Conservation of Energy · Grade 12 Science
Chapter: Work, Energy & MomentumConservation of Energy
Learning goals
- I can apply conservation of mechanical energy (KE + PE = constant) to solve for speeds and heights in ideal systems.
- I can account for energy dissipated by friction and calculate the efficiency of a real device.
Readings for this unit
Before this unit
Work, energy types, and system boundaries.
Use the detailed parent plan: two core 50–60-minute sessions per focused unit, with a third assigned for selected calculation/modeling topics. First study the core explanation and Parts 1–2 with their figures/checks; next study Parts 3–4, the whole-unit application, worked cases, varied practice and exit check. Attempt reasoning before opening guidance. Extend time when actual understanding requires it; investigations and chapter assessments have separate sessions.
Start with something to explain
Does friction destroy energy?
Illustrative evidence case. A sliding cart slows while the surface warms and some energy paths remain unmeasured.
Notice and wonder
Record two observations using the stated quantities or relationships. Keep observations separate from possible causes. Write one question that a new measurement could help answer.
Does friction destroy energy?
Your first model
Separate mechanical stores from the complete boundary account.
Build the language from the idea
- mechanical energy
- The sum of an object's kinetic and potential energy: E = KE + PE.
- conservation of mechanical energy
- In the absence of friction and other non-conservative forces, KE + PE stays constant: KE_i + PE_i = KE_f + PE_f.
- law of conservation of energy
- The total energy of an isolated system never changes; energy is only transformed or transferred, never created or destroyed.
- dissipation
- The spreading of useful mechanical energy into scattered thermal energy and sound, usually through friction or drag.
- efficiency
- The fraction of input energy that comes out as useful output: efficiency = (useful output ÷ total input) × 100%.
Mechanical energy includes kinetic and appropriate potential-energy stores for a selected system. It can remain constant when external transfers and nonconservative effects are negligible in the stated account. Absence of friction alone is not sufficient if an external motor does work or energy leaves by another pathway. For an ideal loss-free descent, gravitational potential energy can transfer into kinetic energy while their sum stays constant.
This is not just elegant; it is a ruthlessly efficient problem-solving tool. Suppose a ball is dropped from a height h and you want its speed at the bottom. Setting mgh = ½mv², the mass cancels and you get v = √(2gh) — the answer in a single line, with no need to track acceleration or time at all. Conservation lets you leap straight from a starting condition to a final one, ignoring the complicated journey in between, which is why physicists reach for it constantly.
Friction and other interactions can reduce translational mechanical energy while increasing thermal, sound, deformation, rotational, or other stores. A complete energy account includes transfers across the selected boundary. For an isolated system, total energy is conserved; for an open energy account, initial plus incoming equals final plus outgoing. An unexplained residual identifies missing information or a model limitation rather than proving energy has been destroyed.
Efficiency is useful output divided by input for a specified service and boundary. A motor receiving 800 J and delivering 600 J useful work is 75% efficient; account for the remaining 200 J as other transfers or storage rather than assuming its exact destination without evidence. Thermal engines have a second-law limit under their operating conditions, but it is too broad to say every kind of defined efficiency must always be below 100%. An ideal electrical heater can convert electrical input to thermal output with a near-unity device account; a heat pump’s coefficient of performance is a different ratio that includes transferred environmental heat.
A Newton's cradle captures the whole story in one desktop toy. Lift a steel ball and release it, and the energy races through the row and out the far side, swing after swing, almost perfectly conserved. Almost — each click you hear is a sliver of energy leaving as sound, and each collision warms the steel a touch, so the swings slowly shrink and finally stop. Nothing was destroyed; the energy merely dissipated. The same accounting lets engineers design hydroelectric dams, regenerative brakes, and power plants, all of which live or die by how much of their energy they can steer into a useful form instead of losing it to heat.
Learning to write down every joule at the start, track where each one goes, and insist that the books balance at the end is the heart of energy analysis. Whether the 'missing' energy went into motion, into height, into a stretched spring, or into scattered heat, conservation guarantees it went *somewhere* — and that certainty is what turns a tangle of forces into a solvable problem.
Model check: Mechanical energy can decrease while total energy remains conserved. Energy transfers into thermal energy, deformation, or sound of the chosen larger system; account for boundaries.
Build your understanding
Use the detailed parent plan: two core 50–60-minute sessions per focused unit, with a third assigned for selected calculation/modeling topics. First study the core explanation and Parts 1–2 with their figures/checks; next study Parts 3–4, the whole-unit application, worked cases, varied practice and exit check. Attempt reasoning before opening guidance. Extend time when actual understanding requires it; investigations and chapter assessments have separate sessions.
Part 1 · State an energy boundary before balancing
Energy conservation is a complete accounting principle: the change of energy stored in a system equals transfers into it minus transfers out. A boundary including Earth represents gravity with potential energy; a boundary around the moving object alone may instead count gravity’s external work. These are compatible choices when used consistently. Do not count the same gravitational transfer both as external work and as a potential-energy decrease in the same balance.
Read the visual model
Identify the labeled structures and units, then trace the arrows or compare values under the caption’s stated conditions. Separate the displayed evidence from a further inference. Write your reasoning before opening the guidance.
Why must the system be stated before listing work and energy stores?
Visual reasoning — after your attempt
The boundary decides which interactions are internal stores and which are external transfers, preventing omissions or double counting.
Energy conservation applies to a complete account including stores and transfers across the chosen boundary. A system closed to matter can still exchange energy. Write initial energy plus incoming transfers equal to final energy plus outgoing transfers; identify what has deliberately been omitted.
A cooling sealed bottle loses energy to its surroundings although it retains its matter. Calling it closed does not make its energy constant.
Pause and explain
When can the selected total energy be treated as constant?
Check your reasoning after answering
When the account is complete and no net energy crosses the relevant boundary over the interval, or incoming and outgoing transfers balance.
Part 2 · Distinguish total and mechanical conservation
Mechanical energy combines selected kinetic and potential stores. Friction can reduce this sum while increasing internal thermal energy of bodies and surroundings. The 100 J account here ends with 70 J mechanical and 30 J thermal increase. Saying energy was “lost” is only shorthand for leaving the selected useful or mechanical category; it is not destruction. A complete account also checks sound, deformation and transfer outside the boundary where relevant.
Read the visual model
Identify the labeled structures and units, then trace the arrows or compare values under the caption’s stated conditions. Separate the displayed evidence from a further inference. Write your reasoning before opening the guidance.
Can a 30 J mechanical decrease by itself establish where all 30 J went?
Visual reasoning — after your attempt
No. Additional evidence is needed to apportion thermal, sound, deformation and external transfers.
Mechanical energy K+U can remain constant when nonconservative transfers are negligible in the selected model. Friction can reduce mechanical energy while increasing thermal energy elsewhere. Total energy conservation and mechanical-energy conservation are different claims.
A sliding block stopping does not destroy its kinetic energy; contact interactions transfer energy into other stores.
Pause and explain
Does friction violate energy conservation?
Check your reasoning after answering
No. It can change the distribution among stores and transfers while reducing the part labeled mechanical energy.
Part 3 · Use the work-energy connection correctly
The work–energy theorem states Wnet=ΔK in this ordinary mechanics model. It concerns the sum of all forces’ work on the selected object, not necessarily total energy or one applied force’s contribution. A driving force doing 50 J while resistance does−20 J yields a 30 J kinetic increase. If gravity is among the forces, its work enters this net-work account; a potential-energy formulation reorganizes that term instead of adding it twice.
Read the visual model
Identify the labeled structures and units, then trace the arrows or compare values under the caption’s stated conditions. Separate the displayed evidence from a further inference. Write your reasoning before opening the guidance.
If initial K is 10 J, what is final K here?
Visual reasoning — after your attempt
40 J, because net work is+30 J.
Net work on a particle-like object equals its kinetic-energy change in the ordinary mechanics model. Individual forces may do nonzero work while net work is zero. Potential-energy formulations move suitable conservative interactions into an energy-store account, so avoid counting the same transfer twice.
An ideal constant-speed lift has positive work by support and negative work by gravity, leaving zero net work on the load.
Pause and explain
What error occurs if gravity’s work and the same mgΔh are both counted as separate losses?
Check your reasoning after answering
The same conservative interaction is double-counted. Use either a force-work account or a compatible system potential-energy account consistently.
Part 4 · Use evidence to account for nonmechanical outputs
A residual is the difference between a predicted account and the terms currently measured or estimated. If mechanical energy falls 35 J while only 28 J thermal increase is recorded,7 J remains unaccounted for in those terms. Sound, deformation, heat leaving the boundary or measurement bias are possible explanations. Do not force closure by inventing an observation; report uncertainty and identify a measurement that can distinguish plausible missing pathways.
Read the visual model
Identify the labeled structures and units, then trace the arrows or compare values under the caption’s stated conditions. Separate the displayed evidence from a further inference. Write your reasoning before opening the guidance.
Does a 7 J residual prove energy conservation failed?
Visual reasoning — after your attempt
No. It may reflect omitted transfers, storage or uncertainty; test the account before making that inference.
A real-device energy budget may need thermal, sound, deformation, and external work terms. A residual can be recorded as unaccounted energy rather than assigned to an unsupported destination. Compare measured quantities with uncertainties and a clear interval.
If 100 J enters a device and 80 J is measured in named outputs, the remaining 20 J needs a store or transfer account, not a claim of destruction.
Pause and explain
How should an unexplained residual be reported?
Check your reasoning after answering
Keep it explicit, identify measurement limits and possible omitted pathways, and propose observations that can distinguish them.
Apply the whole unit · Account for a descending cart
Illustrative 2.00 kg cart starts at rest 1.00 m above a reference and reaches speed 3.00 m/s at that reference. Use g=10.0 m/s². Ignore cart rotation in the first account.
Your task
Find initial gravitational potential energy, final translational kinetic energy, and residual. Explain why the residual is not automatically all frictional heating and improve the measurement plan.
What your response should show
- Define the system, quantities, units, and relevant conditions before selecting a model.
- Show intermediate calculations or a labeled evidence account, including signs or denominators.
- Explain how the supplied evidence supports the conclusion rather than reporting only an answer.
- Identify a meaningful uncertainty, model limit, or measurement needed to test the conclusion.
Response guidance — after your attempt
Initial U=2×10×1=20.0 J; initial K=0.
Final translational K=½×2×3²=9.00 J.
Residual=20.0−9.00=11.0 J in omitted stores, transfers, or measurement discrepancies.
Wheel rotation can be a significant store for a cart, so assigning all 11 J to friction without investigating it would be unjustified. Sound, track heating, deformation, and uncertainties may also contribute. A fuller model should include rotation or justify neglecting it, rather than treating a translational-only account as total mechanical energy.
Scientific references and source-model foundations
Parent / teacher teaching notes
Ask which energy stores belong to a cart rather than an ideal sliding point mass. Reward an explicit unresolved residual with a useful next measurement. Ask the learner to explain each symbol, axis, unit and arrow before calculating. Calculated examples, supplied illustrative records and actual historical observations are labeled separately. Selected public OpenSciEd middle-school source figures are foundation examples, not complete high-school editions or an endorsement; all advanced explanations are supplied locally.
A Newton's cradle transferring energy from one swinging steel ball through a row to the ball on the far end.
The key idea
Total energy follows a complete boundary account. Mechanical energy remains constant only under appropriate transfer conditions; useful efficiency needs a defined service.
Mechanical Energy
E = KE + PE
the sum of an object's kinetic and potential energy.
Conservation of Mechanical Energy
KE_i + PE_i = KE_f + PE_f
valid when friction and other non-conservative forces are absent; the total of kinetic plus potential energy stays constant.
Efficiency
efficiency = (useful output ÷ total input) × 100%
the percentage of input energy that emerges as useful output; always less than 100% for a real device.
Worked examples
Example 1
- All the gravitational potential energy converts to kinetic energy: mgh = ½mv².
- PE = mgh = 1.5 × 9.8 × 10 = 147 J, so 147 = ½ × 1.5 × v² = 0.75v².
- v² = 147 ÷ 0.75 = 196, so v = √196 = 14.
Example 2 · Worked application
- Lost PE=mgΔh=50 J.
- Conservation gives ½mv²=50.
- v=√100=10 m/s.
Mass cancels in this ideal gravitational-energy calculation.
Example 3
- Use efficiency = (useful output ÷ total input) × 100%.
- efficiency = (600 ÷ 800) × 100%.
- = 0.75 × 100% = 75%.
Example 4 · Find and fix the mistake
- Test the claim against the stated conditions: Mass cancels in this ideal gravitational-energy calculation.
- Total energy is conserved, but mechanical energy can transfer to thermal energy or other stores.
Mass cancels in this ideal gravitational-energy calculation.
Example 5 · Reason from the visual model
- The work–energy theorem states Wnet=ΔK in this ordinary mechanics model. It concerns the sum of all forces’ work on the selected object, not necessarily total energy or one applied force’s contribution. A driving force doing 50 J while resistance does−20 J yields a 30 J kinetic increase. If gravity is among the forces, its work enters this net-work account; a potential-energy formulation reorganizes that term instead of adding it twice.
- 40 J, because net work is+30 J.
Original signed-work account on one object; internal-energy accounting may use a larger boundary.
Avoid these mistakes
- Mechanical energy can decrease while total energy remains conserved.
Check your understanding
A cart loses 20 J mechanical energy due to friction. Where is it?
Discuss the answer
Energy transfers into thermal energy, deformation, or sound of the chosen larger system; account for boundaries.
Return to the phenomenon
Return to the initial case. Does friction destroy energy? Revise your drawing or explanation using two named ideas and one stated limitation.
Compare explanations after your attempt
Mechanical energy can decrease while total energy is conserved through thermal changes and transfers; residuals need evidence.
What should we investigate next?
Which comparison would test a remaining uncertainty? State what you would change, keep comparable and measure.
Investigate, calculate and revise
Keep a first model and an evidence-supported revision. Attempt each task before opening guidance; all required explanations and supplied-data routes are available locally.
Activity 1 · Return to the opening case
Use the opening phenomenon, unit definitions and explained Parts 1–4.
Your task
Return to the initial case. Does friction destroy energy? Revise your drawing or explanation using two named ideas and one stated limitation.
Success criteria
- Use the stated observations and appropriate units.
- Connect a labeled model with the physical/environmental mechanism.
- State assumptions and an evidence limitation.
- Explain a substantive revision or next test.
Hint
Keep object/material identity, material transfer and energy account distinct.
Response guidance — discuss after your attempt
Mechanical energy can decrease while total energy is conserved through thermal changes and transfers; residuals need evidence.
Activity 2 · Transfer to a changed case
Cover the worked answer and use the explained models, relationships and notebook.
Your task
A system begins with 200 J mechanical energy. It ends with 120 J mechanical,50 J thermal increase and 20 J transferred outward as sound/heat. Find the unaccounted energy and a useful next check.
Success criteria
- Use the stated measurements or conditions, with correct units.
- Connect the result to the physical or environmental mechanism or evidence relationship.
- Identify the assumption, model limit or further evidence requested.
Hint
Name known quantities and conditions, show intermediate steps and check conservation.
Response guidance — discuss after your attempt
The accounted final/output total is 190 J, leaving 10 J. Check boundary heat transfer, deformation/storage and instrument uncertainty rather than claiming energy was destroyed.
Words to know
- mechanical energy
- The sum of an object's kinetic and potential energy: E = KE + PE.
- conservation of mechanical energy
- In the absence of friction and other non-conservative forces, KE + PE stays constant: KE_i + PE_i = KE_f + PE_f.
- law of conservation of energy
- The total energy of an isolated system never changes; energy is only transformed or transferred, never created or destroyed.
- dissipation
- The spreading of useful mechanical energy into scattered thermal energy and sound, usually through friction or drag.
- efficiency
- The fraction of input energy that comes out as useful output: efficiency = (useful output ÷ total input) × 100%.
Joule's paddle wheel: proving heat is energy
In the 1840 s, English brewer and physicist James Prescott Joule ran a famous experiment: a falling weight turned a paddle wheel inside an insulated barrel of water, and Joule measured the tiny temperature rise as the paddles churned. He found a fixed exchange rate between mechanical energy lost and heat gained — the 'mechanical equivalent of heat.' This showed that the energy 'lost' to friction was not gone but had become thermal energy, and it was decisive evidence for the law of conservation of energy. Fittingly, the SI unit of energy, the joule, is named after him.
Why we can't recycle waste heat: the second law
Conservation of energy guarantees the total never changes, which raises a puzzle: if the heat wasted by an engine is still energy, why can't we just collect it and reuse it to get 100% efficiency? The answer is the second law of thermodynamics: energy spontaneously spreads from concentrated, organized forms into disordered, scattered heat, and reversing that spread perfectly is impossible. This is why every perpetual motion machine ever built has failed, and why patent offices now refuse to even consider them. Energy is always conserved (first law), but its usefulness steadily degrades (second law).
Conservative vs non-conservative forces
Mechanical energy is only conserved when the forces at work are conservative — forces like gravity and ideal springs, for which the energy depends only on position, not on the path taken. Lift a book straight up or along a winding ramp to the same shelf, and gravity stores the same PE either way. Friction and air resistance are non-conservative: they always oppose motion, drain energy along the whole path, and turn it into heat that cannot be recovered as motion. Recognizing which forces are present tells you instantly whether you can use the clean 'KE + PE = constant' shortcut or must also subtract the energy dissipated.
Regenerative braking: conservation of energy on the road
An ordinary car brake throws away kinetic energy as useless heat in the brake pads — you can smell it after a long downhill. Electric and hybrid vehicles do something cleverer with regenerative braking: the electric motor runs backward as a generator, using the car's kinetic energy to recharge the battery instead of just heating the brakes. It is conservation of energy turned into engineering profit, recovering a large fraction of the energy that would otherwise be lost and noticeably extending an EV's range, especially in stop-and-go city driving.
In the real world
Electric and hybrid cars use regenerative braking to capture the kinetic energy that would normally be wasted as brake heat, converting it back into stored electrical energy in the battery — a direct application of conservation of energy that noticeably extends driving range.
Exit check
A cart loses 20 J mechanical energy due to friction. Where is it?
Check your explanation after answering
Energy transfers into thermal energy, deformation, or sound of the chosen larger system; account for boundaries.
Explain your reasoning without notes. If a key idea is missing, revisit the lesson and try again.