated energy at initial and final states. | Every arrow, sign, axis, node, surface, path, or reference has an explicit meaning. The diagram must contain enough information to determine the relevant components or constraints. |
| Primary model | E_mech = K + U | mechanical-energy inventory for a selected system. Preserve the variables and units long enough to see dependence before inserting numerical values. |
| Second relationship | K_i + U_i = K_f + U_f | mechanical-energy conservation when no energy is transferred across the boundary and no nonconservative interaction reduces mechanical energy. Combine relationships only after confirming they describe the same system, interval, frame, or geometry. |
| Calculus operation | Combine the work–energy theorem with potential energy. Write ΔK=W_c+W_other and use W_c=−ΔU. Then ΔK+ΔU=W_other, so Δ(K+U)=W_other. Which terms belong in W_other depends on the chosen system boundary. If there is no energy transfer across the boundary and no nonconservative interaction reducing mechanical energy, Δ(K+U)=0. Enlarging the system can move terms from “external work” into internal energy changes without violating total-energy conservation. | State what the derivative, integral, differential equation, or limiting process represents physically; do not treat calculus notation as an algebraic decoration. |
| Physical check | Return to the original words and representation. | The final sign, direction, units, endpoint behavior, and functional dependence must all agree. A correct-looking expression that violates one of these checks needs repair. |
Advanced synthesis: how to handle an unfamiliar version of this topic
A cart starts from rest with 18 J of spring potential energy. It crosses a rough patch that converts 5 J of mechanical energy to thermal energy, then climbs to a point where the cart–Earth system has 8 J of gravitational potential energy relative to the chosen reference. Taking no other external transfer, the mechanical ledger gives E_mech,f=18−5=13 J. Therefore K_f=13−8=5 J. If the cart mass is 0.40 kg, v_f=√(2K_f/m)=5.0 m/s. Total energy was not “lost”: the 5 J missing from mechanical energy became nonmechanical energy in the larger physical system. To deepen this example, change one assumption or parameter, predict the effect before recomputing, and then use the symbolic dependence to verify the prediction.
FRQ-quality response architecture for Topic 3.4
Mathematical routines
Start from the governing law or definition, keep the derivation symbolic, identify the condition that permits any simplification, and finish with units plus a limiting-case check. For this topic, the relationship E_mech = K + U should appear only when its physical conditions have been established.
Translation between representations
Move deliberately between Energy bar charts comparing K, U, and nonmechanical/dissipated energy at initial and final states. and System-boundary diagrams showing which interacting objects are inside the system and which forces transfer energy across the boundary.. Explain what a slope, area, direction, curvature, intercept, node value, field orientation, or other feature means instead of merely redrawing it.
Experimental design and analysis
Turn the topic into a measurable relationship. State the independent and dependent variables, important controls, repeated/ranged trials, and the graph or transformed graph whose slope/intercept would test the model or extract a physical parameter.
Qualitative/quantitative translation
Make the comparison or direction claim first, justify it through the model, derive the symbolic dependence independently, and then reconcile the derivation with the original qualitative claim.
Do not let an equation stand in for an explanation. A complete response names what changes, identifies the governing physical relationship, follows the causal chain, and states the requested outcome explicitly.
Experiment and data-analysis upgrade
Launch a cart with a compressed spring along a track that contains both smooth and rough regions. Measure position and speed, estimate spring and gravitational potential energies, and compare energy bar charts for two system choices: cart only versus cart+spring+Earth. On the rough section, quantify the decrease in mechanical energy and discuss where that energy appears, including uncertainty from friction variability and speed measurement.
Operationalize the variables
Say exactly what instrument or observable supplies each measured quantity. Distinguish a directly measured variable from a quantity calculated later from those measurements.
Predict the graph before collecting data
Use the governing relationship to decide what should be plotted. If the raw relationship is nonlinear, choose a transformation that produces a line whose slope or intercept has a defined physical meaning.
Control and repeat for a reason
Hold variables fixed when they would change the predicted relationship. Repeat trials to estimate scatter; vary the independent quantity over a useful range to test functional dependence. These solve different experimental problems.
Evaluate the model, not only the number
Compare the best-fit result with the theoretical prediction, inspect residual or endpoint behavior when appropriate, and identify whether a discrepancy is random scatter, instrument resolution, or a systematic violation of an ideal-model assumption.
Error-repair matrix
| Failure pattern | Repair move | Self-check |
|---|---|---|
| Saying “energy is lost” when mechanical energy decreases. Total energy is conserved; mechanical energy may be converted to thermal, sound, or other forms. | Return to the energy/work representation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Using potential energy for a single-object system without including the interacting source object. Potential energy belongs to the configuration of a multi-object conservative system. | Return to the work-energy or conservation equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Writing K_i+U_i=K_f+U_f automatically whenever the word “energy” appears. Check for external transfer and nonconservative interactions first. | Return to the energy/work representation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Changing the system boundary midway through the solution. State the system once, then keep work, potential-energy, and dissipation terms consistent with that boundary. | Return to the work-energy or conservation equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Treating friction as if it had an ordinary mechanical potential-energy function. Friction is nonconservative and usually reduces mechanical energy. | Return to the energy/work representation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Double-counting friction by subtracting a friction work term and also subtracting a separate dissipated-energy term for the same interaction. Choose one bookkeeping representation consistent with the system boundary. | Return to the work-energy or conservation equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
Spiral review: make this topic transferable
A topic is exam-ready only when it can be recognized inside a mixed problem. Use the connections below as transfer tests rather than as optional reading.
Transfer 1
Topic 3.1 supplies K, the motion-energy term in E_mech=K+U. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 2
Topic 3.3 supplies U and the relation between conservative work and stored configuration energy. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 3
Topics 2.7 and 2.9 provide nonconservative interactions that can reduce mechanical energy. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Formula-condition retrieval
From memory, write E_mech = K + U and one condition that must be true before using it. Then state one situation where a more general derivative, integral, conservation law, or field/force model is required.
Topic 3.4 mastery ladder
Interactive mechanical-energy ledger
Track mechanical energy with E_mech,f=Kᵢ+Uᵢ+W_transfer−E_diss. Then subtract U_f to find K_f. Dissipated mechanical energy may appear as thermal or sound in a larger total-energy system.
Experimental connection
Launch a cart with a compressed spring along a track that contains both smooth and rough regions. Measure position and speed, estimate spring and gravitational potential energies, and compare energy bar charts for two system choices: cart only versus cart+spring+Earth. On the rough section, quantify the decrease in mechanical energy and discuss where that energy appears, including uncertainty from friction variability and speed measurement.
Connections across Physics C
Connection 1
Topic 3.1 supplies K, the motion-energy term in E_mech=K+U.
Connection 2
Topic 3.3 supplies U and the relation between conservative work and stored configuration energy.
Connection 3
Topics 2.7 and 2.9 provide nonconservative interactions that can reduce mechanical energy.
Connection 4
Topic 3.5 asks how quickly energy is transferred or converted, adding the rate concept of power.
Related lessons and resources
Use these links when the current model depends on a prerequisite or becomes the foundation for the next model.
Common mistakes
Avoid this
Saying “energy is lost” when mechanical energy decreases. Total energy is conserved; mechanical energy may be converted to thermal, sound, or other forms.
Avoid this
Using potential energy for a single-object system without including the interacting source object. Potential energy belongs to the configuration of a multi-object conservative system.
Avoid this
Writing K_i+U_i=K_f+U_f automatically whenever the word “energy” appears. Check for external transfer and nonconservative interactions first.
Avoid this
Changing the system boundary midway through the solution. State the system once, then keep work, potential-energy, and dissipation terms consistent with that boundary.
Avoid this
Treating friction as if it had an ordinary mechanical potential-energy function. Friction is nonconservative and usually reduces mechanical energy.
Avoid this
Double-counting friction by subtracting a friction work term and also subtracting a separate dissipated-energy term for the same interaction. Choose one bookkeeping representation consistent with the system boundary.
Mastery check
- Define the main quantity without relying on an equation.
- Choose the correct representation and sign convention.
- Explain why the calculus relationship matches the physics.
- Design or interpret an experiment that could test the model.
Frequently asked questions
Is mechanical energy always conserved?
No. Mechanical energy K+U is constant only when the selected system has no relevant energy transfer across its boundary and no nonconservative interaction reducing mechanical energy. Total energy is conserved more generally.
Where does energy go when friction acts?
Mechanical energy is commonly converted to thermal energy and sometimes sound. If the interacting surfaces are inside the system, total system energy can remain constant while mechanical energy decreases.
Why does system choice matter?
Including or excluding an interacting object changes whether an interaction is represented as potential energy inside the system or as work/energy transfer across the boundary.
Can I use conservation of energy instead of Newton’s second law?
Often yes for state-to-state speed or position questions when the relevant energy terms are known. Force methods remain necessary when you need detailed acceleration, contact forces, or time evolution that energy alone does not determine.