SC07 · SCIENCE THROUGH EVIDENCE

University Physics I with calculus

All 64 destinations in the finite calculus-based SC07 mechanics, oscillations, waves and fluids plan are authored with original derivations, bounded investigations and cumulative evidence workflows. Independent full-course source/action review passed; authored coverage is not practical certification or independent learner mastery.

This is the course outline. Open a unit to choose a lesson; every lesson opens on its own page. Explanations and experiments live inside those lesson pages.

Preparation and course boundaries

Preparation: Algebra, trigonometric components and an introductory understanding of derivatives/integrals. Power/chain rules, definite integrals and vector dot/cross meanings are refreshed locally. Prior course completion is not required to open any lesson.

Course outcome: Derive and test dimensioned differential and integral motion and force models; reconcile line work, potential, momentum and frame-dependent energy accounts; integrate mass moments and rotational/orbital constraints; analyze oscillations, wave interfaces and fluid balances; distinguish truncation, observation error and model inadequacy through withheld predictions, admissibility bounds and independent cumulative evidence

All lessons are directly accessible. Reused foundation explanations are identified where relevant; completing a different course does not award this course's evidence. The 64 registered lessons keep separate predictions, captures, practice responses and reflections.

Differential and integral motion evidence9 lessons

Differentiate/integrate scalar and vector trajectories, compare analytic and numerical approximations and test motion models on withheld observations.

Differentiate a trajectory without losing its units

Compare β<0, β=0 and β>0. Retain a turning case and transform the same derivative into minute units without changing the physical motion.

Choose derivative spacing using an error budget

Compare three spacings at one error bound, then zero versus nonzero error. Retain the error terms separately and choose a grid spacing by the combined bound.

Integrate acceleration twice and locate a reversal

Compare an interval ending before reversal, exactly at reversal where supported, and after it. Preserve both integration constants and piecewise distance calculations.

Separate changes in speed from changes in direction

Vary u at fixed b and then b at fixed u. Reconstruct dot and cross products, verify the component-square identity and explain the equal-scale path.

Differentiate the basis vectors of circular motion

Retain two times at the same α and two α values at one time. Derive Cartesian acceleration from the moving basis and check its magnitude.

Test numerical update order against an exact trajectory

Compare all three update rules at one N, then double N. Preserve exact and computed endpoints, error sign and the distinction between trajectory error and an impact model.

Compare integration rules with an exact displacement

Compute one small grid by hand, compare all three methods and double N. Retain signed errors and state precisely why Simpson is exact for this supplied function.

Differentiate the observer transformation

Compare two observers at the same event time. Differentiate the transformation twice and preserve vector components before comparing magnitudes.

Challenge an exact fit with a withheld trajectory point

Compare zero and nonzero jerk with the same offset, then change only the offset. Freeze the first three fitting points before revealing the fourth and retain separate deterministic observation-bound versus omitted-model limits.

Force equations, resistance and constraint events11 lessons

Derive and solve bounded force-based differential equations, respect geometric/unilateral constraints and distinguish numerical error from a challenged force model.

Infer a force sum without inventing its agents

Compare two masses and two times. Draw the disclosed agent forces, sum their components and name each third-law partner recipient separately.

Integrate a force ramp with the original initial conditions

Integrate two ramps, verify each derivative and initial condition, and compare an endpoint-force shortcut with the actual time integral.

Derive terminal motion and its approach time separately

Compare masses at fixed drag, then drag coefficients at fixed mass. Retain force balance, finite-time force residual and integrated displacement separately.

Solve a nonlinear falling branch by separating variables

Compare two k values and two times, check velocity and displacement derivatives, and show the initial acceleration and terminal limit.

Use an unused speed to challenge a resistance law

Fit at the common reference, freeze a higher-speed prediction and evaluate the withheld residual for both exponents. Preserve coefficient units.

Solve both velocity components before integrating the path

Compare two gamma values at the same time. Recover initial conditions and derivative identities, then distinguish vertical velocity sign from height.

Derive the coordinate constraint before eliminating tension

Retain both recipient diagrams and the differentiated cord constraint. Compare unequal masses in both orders and the equal-mass limit.

Carry position and velocity across a contact-loss event

Locate separation analytically, compare a pre-event, exact-boundary and post-event state, and retain continuous position/velocity plus the growing support gap.

Use v dv/dx without crossing an inaccessible endpoint

Derive the spatial relation from the time equation, test an exact turn and an inaccessible target, and explain where division by velocity would fail.

Derive the unstable mode of an otherwise decaying equation

Compare r below1, between1 and2, at2 and above2. Derive the amplification inequalities and retain intermediate signs, endpoint error and exact decay.

Converge a numerical method without confusing it with validation

Compare two grids against the exact assumed endpoint, then compare that exact endpoint with an independent changed-coefficient packet. Preserve signed error decomposition.

Line work, potential structure and energy evidence10 lessons

Integrate work and derive conservative force, classify stable/accessibility structure, and reconcile frame/boundary choices with independent numerical and model checks.

Integrate a force along two genuinely different paths

Compute each segment integral, reverse one path to form a loop and compare unequal versus equal coefficients. Verify the explicit potential only in the conservative case.

Use the force derivative without discarding a flat minimum

Derive the force and stationary points for positive, zero and negative a. Check second derivatives and directly inspect the potential where that test vanishes.

Distinguish allowed positions from a finite-time crossing

Solve turning roots for below-barrier, exact-barrier and above-barrier energies; retain disconnected intervals and the selected initial well. Explain the separate timing limitation.

Check every direction before calling an equilibrium stable

Compare all sign combinations and one zero coefficient. Derive forces from the potential and classify using both slices, naming exactly what the quadratic model omits.

Evaluate power on the same trajectory whose work you claim

Derive velocity first, integrate Fv and compare with kinetic energy. Retain a counterexample to the endpoint-force-times-distance shortcut.

Integrate mechanical loss once within a declared system

Integrate drag power, compare with endpoint kinetic change and write both point-only and point-plus-medium balances without double counting.

Transform displacement and kinetic energy in the same frame

Choose observers giving positive, zero and negative work where supported. Transform both displacements and velocities, and verify the common−UΔp correction.

Check the motion before assigning zero work to a normal force

Compare a stationary and an accelerating support, draw normal and weight, and reconcile point-only work with the gravitational-potential boundary.

Use an energy plot as one diagnostic, not a full certificate

Compare all three updates on two grids; retain energy extrema and endpoint phase-space error, not just one reassuring scalar value.

Test a potential shape with evidence not used to fit it

Fit only the first difference, predict the withheld energy and independent force, then vary the common offset. Preserve which evidence was used and which was withheld.

Momentum boundaries and extended-system moments9 lessons

Integrate force and mass moments, derive open-system flux, separate center and relative energy, and reject underdetermined or energy-incompatible inferences.

Integrate a smooth force without confusing impulse and work

Derive both integrations and compare the two equal-impulse packets. Retain signed displacement and kinetic change separately.

Account for mass crossing a momentum boundary

Derive the differential boundary balance before integrating it. Compare two initial masses at equal fractions, and then compare constant thrust with increasing acceleration.

Normalize a density before integrating its center

Integrate total mass, whole first moment and one segment first moment independently. Check the uniform limit and preserve the coordinate origin.

Use both area integrals for a nonuniform lamina

Reconstruct each double integral and compare a single-gradient case with the uniform limit. Record density normalization and both first moments.

Cancel internal impulses only inside the correct system

Construct each particle ledger before adding it. Repeat after excluding one internal recipient and identify the impulse that has become external to that smaller boundary.

Separate center motion from collision energy

Derive the energy split, reconstruct both outgoing velocities from center and relative motion, and check momentum and kinetic loss independently.

Do not mistake a balanced vector ledger for a passive collision

Derive the admissible velocity disk by completing the square. Test one interior, boundary and exterior point with separate vector and energy ledgers.

Transform relative walking into two ground displacements

Solve the weighted center constraint and independently verify the relative-displacement relation. Vary mass at fixed relative displacement.

Break a density-gradient and origin-offset ambiguity

Invert the independent fraction, propagate its deterministic endpoints in the correct order and then infer offset bounds. Preserve the original assumed density family.

Rotational moments and central-field evidence11 lessons

Derive distributed inertia, torque and angular conservation, reject infeasible rolling, and compare orbital energy/angular constraints with independent numerical and observational evidence.

Derive an axis-dependent inertia from the actual density

Derive the normalized second moment and compare direct integration with the parallel-axis reconstruction for three axes.

Include the ring area when deriving disk inertia

Normalize the density, derive mass and inertia from separate ring integrals, and compare the uniform and graded cases.

Integrate rotational relaxation and its work budget

Derive speed and angle from the torque equation, then close drive work as kinetic gain plus viscous dissipation on two time intervals.

Test a rolling constraint before using its acceleration

Derive translation and torque equations, evaluate the friction inequality, and retain one accepted and one rejected branch.

Distinguish conservation from independence of origin

Evaluate both cross products at two times and verify the origin-shift identity before interpreting signs.

Derive a radial barrier from angular momentum

Derive Ueff, its stationary radius and curvature, then compare an actual circular energy with a higher-energy passage and a forbidden radius.

Reconstruct an orbit from two turning radii

Derive energy and angular momentum from both apsides, check both energy substitutions, and identify the circular limiting case.

Keep the zero-energy escape limit distinct from finite turning

Derive and compare sub-escape, exact-escape and above-escape branches, retaining the infinity reference and the absent surface model.

Propagate orbital observations without hiding a clock gain

Derive the circular inference, evaluate its monotone endpoint bounds and compare a repeated shared-clock record with genuinely independent calibration.

Test orbital invariants against an independent exact trajectory

Compare Euler and Verlet on two grids, saving invariant extrema and endpoint position/velocity errors with the exact reference.

Distinguish radial laws that share angular conservation

Fit only the first radial acceleration, predict the unused acceleration and independent potential difference, then vary the common potential offset conceptually.

Differential oscillations, waves and fluid profiles12 lessons

Derive initial/boundary-value oscillations and waves, integrate fluid profiles, and validate a coupled hydrostatic oscillator with independent domain and calibration checks.

Recover oscillator phase from both initial conditions

Derive both quadratures and evaluate the same solution through position, velocity and conserved energy. Include the zero-amplitude phase boundary.

Derive the period of an extended pendulum from its torque

Derive the small-angle equation, compare the point-mass approximation, and differentiate the period-squared expression to locate its minimum.

Keep the same initial conditions across three damping regimes

Derive roots and coefficients for one case in each regime, then check initial position, initial velocity and the energy-rate identity.

Use the driven phase to close the average-power account

Derive both quadratures, verify the average input/loss identity and distinguish a displacement maximum from other response measures.

Derive independent modes before interpreting energy exchange

Derive sum and difference equations, reconstruct the physical coordinates and audit every energy term at two times.

Verify a traveling wave through derivatives and energy flux

Check the wave equation by two derivatives, compare phase and material velocities and derive the amplitude-squared mean-power scaling.

Match a wave at a boundary before interpreting transmitted power

Derive both coefficients from the two boundary equations, retain reflection sign and independently verify the power sum for matched, higher and lower impedances.

Derive allowed modes without mistaking harmonic ratios for calibration

Derive fixed-end modes, compare a fixed-free boundary and propagate an elapsed-time gain through frequency and density inference.

Integrate a density profile before evaluating pressure and buoyancy

Integrate pressure, compute opposite-face forces and independently integrate the displaced-fluid weight. Compare the uniform limit.

Derive a pipe profile and then test its admissibility

Derive the shear and velocity profiles, integrate their flow and independently check the regime before accepting any radius extrapolation.

Terminate a draining solution at its physical endpoint

Derive the separated solution, identify its endpoint and compare Euler event times on two grids without hiding the signed error.

Couple buoyancy stiffness to a bounded oscillation model

Derive the hydrostatic stiffness from displaced volume, combine it with spring stiffness and test the full amplitude range before assigning a period.

Cumulative calculus evidence and independent model comparison2 lessons

Connect all six finite categories through derivations, independent conservation/measurement checks and retained unfamiliar multi-domain transfer.

Reconcile generalized force with a first-branch energy integral

Derive force/torque effective inertia and independently integrate net work. Retain one accelerating, one decelerating and one unreachable requested state, preserving actual versus requested coordinates.

Separate physical parameters from a shared measurement gain

Freeze used and withheld data, derive identifiable combinations, add an independent static calibration, and propagate the stated bounded errors only under their declared mass condition.

Investigation portfolio

Keep predictions before comparison, complete raw records and units, explain competing models and test a changed case. Supplied and simulated data are labeled in each lesson. A saved notebook is not proof of experimental technique or correct reasoning.

Generalized first-branch mechanics

Evidence to retain: Derive a translation/rotation effective inertia and an independent position-force integral. Identify accelerating, decelerating and unreachable states, and distinguish bilateral linkage from tension-only contact.

Boundary: Original rigid-coupling model with a first-forward-branch restriction; no machine construction, real thermal measurement or future friction continuation.

Reconcile generalized force with a first-branch energy integral

Independent calibration and identifiability portfolio

Evidence to retain: Fit loaded-period and independent static data, derive identifiable parameter combinations, predict withheld records and propagate explicit deterministic bounds under the stated independent-mass condition.

Boundary: Supplied linear generator and bounded alternatives; repeated shared-gain evidence is not independent calibration or statistical confidence.

Separate physical parameters from a shared measurement gain

Initial conditions, distributed inertia and damping branches

Evidence to retain: Reconstruct harmonic quadratures, derive a physical-pendulum torque equation and optimum pivot, then match identical release conditions across under/critical/overdamped roots.

Boundary: Linearized and constant-coefficient mathematical models, not physical pendulum motion or observed decay.

Driven phase and coupled normal-mode evidence

Evidence to retain: Close average input/dissipation using the phase quadrant, derive sum/difference modes and retain all kinetic, wall and connector energy terms.

Boundary: Steady driven or separately conservative two-mass models; displacement envelopes alone do not establish energy transfer.

Wave-equation and interface boundary audit

Evidence to retain: Verify partial derivatives, compare material and phase speeds, derive displacement/force matching and impedance-weighted power, then test standing-boundary and clock assumptions.

Boundary: Small-slope ideal strings with disclosed tension and densities; no sound exposure, real impedance or instrument calibration claims.

Integrated fluids and coupled-model domain validation

Evidence to retain: Integrate stratified pressure/buoyancy and a viscous profile, recheck Reynolds validity, compare analytic/numerical draining endpoints and derive a hydrostatic spring stiffness only inside its immersion domain.

Boundary: Stipulated density, viscosity, discharge and preload; rejected domains do not receive invented continuations. No liquids, apparatus or clinical inference.

Axis and density-dependent rotational moments

Evidence to retain: Normalize rod and disk distributions, integrate second moments about declared axes and independently verify a parallel-axis reconstruction. Predict an isolated redistribution using angular rather than kinetic conservation.

Boundary: Prescribed thin rigid distributions and ideal axial redistribution, not material strength, mechanism design or physical handling.

Torque, constraint and origin audit

Evidence to retain: Solve rotational relaxation and close work/loss, derive rolling friction before accepting no-slip acceleration, and verify angular-momentum origin shifts plus a separate nonzero vector-torque derivative.

Boundary: Fixed-axis linear viscous model, bounded ideal contacts and specified inertial origins. Rejected rolling states have no invented slipping continuation.

Central-field accessibility and orbital inference

Evidence to retain: Derive effective-potential stability, reconcile both apsides, retain the exact zero-energy escape boundary and propagate deterministic circular-observation bounds separately from clock gain.

Boundary: Newtonian test particles and normalized hypothetical fields; no finite source surface, atmosphere, relativity, mission planning or physical launch.

Independent numerical and radial-law tests

Evidence to retain: Compare two integrators and step counts using energy/angular extrema and exact endpoint errors. Fit one radial acceleration, then test withheld acceleration and independently supplied potential differences.

Boundary: Synthetic central laws and a noiseless circular benchmark; invariant preservation alone is not accuracy or empirical model identification.

Force histories and subsystem flux

Evidence to retain: Integrate two equal-impulse pulses through velocity and displacement; derive the changing-mass differential momentum account before its logarithmic integral.

Boundary: Original smooth force and abstract force-free mass-expulsion packets; no impact measurements, fuel, apparatus, targeting or physical propulsion procedures.

Normalized extended-mass and independent-observable inference

Evidence to retain: Normalize rod and lamina densities, retain whole/segment first moments and origins, then use an independent mass fraction to infer gradient and bounded coordinate offset.

Boundary: Prescribed positive density families and deterministic error packets, not measured specimens or statistical confidence.

Center motion, relative energy and admissibility

Evidence to retain: Retain internal/external impulse recipients, reconstruct collision velocities from center and relative motion, test planar passive-energy admissibility, and convert relative displacement into ground shifts.

Boundary: Ideal fixed-mass and translation-only models; restitution is supplied, and endpoint constraints do not establish microscopic conversion or motion histories.

Path integrals and potential reconstruction

Evidence to retain: Retain segment integrals, loop orientation and an explicit potential in the conservative subfamily. Derive nonlinear equilibrium/curvature and preserve the zero-quadratic-curvature exception.

Boundary: Original full-plane and one-dimensional mathematical potentials; no inferred physical actuator or material system.

Multidirectional stability and allowed energy sets

Evidence to retain: Solve turning roots and distinguish disconnected wells, exact-barrier asymptotics and above-barrier accessibility. Compare two independent coordinate slices rather than certifying stability from one.

Boundary: Exact ideal potentials; accessibility is not a numerical time trajectory, and no quantum tunneling or finite-temperature activation is included.

Work, rate, frame and system-boundary accounts

Evidence to retain: Integrate trajectory power, close one drag conversion, transform work and kinetic changes together and test a moving normal-force work claim with an actual agent diagram.

Boundary: Prescribed force/medium/support models, no device efficiency, observed heat distribution, real lifting procedure or safety rating.

Energy diagnostics and genuinely withheld model evidence

Evidence to retain: Compare numerical energy and phase-space errors on two grids, then fit one potential difference before predicting an unused energy difference and separately supplied force.

Boundary: Finite harmonic numerical benchmark and synthetic independent packets; conservation diagnostics alone do not establish physical or numerical accuracy.

Recipient equations and differential resistance evidence

Evidence to retain: Retain agent/vector sums, integrate a time ramp with initial conditions, derive linear and quadratic resistance solutions and discriminate laws using a genuinely unused speed. Compare a two-component drag trajectory with its derivative and initial-state constraints.

Boundary: Original mathematical/synthetic force packets, not measured actuator identities, physical falls or universal drag regimes.

Geometric and unilateral constraint transitions

Evidence to retain: Differentiate the cord-length constraint, solve both recipients, carry continuous state across a nonnegative-normal boundary and reject an inaccessible first-branch spatial target.

Boundary: Massless ideal cord, one-sided support and bounded monotonic branch; no physical pulley, drop or force apparatus.

Solver versus model discrepancy

Evidence to retain: Derive Euler amplification limits, retain intermediate sign errors and compare two Heun grids with both an exact assumed-model solution and an independent changed-coefficient record.

Boundary: Finite CPU teaching algorithms. Stability and convergence do not authenticate force coefficients or prove practical accuracy.

Derivative and integration evidence with uncertainty

Evidence to retain: Retain dimensioned derivatives and time-coordinate rescaling, three finite-difference spacings with separate truncation/noise bounds, and a reversal-split absolute integral with both initial conditions.

Boundary: Supplied smooth polynomials and bounded synthetic observations, not validated motion sensors or measured experimental forces.

Vector geometry and observer comparison

Evidence to retain: Derive dot/cross projections and verify component-square identities; differentiate moving circle basis vectors; compare the same events under constant-velocity translation without treating overlaid paths as two objects.

Boundary: Nonzero-speed tangent geometry and Newtonian inertial observers only; rotating observers and physical force constraints are not simulated.

Algorithm order and held-out model testing

Evidence to retain: Preserve two grids and three update/quadrature rules, exact benchmark errors and convergence ratios. Freeze three fitted observations before evaluating a withheld fourth; separate common offsets, independent bounded errors and omitted jerk.

Boundary: Noise-free quadrature and declared synthetic errors; numerical exactness in a special polynomial does not establish universal accuracy, impact physics or practical proficiency.

Mixed assessment and cumulative project

Open the retained response and project workspaces. Practice, saved work and independently reviewed proficiency remain separate.

Mixed assessment and retained reasoning

These 39 course-specific public tasks are practice, not a secure examination. Each retains up to three immutable written responses, known feedback exposure before each response, and human rubric critiques. Drafts remain editable. No text-length check or self-reported review certifies correctness, independence, laboratory skill or mastery.

1. A reversal, a quadrature error and a moving tangent

In consistent SI, x(t)=t³−3t on0≤t≤2. Derive velocity, reversal time, signed displacement and total distance. Apply a composite trapezoid to velocity at t0,1,2 and compare its displacement with the analytic integral. Separately r(t)=(t,t²/2) m in its dimensioned polynomial convention: find speed and tangential/normal acceleration magnitudes at2s.

Course-specific review criteria

Differentiate and split the absolute-velocity integral at the actual reversal.
Distinguish quadrature truncation from observation/model error.
Resolve acceleration along and perpendicular to the instantaneous velocity.

Your solution draft for A reversal, a quadrature error and a moving tangent

Include calculations with units, source or dataset provenance, assumptions, competing explanations and limitations as relevant. The draft is saved but is not a locked response until submitted.

0/3 responses retained. No recorded feedback exposure for this task version. Earlier versions may already be familiar. Public inspectability still prevents certification of unfamiliarity.

2. A resistance solution and a numerical stability claim

Solve2dv/dt=4−v from rest and integrate displacement from zero. Evaluate at2s. For the relaxation error equation eprime=−e/2, compare the exact one-step amplification with forward Euler at step3s; is the Euler error decay monotone in sign? Separately a contact model would require normal force−2N: can it be accepted by clipping the force while retaining the constrained trajectory?

Course-specific review criteria

Apply initial conditions to both velocity and position.
Separate magnitude stability from sign-preserving approximation.
Reject an unavailable unilateral force before extending its trajectory.

Your solution draft for A resistance solution and a numerical stability claim

Include calculations with units, source or dataset provenance, assumptions, competing explanations and limitations as relevant. The draft is saved but is not a locked response until submitted.

0/3 responses retained. No recorded feedback exposure for this task version. Earlier versions may already be familiar. Public inspectability still prevents certification of unfamiliarity.

3. Path dependence and a disconnected accessible set

Integrate F=(2y,x) from(0,0) to(1,2) on east-then-north and north-then-east paths. Find the loop using first forward/second backward. Separately U=x⁴/4−x² at E−.75: find the right-well interval and explain the exactE0 barrier timing. Under an inertial frame shift Uobserver=2m/s and net impulse3kg m/s, how does net work change?

Course-specific review criteria

Retain segment orientations and field components.
Solve turning roots before assigning a connected trajectory.
Distinguish energy accessibility from finite-time passage.
Transform work and kinetic energy consistently.

Your solution draft for Path dependence and a disconnected accessible set

Include calculations with units, source or dataset provenance, assumptions, competing explanations and limitations as relevant. The draft is saved but is not a locked response until submitted.

0/3 responses retained. No recorded feedback exposure for this task version. Earlier versions may already be familiar. Public inspectability still prevents certification of unfamiliarity.

4. A normalized rod and a collision energy split

Normalize a3kg,2m rod with lambda=lambda0(1+x/L); find center and inertia about its left end and center. Two particles of masses2 and1kg approach at3 and−1m/s with supplied restitution.5: find final velocities and kinetic loss. Separately derive the academic force-free mass-flux speed change for relative expelled speed100m/s and final mass fraction.8.

Course-specific review criteria

Integrate normalized moments with the correct domain.
Reconstruct final velocities from center and separation speed.
Do not confuse linear momentum with translational energy conservation.
Include expelled matter in the closed momentum boundary.

Your solution draft for A normalized rod and a collision energy split

Include calculations with units, source or dataset provenance, assumptions, competing explanations and limitations as relevant. The draft is saved but is not a locked response until submitted.

0/3 responses retained. No recorded feedback exposure for this task version. Earlier versions may already be familiar. Public inspectability still prevents certification of unfamiliarity.

5. Distributed rotation and two orbital turning points

A4kg disk radius1m has density sigma0(1+r²/R²). Normalize and derive its axial inertia. A constant net torque1N m acts through2rad from rest: find work and final angular speed. Separately an ideal mu1 orbit has rp2,ra4: find h,E,e, both apsidal speeds and period; explain why conserved numerical angular momentum alone does not certify an exact orbit after that period.

Course-specific review criteria

Include annular area in both normalization and inertia.
Integrate net torque over angle.
Use common apsidal energy and angular momentum with center-distance radii.
Keep a numerical invariant distinct from exact time-parametrized motion.

Your solution draft for Distributed rotation and two orbital turning points

Include calculations with units, source or dataset provenance, assumptions, competing explanations and limitations as relevant. The draft is saved but is not a locked response until submitted.

0/3 responses retained. No recorded feedback exposure for this task version. Earlier versions may already be familiar. Public inspectability still prevents certification of unfamiliarity.

6. Three oscillator boundaries and an impedance check

For critical free damping c4,m1,k4,x0=1,v0=0, find x,v at.5s. For a separate steady drive.2cos(2t) with c.5,m1,k4, find amplitude,lag and average dissipated power. A string interface has impedances2 and1: derive signed displacement coefficients and power fractions. What density is inferred from true.1kg/m if the elapsed-time gain is.9 in the fixed-string period method?

Course-specific review criteria

Keep free initial-value and driven steady-state systems separate.
Include the response phase when checking power.
Weight transmitted power by impedance, not amplitude alone.
Propagate elapsed-time gain in the right direction.

Your solution draft for Three oscillator boundaries and an impedance check

Include calculations with units, source or dataset provenance, assumptions, competing explanations and limitations as relevant. The draft is saved but is not a locked response until submitted.

0/3 responses retained. No recorded feedback exposure for this task version. Earlier versions may already be familiar. Public inspectability still prevents certification of unfamiliarity.

7. Integrated profiles and a mixed-model limit

For rho=1000(1+.4z),g10, find buoyancy of a.01m² slab from depth.5 to1m. For a laminar tubeR.003,L1,Delta p100,eta.001,rho1000 in SI, derive mean speed andRe. Find exact drain time at area.003 using the taught tank packet. Finally derive combined stiffness and frequency of the taught partially immersed oscillator at density1200 and amplitude.1; explain why a larger amplitude crossing.15 needs a new branch.

Course-specific review criteria

Integrate pressure and compare displaced-fluid weight.
Recheck a conditional laminar profile against its regime.
Respect the draining endpoint and distinguish numeric/model changes.
Check full immersion range before applying a linear oscillator.

Your solution draft for Integrated profiles and a mixed-model limit

Include calculations with units, source or dataset provenance, assumptions, competing explanations and limitations as relevant. The draft is saved but is not a locked response until submitted.

0/3 responses retained. No recorded feedback exposure for this task version. Earlier versions may already be familiar. Public inspectability still prevents certification of unfamiliarity.

8. Generalized work and independently identifiable calibration

For the taught generalized translation/rotation model, use drive gain1.1 and requested endpoint2m. Derive net kinetic energy, speed,drive/resisting works and first turning point; identify the bilateral-force premise. Separately static readings3 and5.5N at extensions.1,.2m accompany loaded-period data generated byk25,g.95,m2. Derive the static parameters, the dynamic identifiable combinations and how independent stiffness recoversg. State the extra mass premise behind the displayed deterministic clock bounds.

Course-specific review criteria

Reconcile integrated work with effective inertia and branch accessibility.
Distinguish rigid bilateral coupling from a pull-only cord.
Derive parameter combinations before claiming separate identification.
State error-bound independence and the additional exact-mass condition.

Your solution draft for Generalized work and independently identifiable calibration

Include calculations with units, source or dataset provenance, assumptions, competing explanations and limitations as relevant. The draft is saved but is not a locked response until submitted.

0/3 responses retained. No recorded feedback exposure for this task version. Earlier versions may already be familiar. Public inspectability still prevents certification of unfamiliarity.

9. Initial conditions and distributed-pivot period

For m2,k8,x0=.1,v0=.2 in SI, derive x(t),v(t), amplitude, cosine phase and energy; find position/velocity after a quarter period. Separately derive a uniform L2,M3 physical pendulum with pivot distance .5 from its center at g10, and find its small-angle period. Find the pivot distance minimizing that period and explain why a point-mass approximation fails.

Course-specific review criteria

Match both initial quadratures and phase quadrant.
Check conserved energy independently of the phase form.
Use full pivot inertia in the torque equation.
Differentiate the period-squared expression with a domain-qualified minimum.

Your solution draft for Initial conditions and distributed-pivot period

Include calculations with units, source or dataset provenance, assumptions, competing explanations and limitations as relevant. The draft is saved but is not a locked response until submitted.

0/3 responses retained. No recorded feedback exposure for this task version. Earlier versions may already be familiar. Public inspectability still prevents certification of unfamiliarity.

10. A repeated root is not a steady driven response

Solve xddot+4xdot+4x=0 with x0=1,v0=0 and evaluate x,v at1s. Derive its energy-rate sign. Separately for xddot+xdot+4x=.2cos(2t) after transients, find amplitude, phase lag, average input and dissipated power. Explain why the critical free solution and driven sinusoid cannot be combined as if they were the same experiment.

Course-specific review criteria

Use the repeated-root polynomial factor and both initial conditions.
Derive the energy-rate identity.
Preserve force-relative phase convention and quadrant.
Separate free initial-value and steady externally driven boundaries.

Your solution draft for A repeated root is not a steady driven response

Include calculations with units, source or dataset provenance, assumptions, competing explanations and limitations as relevant. The draft is saved but is not a locked response until submitted.

0/3 responses retained. No recorded feedback exposure for this task version. Earlier versions may already be familiar. Public inspectability still prevents certification of unfamiliarity.

11. Normal coordinates and a traveling wave are different representations

For two1 kg masses with wall springs4 and connector1 N/m, derive symmetric/antisymmetric frequencies. Initial positions(.1,0), velocities0: reconstruct both positions and total initial energy including connector. Separately a string T40,mu.1 has y=.02cos(x−20t). Verify its wave equation, maximum material speed and average traveling power. Explain why a displacement zero alone does not certify zero energy in either model.

Course-specific review criteria

Derive sum and difference equations rather than guess mode frequencies.
Retain connector energy and both kinetic terms.
Use partial derivatives and distinct phase/material velocities.
Do not infer energy from displacement alone.

Your solution draft for Normal coordinates and a traveling wave are different representations

Include calculations with units, source or dataset provenance, assumptions, competing explanations and limitations as relevant. The draft is saved but is not a locked response until submitted.

0/3 responses retained. No recorded feedback exposure for this task version. Earlier versions may already be familiar. Public inspectability still prevents certification of unfamiliarity.

12. Amplitude matching and a harmonic calibration alternative

Two strings share tension40 with linear densities.1 and.4 kg/m. Derive signed displacement reflection/transmission coefficients and both power fractions. For incident amplitude.03 m find reflected/transmitted displacements. Separately a fixed-fixed2 m string with T40,mu.1 has third-mode frequency read by an elapsed-time gain1.1 clock. Find true/read frequency, node spacing and uncorrected density; compare the first fixed-free mode at the same parameters.

Course-specific review criteria

Match displacement and transverse force before power.
Include impedance ratio in transmitted flux.
Propagate elapsed-time gain inversely into frequency.
Keep distinct endpoint conditions and independent density reference.

Your solution draft for Amplitude matching and a harmonic calibration alternative

Include calculations with units, source or dataset provenance, assumptions, competing explanations and limitations as relevant. The draft is saved but is not a locked response until submitted.

0/3 responses retained. No recorded feedback exposure for this task version. Earlier versions may already be familiar. Public inspectability still prevents certification of unfamiliarity.

13. Integrated pressure and a conditional velocity profile

Take rho(z)=1000(1+.2z),g10 with depth downward and surface gauge zero. Find gauge pressures at1 and1.5m and buoyancy of a.01m² vertical slab between them. Separately derive the laminar profile for R.002m,Delta p200Pa,L1m,eta.001,rho1000; find mean/center speed,Q and Reynolds number. Double radius at fixed other parameters and audit the model premise.

Course-specific review criteria

Integrate nonconstant density and match displaced volume.
Derive shear/profile with no-slip condition.
Integrate annular flow and distinguish center/mean speed.
Recheck Reynolds scaling before accepting the extrapolation.

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14. A draining endpoint and an immersion-domain limit

Derive the emptying time for H0=1,Atank.1,outlet.002,Cd.6,g10 using the stated quasisteady model. Explain why continuing the squared height after emptying is invalid and how a numerical event differs from the exact event. Separately a2kg cylinder with area.01,height.3,equilibrium depth.15 and spring20 N/m is in density1000 liquid. Derive incremental stiffness, frequency and energy at amplitude.1; decide whether the same linear model covers amplitude.2.

Course-specific review criteria

Separate the ODE with the correct sign and domain endpoint.
Retain numerical event error rather than overwrite it with the exact answer.
Derive buoyancy-force change about equilibrium.
Check the whole immersion range and calibration identifiability.

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15. Two different second-moment integrals

For a3 kg,2 m rod with density lambda0(1+2x/L), normalize lambda0 and derive inertia about x=0, the mass center, and x=1 m. Separately a4 kg disk of radius2 m has density sigma0(1+2r²/R²); derive axial inertia and mass inside half radius. At fixed mass/radius, an ideal isolated redistribution changes this disk from beta0 to beta2. Find the angular-speed and rotational-energy ratios if external axial torque is zero.

Course-specific review criteria

Normalize density before taking second moments.
Use the correct axis and annular area factor.
Verify parallel-axis reconstruction.
Distinguish angular conservation from rotational kinetic conservation during internal redistribution.

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16. Rotational loss is not unavailable rolling friction

For I2, drive torque4 and viscous coefficient1 in SI, derive speed and angle from rest and evaluate them at2 s; close drive work as rotational gain plus loss. Separately a body with I=.5mR² is released on a40 degree stationary incline with g10 and static coefficient.2. Derive the coefficient required for no-slip rolling and decide whether its acceleration can be accepted. Explain why zero contact work in feasible ideal rolling does not mean zero center torque.

Course-specific review criteria

Derive the first-order response with initial conditions.
Integrate angle before multiplying by constant torque.
Test the contact-force bound before imposing no slip.
Separate total contact work from center-translation and rotation accounts.

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17. Conserved quantities still need an origin

A force-free2 kg point has r=(1,3t,0) m. Compute angular momentum about fixed origins(0,0,0) and(2,0,0). Explain why different results do not require torque. Separately use dimensioned polynomial r=((1 m/s)t,(1 m/s²)t²,0) for2 kg: derive momentum, angular momentum and net torque about the origin and verify dL/dt at t=1s.

Course-specific review criteria

Use the vector cross product and named fixed origins.
Apply the origin-shift relation with the momentum vector.
Differentiate the dimensioned trajectory before forming force.
Compare torque with the angular-momentum derivative, not L itself.

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Related notebook: Distinguish conservation from independence of origin

18. A potential minimum and two apsides

For mu1 and specific angular momentum h1, derive Ueff, stationary radius and its curvature. At r1 compare radial motion for energies−.5 and−.2. Separately reconstruct an ideal ellipse with rp1,ra3: find h,E,e, both apsidal speeds and period. Verify energy at both apsides rather than relying only on a geometric sketch.

Course-specific review criteria

Include angular kinetic energy once in Ueff.
Distinguish potential minimum from the actual energy/initial state.
Use common energy and angular momentum at both apsides.
Keep geometric ellipse parameters distinct from uniform elapsed time.

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19. An exact energy boundary and an imperfect period

For radial outward motion in mu1 at r0=2, compare speed fractions.5,1,1.5 of local escape speed: give energy, finite turning radius or asymptotic speed as applicable. Separately a circular orbit with true radius2 and mu1 has period read by a gain1.1 clock. Find uncorrected inferred mu. Explain how to propagate radius2±.02 and indicated period±.1 deterministically, and why those bounds do not automatically cover the omitted gain.

Course-specific review criteria

Handle exact escape analytically rather than by an arbitrary tolerance.
Separate finite turning from asymptotic speed and travel time.
Derive circular period inference and its clock scaling.
Use monotone corners and disclose the omitted systematic effect.

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20. Conservation cannot replace a withheld prediction

For the unit circular benchmark, name its exact one-period endpoint, energy and angular momentum and explain why preserving the last one is insufficient for phase accuracy. Separately a synthetic inward law is1/r²+.2/r³. Fit a pure inverse-square coefficient at r1, predict inward acceleration at r2 and compare with the generator. Derive the independent potential difference1→2 and its fitted residual. Explain why both fields conserve central angular momentum.

Course-specific review criteria

Keep independent exact reference separate from computed invariants.
Fit only the stated calibration datum.
Integrate the correct potential sign and cancel common offsets.
Distinguish centrality from the inverse-square radial form.

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21. Equal impulse and changing subsystem mass

A 2 kg point starts at −1 m/s. Compare net pulses A sin(pi t/T) with (A,T)=(4 N,1 s) and (2 N,2 s). Integrate impulse, final velocity and displacement; explain what equal impulse does not establish. Separately derive the force-free academic mass-flux velocity change for relative expelled speed 100 m/s and retained fraction 0.8. Explain why zero external force does not keep the retained subsystem velocity fixed.

Course-specific review criteria

Integrate signed pulse and velocity separately.
Distinguish endpoint momentum from trajectory.
Derive the open-subsystem flux term before integrating.
Name the closed retained-plus-expelled boundary.

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22. A partial rod and a whole lamina

Normalize a 3 kg,2 m rod with density lambda0(1+2x/L). Find its full center, left-half mass and left-half center in the original coordinate. Separately normalize a 4 kg rectangle x=0..2,y=0..1 with density sigma0(1+x+y), interpreting inverse-metre gradient coefficients. Find both center coordinates. Explain why the segment center cannot use full mass as its denominator.

Course-specific review criteria

Normalize each density using its specified total mass.
Integrate first moments over the correct domain.
Use the segment mass for its own center.
Retain units and coordinate origins.

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23. Internal impulses do not certify elastic collisions

Masses1,2,3 kg initially move at1 m/s. Give the first pair internal impulses+2 and−2 N s and the third external impulse3 N s. Find velocities, total momentum and center velocity. Separately two2 kg particles approach at3 and−1 m/s with supplied e=.5. Find final velocities and decompose kinetic energy before/after into center and relative terms. State what is not known about the loss mechanism.

Course-specific review criteria

Include both internal recipients in cancellation.
Compute center velocity from total momentum.
Use separation speed e times approach speed.
Do not infer a microscopic mechanism from translational loss alone.

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24. Independent constraints on outgoing vectors and relative shifts

A2 kg point enters at(3,0) m/s and a1 kg point rests. Test proposed first outgoing velocity(3,1) m/s against momentum and a passive no-stored-energy-release premise. Derive the full admissible disk for that first velocity. Separately a40 kg point moves3 m relative to an80 kg translating base with stationary whole-system center. Find both ground displacements and explain why work cannot be inferred.

Course-specific review criteria

Solve both momentum components.
Check energy independently and complete the square.
Convert relative displacement using the weighted center constraint.
Separate endpoint constraints from interaction histories.

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25. Propagate a mass-fraction interval before inferring origin shift

For the normalized2 m linear-density rod, an independent left-half fraction is .375±.005 and the reported center is19/15 m, exact except for unknown additive offset. Derive the inverse alpha(f), propagate its bounds, then infer offset bounds. State the fixed-family and sensor assumptions; distinguish this from a probability interval.

Course-specific review criteria

Derive inverse from the mass fraction rather than reuse the coordinate.
Reverse endpoints for the decreasing inverse.
Propagate gradient bounds through the increasing center function.
Disclose deterministic bounds and gain/tare assumptions.

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26. A loop and a quartic equilibrium are different tests

For F=(−y,x) N with coordinates in metres, integrate work from(0,0) to(1,2) east-then-north and north-then-east. Find loop work using the first forward and second backward. Explain why an endpoint-only potential cannot represent this full field. Separately U=−x²/2+x⁴/2 J: derive force, equilibrium positions, potential minima and their curvatures. What changes at U=x⁴/2?

Course-specific review criteria

Parameterize each segment and retain signed orientation.
Distinguish disproof by nonzero loop from proof by one accidental equality.
Derive force and test stationary-point curvature.
Inspect higher-order terms when the second-derivative test vanishes.

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27. Two wells and two directions

For U=x⁴/4−x² and unit mass, find turning magnitudes at E−.5 J, identify the accessible right-well interval and initial speed at its minimum. Contrast E = 0 and E+.25 for central-barrier crossing. Separately classify U=(x²−y²)/2 at the origin and calculate force at(1,1). Does a positive x curvature prove planar stability?

Course-specific review criteria

Solve the quadratic in x² and preserve disconnected intervals.
Separate kinetic nonnegativity from finite-time separatrix crossing.
Check both coordinate directions and derive gradient force.
Avoid inferring a full trajectory or stable plane from one favorable scalar slice.

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28. Integrate changing power and count one dissipation

A unit-mass point starts from rest with net force F=2t N. Derive v,x,P and net work through2 s; compare endpoint and average power and the incorrect endpoint-force-times-distance product. Separately m2,bdrag1,v(0) = 4 undergoes free linear decay for2 s: find final kinetic energy and integrated drag work, then close a point-plus-medium account.

Course-specific review criteria

Use the same trajectory in both Newton-law and power calculations.
Distinguish rate from interval energy and endpoint from average.
Integrate dissipative power with a declared stationary medium.
Count mechanical loss/internal conversion once for the chosen system boundary.

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29. The sign of work depends on motion and frame

For netF2 N,m2,T = 2 s,v(0) = 0, compare ground work and work seen by observerU2 m/s; verify both from endpoint kinetic energies. Separately a = 2 kg point rests initially on a support accelerating upward1 m/s² for2 s under g10. Calculate normal work, gravitational gain and kinetic gain. Explain why neither a translating frame nor a normal-force label permits dropping a required energy term.

Course-specific review criteria

Transform displacement and both endpoint velocities consistently.
Derive the common−UΔp correction rather than mixing frame terms.
Use actual contact-point displacement for constraint work.
Close gravitational and kinetic accounts without double counting weight work.

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30. A stable-looking energy record is not independent validation

For xdot=v,vdot=−x with x(0) = 1,v(0) = 0, derive the forward-Euler energy factor and its energy after10 steps of1 s. Why should a bounded or nearly exact endpoint energy from another method still be checked against exact phase? Separately U=x²+.2x⁴+C: fit a quadratic from U = 1−U = 0, predict U = 2−U = 0 and compare the separately supplied force atx1. State what changing C can repair.

Course-specific review criteria

Derive energy growth from the actual update rather than calling it a physical source.
Keep endpoint, interval maximum and phase errors distinct.
Freeze the energy fit before using withheld energy and force packets.
Distinguish additive gauge freedom from wrong spatial dependence.

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31. One recipient and a separate time-ramped drive

A3 kg point follows r=(t²,t³/2) with gravity(0,−30) N. At2 s infer net force and the one-actuator remainder. Explain where its third-law partner acts. Separately a2 kg point with x0=0,v0=1 has net force F=−2+2t N: derive v,x and find both at2 s; compare average force and endpoint force.

Course-specific review criteria

Differentiate a vector trajectory twice and sum only forces on the selected recipient.
Keep the inferred remainder conditional on the given agent list.
Integrate the force with original initial conditions and coefficient units.
Distinguish time-average and endpoint forces.

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32. Terminal balance, a transient and a withheld law test

For m2 kg,F10 N,b1 kg/s and rest initial condition, derive linear-resistance v,x, tau and terminal velocity; evaluate at2 s. Separately a1 kg point falls from rest with k.1 kg/m,g10: find terminal speed and its speed at1 s using the supplied tanh solution. Two candidate resistance laws F=Fref(v/1 m/s)^p share Fref1.5 N at the reference; predict the withheld force at2 m/s for p1 and p2. Explain why their coefficients have different units if written directly as powers of v.

Course-specific review criteria

Derive the transient with mass and initial condition rather than applying terminal balance at every time.
Keep linear and quadratic force laws and coefficient dimensions distinct.
Use the declared falling sign branch.
Freeze the reference fit before comparing the unused speed.

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33. Independent component equations versus a shared geometric constraint

A linear-drag trajectory begins at(0,5), velocity(4,6),g10,gamma.5. Evaluate position and velocity at1 s from the derived component solutions and state why downward velocity need not mean below-ground position. Separately ideal pulley masses3 and2 kg use first-upward/second-downward positive coordinates. Find shared signed acceleration and common tension; identify the assumption that makes both tensions equal.

Course-specific review criteria

Carry both position initial conditions through component integration.
Distinguish vector velocity signs from position coordinates.
Derive the cord constraint and separate recipient equations.
State massless/ideal pulley assumptions rather than assuming any real pulley has equal tension.

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34. Two different branch boundaries

A2 kg point starts on support ys=−10t³/6 with g10,upward positive. Find separation time and carry its position/velocity to t2 under gravity only; compare support position and gap. Separately a2 kg point starts x0=0,v0=3 under F=−4x. Derive v²(x), find the first rightward turn and compare requested targets2 and2.5 m. Explain why neither a negative normal nor negative squared speed is an allowed state.

Course-specific review criteria

Solve the nonnegative-contact condition before switching equations.
Carry continuous event position/velocity rather than resetting initial conditions.
Derive the spatial chain rule only on a checked monotonic branch.
Report rejected endpoints instead of assigning physically impossible values.

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35. A stable algorithm may still solve the wrong equation

For dv/dt=−v,v0=10,derive the Euler amplification and classify r=.5,1.5,2,2.5. For r1.5 and six steps compare the numerical endpoint with exact velocity at the same elapsed time. Separately Heun with four steps to2 s assumes coefficient1 and gives1.52587890625 m/s; the exact assumed endpoint is10e⁻² and independent observation uses coefficient1.25,10e⁻²·⁵. Split total signed discrepancy into numerical and model parts. What can further grid refinement establish?

Course-specific review criteria

Distinguish monotone stable, oscillatory stable, neutral and growing numerical behavior.
Compare exact and numerical results at the same elapsed time.
Keep discretization and independent-model discrepancy separate with signed addition.
Do not claim convergence proves the physical coefficient or force law.

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36. Two derivatives, two integration constants and a reversal

For x=1−2t+.5t³ in metres with t seconds, find x,v,a at2 s and rewrite x using tau in minutes. Separately a=2t m/s², v0=−2 m/s and x0=1 m: derive v,x and find displacement and traveled distance over0–2 s. Explain why the two trajectories are not the same despite both containing cubic terms.

Course-specific review criteria

Preserve coefficient dimensions and rescale all time powers.
Differentiate each term and distinguish position from speed.
Retain both initial conditions during integration.
Locate the velocity zero and split the distance integral.

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37. A derivative estimate needs an error and reference model

For x=t³ at t2 s, use symmetric h=.5 s with independent endpoint errors bounded .05 m. Derive the noiseless estimate and feasible measured interval; compare h=.1. Separately fit a quadratic to t0,1,2 for x=1+2t+1.5t²+.6t³/6−.1. Predict the withheld t3 residual and state what a common position offset can change. Is the ±.04 m/s² bound from independent .01 m fitting errors a bound on the omitted jerk bias?

Course-specific review criteria

Derive truncation and independent endpoint-error terms separately.
Compare the full bound rather than only h.
Use only fitting points for fitted coefficients and retain the withheld residual.
Keep common offset, independent bounded errors and model bias distinct.

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38. Project a vector derivative and differentiate the basis

For r=(3t,t²/2), find speed, tangential and normal acceleration at2 s and verify their squared sum. Separately radius2 m and theta=t²/2: find speed and the tangent/inward acceleration magnitudes at1 s. Finally a different ground trajectory r=(t²,t³/2) is observed from constant U=(2,2): find relative velocity and acceleration at2 s. State which claims would fail for rotating axes.

Course-specific review criteria

Use dot and cross projections rather than acceleration magnitude as speed derivative.
Differentiate changing unit vectors with the correct inward sign.
Apply the observer transformation to components before magnitudes.
State nonzero-speed and inertial/parallel-axis assumptions.

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39. Error order is a conditional benchmark claim

For x0=0,y0=1,v0=(3,4),a=(0,−10), compare endpoint height at1 s using N20 forward Euler, velocity-first and midpoint-velocity updates. Find exact final vertical velocity and explain why it cannot validate position accuracy alone. Separately integrate v=1+t² over0–2 with N2 then N4 trapezoids and find the error ratio. Explain the scope of Simpson exactness.

Course-specific review criteria

Use the actual update order and retain signed position errors.
Distinguish exact endpoint/event benchmark from a physical impact response.
Compute a quadrature error against an independent analytic integral.
Qualify exactness and convergence by the supplied function and observation model.

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Retain a criterion-by-criterion human review

Reviewer identity and role are declarations, not authenticated credentials. Unsaved review fields below are lost on navigation. Review every criterion with an exact quotation, calculation or evidence citation and a next step.

Task to review

University Physics I final calculus evidence portfolio

Assemble a versioned first report spanning all six finite SC07 categories. Use these concrete original packets: x=t³−3t on0..2; resistance2dv/dt=4−v from rest; the F=(2y,x) two-path work; normalized alpha1 rod and supplied restitution collision; rp2,ra4 orbit; critical c4 release and string impedances2/1; beta.4 hydrostatic slab and the taught draining endpoint; generalized drive gain1.1 at2m plus static readings3/5.5N and loaded-period calibration. This64-destination authored scope does not certify practical work or independent learning.

Freeze equations, observations and first predictions
Record initial/boundary conditions, dimensions, frames, axes and observation error assumptions. Label which supplied records calibrate a model and which are genuinely withheld. Derive actual force/torque or field equations before reducing them.
Evidence to retain: Raw/derived tables, recipient diagrams, initial predictions and source/modality declarations.
Reconcile independent analytic accounts
Carry at least one motion integral through work/energy and one extended-system momentum/rotation account. Verify orbit energy/angular constraints and an oscillator or wave boundary using an independent derivation rather than repeated formula substitution.
Evidence to retain: Calculus derivations, conservation checks and explicit finite-domain or unilateral rejection.
Separate numerical, observation and model errors
Use the exact circular or draining benchmark on two grids, preserving its numerical endpoint error. Separately retain the static/dynamic calibration triad and one radial-law or fluid-regime rejection. Do not call a common-gain repeated sample independent evidence.
Evidence to retain: Exact benchmark, two numerical grids, deterministic bounds, rival models and unused-observation residuals.
Human criterion critique and retained revision
A reviewer cites concrete report passages and criteria, identifying an error or explicitly justified no-error result. Preserve the first report and critique before editing a separate revised report; identify changed claims and remaining limitations.
Evidence to retain: Reviewer relationship and expertise limits, criterion-linked feedback, immutable first report and reasoned revision.
Actually frozen unfamiliar cross-category response
A reviewer supplies new numerical observations across at least two categories plus a changed domain, calibration or boundary assumption and allowed resources. Freeze the complete assignment before the response, record assistance and public-key exposure, and retain the criterion review.
Evidence to retain: Actual unfamiliar data, independent response declaration, human review and unresolved gates; a generic instruction to change values is not an assignment.

Project review criteria

Every finite category has concrete derivations and retained actions
Initial/boundary conditions, dimensions, frames and axes are consistent
Independent energy, linear/angular momentum and fluid/wave accounts agree
Unilateral, stability, accessibility and regime limits are checked
Numerical truncation, observation bounds and model inadequacy remain separate
Actual first critique revision and frozen unfamiliar transfer are retained
Authored pages are not practical certification, college credit or independent mastery

Course-specific project report draft

Retain raw-data identifiers, provenance, units, prior predictions, methods, calculations, competing explanations and safety boundaries. State explicitly when data are simulated, supplied or genuinely observed. This saved text cannot verify a real experiment; do not invent measurements or completed supervised work. Maximum 12,000 characters; retain larger evidence files separately and cite them.

A nonempty report is only a saved submission. First and revised reports remain separate and cannot be overwritten. A critique of the first report is required before retaining one revision.

Fresh transfer direction

The new packet must combine actual changed numbers with a meaningful alteration to the force/system boundary, initial state, measurement gain, constitutive law or continuum regime. Keep prior public keys visible as exposure rather than claiming secure testing. Missing review or practical evidence remains missing.

A human must supply the actual new task, evidence and conditions before the learner responds. The transfer-direction text is not itself an unseen assessment. Use the retained assignment form after the first project; exposed public task identifier: expanded-public:university-physics1.

Science backups preserve each course's separate responses, exposure, rubric reviews and project history. Older backups do not reset attempts or erase known exposure. Divergent same-version immutable responses and incompatible same-version project histories are rejected rather than silently replaced; preserve both files for review.

Sources and boundaries

OpenStax University Physics1 — oscillations, waves and fluids: Checked 30 October 2026 with15.4–15.6 initial/forced oscillations,16.2/16.4/16.6 wave equations/power/boundaries and14.1/14.4/14.6/14.7 integrated fluid models. Original functions and data.
MIT 8.03 Lecture4 — coupled normal coordinates: Primary mode-superposition method; original two-mass coefficients and initial conditions derived locally.
Richard Fitzpatrick, University of Texas — string interface matching: Displacement/impedance and energy-flux matching; original signed-amplitude and power packets, no copied exercises.
OpenStax University Physics1 — rotational moments and central orbits: Checked 30 October 2026 with10.8 rotational work,11.1–11.3 rolling/angular momentum and13.3–13.5 gravitational energy/orbits. Original density, effective-potential, inverse-bound and numerical evidence tasks.
OpenStax University Physics Volume 1 — momentum and mass moments: Checked 30 October 2026 with9.2 impulse,9.3 momentum,9.4 collisions and9.7 mass flux; original density, inverse-error and energy-admissibility packets.
Andrew Cumming, PHYS 512 — symplectic integrators: Checked 30 October 2026: velocity-first Euler and kick-drift-kick velocity Verlet update definitions; our harmonic energy/phase packets and numerical derivations are original.
OpenStax University Physics1 — line work and potential structure: Primary potential/stability concepts with7.1 line work,7.4 power and8.2 conservative forces; original multidirectional, separatrix, numerical and withheld-packet derivations.
OpenStax Calculus Volume 2 — Euler method: Checked 30 October 2026: first-order numerical method; stability and Heun comparisons are derived explicitly for the original bounded relaxation model.
OpenStax University Physics1 — force equations and resistance: Primary resistance regimes; original differential solutions, branch transitions and numerical-error derivations with6.1 recipient/constraint reasoning.
OpenStax Calculus Volume2 — numerical integration: Composite trapezoid and Simpson weights/error-order context; original quadratic exact-error derivations and bounded samples.
OpenStax University Physics Volume1 — derivatives and acceleration: Derivative-based kinematics and local versus interval interpretation. All cases and data original.
OpenStax University Physics Volume1 — integral kinematics: Definite accumulation and initial conditions; original exact versus quadrature/integrator comparisons.
OpenStax University Physics Volume1 — vector derivatives: Component derivatives and original tangent/normal extension derived explicitly.
OpenStax University Physics Volume1 — nonuniform circular motion: Moving basis, tangential and inward contributions; no force mechanism claimed.

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