Audit a multi-stage mechanism by changing system and law

1. Frame the question

A successful synthesis begins by dividing the event into intervals and naming a system for each. A spring launches a cart, a rough segment converts mechanical energy into internal energy, a brief inelastic capture joins a second cart, and a different spring stops the joined pair. The same conserved subtotal does not automatically apply across all four intervals.

2. Build the mechanism

During ideal launch, spring potential becomes cart kinetic energy. Across the horizontal rough segment, kinetic friction does negative work on the cart; in a larger cart–track system that loss appears as internal energy. Before predicting any later event, compare available energy with the work needed to traverse the whole segment. An algebraic negative downstream kinetic energy signals that the assumed path is impossible.

During a sufficiently brief capture, external horizontal impulse on the two-cart system is negligible, so their total horizontal momentum is approximately conserved. Mechanical energy decreases in a perfectly inelastic capture. The later spring compression conserves the post-capture mechanical energy under the stated frictionless conditions, not the original launch energy. Spring forces provide external momentum transfer to the carts over that longer interval.

This model uses two equal 2 kg carts, launch stiffness100 N/m, final stiffness50 N/m, a0.20 m friction segment andg10. If initial compression isx and coefficientμ, initial energy is50x² and crossing loss is4μ joules. Positive arrival energyE gives arrival speed√E, joined speed√E/2, post-capture energyE/2 and final compression√(E/50). All quantities refer to the specified stage; no actual spring or collision apparatus is requested.

3. Work an example

At x=0.30 m and μ=0.10, initial energy4.5 J loses0.4 J crossing the rough segment, leaving4.1 J. Arrival speed≈2.025 m/s becomes≈1.012 m/s after equal-mass capture. Capture converts2.05 J internally and leaves2.05 J for final compression≈0.2864 m. The full account is4.5=0.4+2.05+2.05 J. At x=0.10 m and μ=0.20, available0.5 J cannot supply0.8 J crossing loss: stopping distance is0.125 m and downstream events never occur.

4. Predict and investigate

Can a mathematically correct collision formula still be irrelevant because the cart never reaches the collision?

Compare the passing case (x=0.30,μ=0.10), a stopped case (0.10,0.20), and threshold (0.10,0.125). Retain a stage diagram and separate energy/momentum ledgers. Change launch compression and predict the feasibility boundary before capture. Explain why no downstream speed is displayed for a failed path.

5. Transfer to a new case

Try a new practice case

In a changed packet, arrival energy is9 J for a2 kg cart, which captures a stationary2 kg cart. A final spring has k=100 N/m. Find its maximum compression.

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. Audit the evidence

How would rotation of wheels, a rebound, or a non-negligible external impulse change the model? Choose one and identify the missing state or transfer rather than merely saying “friction matters.”

Retain the first prediction, contrasting settings, units, derivation, uncertainty assumptions and a changed-case explanation in the course workspace. Public worked answers do not constitute secure independent assessment. Supplied-data captures are not hands-on laboratory time or observed instrument competence. A practical proposal requires qualified local review before any physical execution.

Sources

Original synthesis cases and code, checked30 October2026 against the current AP Physics1 CED, OpenStax work and energy, inelastic collisions, and pendulum assumptions. The common-calibration counterexample and numerical packets are original, not official exam items.