Conserve total energy when mechanical energy decreases

1. Orient: Conserve total energy when mechanical energy decreases

Mechanical energy is the sum of the kinetic and potential energies tracked in a mechanical model. It need not be conserved when friction or other dissipative interactions convert it into internal or thermal energy. Total energy conservation is broader: energy does not disappear merely because the mechanical subtotal falls.

2. Explain and derive

For a sliding block considered alone, kinetic friction does negative work. For a larger system containing block and surface, the frictional interaction is internal and its effect can be represented as thermal-energy increase. Choose one boundary consistently. Counting negative friction work as an external loss and adding the same thermal conversion again would double-count the transfer.

On the horizontal track, friction magnitude is μmg while the block is sliding. Over actual travel d, its work is−μmgd. Starting from kinetic energy Ki, the stopping distance is Ki/(μmg). If a requested distance exceeds this, the block stops first; continuing the algebra to negative kinetic energy would violate the assumed motion.

The thermal increase belongs to block plus surface in this closed teaching ledger, not necessarily to the block alone. The model cannot infer their separate temperatures without heat capacities, energy partition and transfer information. It also omits sound and other losses; real accounting must identify which terms matter before claiming all lost mechanical energy became one measured thermal store.

3. Work an example

The 2 kg block starts at 6 m/s with Ki=36 J. At μ=0.2 and g=10 m/s², friction is 4 N. After 3 m, kinetic energy is 24 J and the combined thermal increase is 12 J. The stopping distance is 9 m. At μ=0.5 the stopping distance is 3.6 m, so a requested 5 m cannot be reached without additional input.

4. Predict and investigate

Predict first: If the requested travel exceeds the stopping distance, should thermal energy exceed the initial kinetic energy in this no-input model?

Compare μ=0.2 and 0.5 for a requested 3 m, then request 5 m at μ=0.5. Reconstruct Kfinal+ΔEthermal and check it equals 36 J. Distinguish requested from actual travel, and explain why kinetic friction is not applied through a reversal after stopping. Use the lesson controls and retain the settings with your evidence.

5. Transfer and explain

Try a new practice case

A 2 kg block initially moves at 4 m/s. A constant 4 N kinetic-friction force stops it. Find stopping distance.

Your numerical result (m) Check number

Reveal worked check (practice, not mastery)

This extra practice is not saved and does not award mastery. Record your first answer, explanation and whether you revealed help in the lesson notebook. Use the lesson’s saved question separately.

6. Reflect and retain evidence

How would the ledger change if an external motor pulled the block through the full distance at constant speed? State the required input before assuming K changes.

Retain a first attempt, signed ledger, units, system boundary, assumptions and a changed-case prediction. Public worked checks are practice, not independent mastery evidence. Synthetic outputs are not physical measurements.

Conservative return and a longer dissipative route

A conservative interaction gives the same work between fixed configurations regardless of route; on a closed return it gives zero net work. Its potential-energy bookkeeping is possible because the endpoint difference is path-independent. Friction and air resistance are examples of dissipative, generally path-dependent interactions; their effects cannot generally be represented by one potential depending only on position.

Predict and check: a 2 kg object rises 3 m and returns to its starting height in uniform g=10 m/s². Gravity does −60 J upward and +60 J downward, hence zero over the closed return even if the routes differ. A separate sliding packet with constant friction magnitude 4 N dissipates 24 J on a 6 m route and 40 J on a 10 m route between the same endpoints. The object is assumed supplied with enough energy to complete each route. State the larger system that retains that dissipated energy, and explain why a single endpoint potential difference cannot account for both friction amounts.

Framework and source notes

Original teaching prose and numerical cases. College Board AP Physics 1 CED, Unit 3 topic mapping: 3.2; 3.4. Checked 30 October 2026; this is a content cross-reference, not AP authorization.