Investigation inventory: 13 original paper/data cases
25 supplied datasets, 39 level-specific tasks with checked solutions, 82 numerical calibrations and 13 changed-input transfers. Counts describe authored material, not performed experiments.
These are supplied-data and paper-model investigations, not laboratory procedures. No launches, high-speed rotors, elevated loads, pressurized vessels, hydraulic builds or improvised apparatus are assigned. An educator must approve any actual practical, equipment, facilities and accessible alternative separately. Synthetic records do not certify hands-on skill or performed lab hours; public answers are nonsecure practice.
Plan roughly 15-25 minutes of source/model discussion, 25-40 minutes of selected-level analysis and 10-20 minutes of transfer per case, with difficult cases split across meetings. These are unpiloted planning estimates, not a timetable, lab-time log or guarantee of learner readiness.
Local Unit 1: Signed motion areas and a moving observer [motion-graphs]
Given inputs, level tasks, solutions and fresh transfer. Official topic connections: 1.1, 1.2, 1.3, 1.4. Science criteria: 1, 2, 3.
materials: Use the supplied motion-velocity records, assigned source sections, calculator and an accessible graph or diagram. No physical apparatus is required.
record: Retain motion-graphs, the selected level, original inputs and first attempt, source/date, system and axes, units, assumptions, calculation/graph, uncertainty, correction and fresh transfer. Link only to the stated science criteria.
review: An educator reviews prerequisites, source interpretation, reasoning and accessibility before assessment. A paper response does not certify unobserved practical technique; approved hands-on work requires a separate record.
question: How can a trace have zero displacement but positive distance, and what changes when the observer moves?
prerequisites: Read signed axes, triangle/trapezoid areas, slopes and the difference between an interval and an instant.
variables: Compare lab versus constant-speed observer coordinates; the transfer changes only the final velocity sample.
controls: Keep times, positive direction, initial position and straight-segment interpolation explicit in each comparison.
replication: Independent arithmetic/graph checks are not new physical trials; actual sampling would need repeated calibrated observations.
analysis: Plot the trace, sum signed and absolute areas, accumulate position, transform every velocity and split any zero-crossing interval.
Uncertainty: Real measurements require timing, position and calibration uncertainty; a handful of sampled velocities would not by itself establish straight-line behavior between samples. The model supplies that interpolation explicitly. Do not label the signed integral as distance when the object reverses direction.
Assessment evidence: Science criteria 1-3: retain sign conventions, slopes, position/velocity sketches, separate distance and displacement, the frame transformation and the zero-crossing transfer. No timing technique is inferred from supplied samples.
Local Unit 1: Two components share one flight time [projectile-model]
Given inputs, level tasks, solutions and fresh transfer. Official topic connections: 1.5. Science criteria: 1, 3, 4.
materials: Use the supplied projectile-inputs records, assigned source sections, calculator and an accessible graph or diagram. No physical apparatus is required.
record: Retain projectile-model, the selected level, original inputs and first attempt, source/date, system and axes, units, assumptions, calculation/graph, uncertainty, correction and fresh transfer. Link only to the stated science criteria.
review: An educator reviews prerequisites, source interpretation, reasoning and accessibility before assessment. A paper response does not certify unobserved practical technique; approved hands-on work requires a separate record.
question: Which initial conditions control flight time versus horizontal displacement in the stated two-component model?
prerequisites: Use signed acceleration, a square root and a common time in two perpendicular component equations.
variables: Compare changed horizontal speed or height separately while retaining the other stated initial conditions.
controls: Keep the inertial ground frame, constant gravity, zero vertical start speed and drag-free approximation fixed.
replication: These are deterministic model variants; actual trajectories would require repeated calibrated timing and position records.
analysis: Solve the vertical landing equation first, reject the nonfuture root and use its time in both velocity and horizontal displacement.
Uncertainty: Drag, wind, object size, terrain and uncertain initial conditions can change the outcome. A model prediction is not a measured landing location. The independence of perpendicular components follows the specified forces, not a rule that components never interact in other systems.
Assessment evidence: Science criteria 1, 3 and 4: retain axes, independent component equations, physical time-root selection and the changed-height transfer. Calculations do not establish motion-capture or launch technique.
Local Unit 2: One system, two free-body diagrams and friction bounds [force-systems]
Given inputs, level tasks, solutions and fresh transfer. Official topic connections: 2.1, 2.2, 2.3, 2.4, 2.5, 2.7. Science criteria: 1, 2, 3, 4.
materials: Use the supplied connected-bodies, force-settings, incline-inputs records, assigned source sections, calculator and an accessible graph or diagram. No physical apparatus is required.
record: Retain force-systems, the selected level, original inputs and first attempt, source/date, system and axes, units, assumptions, calculation/graph, uncertainty, correction and fresh transfer. Link only to the stated science criteria.
review: An educator reviews prerequisites, source interpretation, reasoning and accessibility before assessment. A paper response does not certify unobserved practical technique; approved hands-on work requires a separate record.
question: How does changing the selected system alter the force ledger without altering the predicted motion?
prerequisites: Understand mass, weight, signed forces, basic trigonometric components, weighted means and static/kinetic distinctions.
variables: Compare individual versus combined systems and a moving versus initially resting friction case.
controls: Keep masses, gravity, contact geometry, ideal string and the appropriate coefficient consistent within each model.
replication: Cross-check acceleration from both individual equations; this is an analytical consistency test, not a repeated force measurement.
analysis: Draw boundaries first, sum external forces, solve the total acceleration, recover tension and check the stationary friction inequality.
Uncertainty: Coefficients and loads are supplied model values, not measurements. A real force record needs sensor calibration, alignment and uncertainty. A stationary combined system fixes total friction but does not generally determine how static friction and tension split between contacts.
Assessment evidence: Science criteria 1-4: separate and combined diagrams, interaction pairs, mass-weighted position, acceleration/tension, incline components and static-versus-kinetic reasoning. No force-sensor technique is certified.
Local Unit 3: Gravity supplies a centripetal requirement, not a second force [circular-orbits]
Given inputs, level tasks, solutions and fresh transfer. Official topic connections: 2.6, 2.9, 6.6. Science criteria: 1, 2, 3, 4.
materials: Use the supplied orbit-inputs records, assigned source sections, calculator and an accessible graph or diagram. No physical apparatus is required.
record: Retain circular-orbits, the selected level, original inputs and first attempt, source/date, system and axes, units, assumptions, calculation/graph, uncertainty, correction and fresh transfer. Link only to the stated science criteria.
review: An educator reviews prerequisites, source interpretation, reasoning and accessibility before assessment. A paper response does not certify unobserved practical technique; approved hands-on work requires a separate record.
question: Which radius relationships follow from a circular gravitational model, and which do not describe an actual transfer?
prerequisites: Use inverse-square reasoning, square roots, circumference, period and a stated angular-momentum axis.
variables: Compare circular radius with the same central gravitational parameter; distinguish state comparison from time evolution.
controls: Retain central reference, circularity, negligible satellite mass and the same potential-energy zero.
replication: These deterministic reference states are not repeated observations of a satellite or independent evidence for a trajectory.
analysis: Equate gravity with radial mass times acceleration, derive speed and period, and compare L and energy before making a conservation claim.
Uncertainty: Real orbits may be eccentric, perturbed or thrust-driven; uncertain radius and time affect inference. Constant speed in a circle does not mean zero acceleration. Momentum/energy for a maneuver require the complete force and work history; two endpoint circular formulas do not supply a transfer trajectory.
Assessment evidence: Science criteria 1-4: radial force diagram, center-based radius, speed/period/field derivation and a qualified angular-momentum/energy comparison. This paper orbit model does not demonstrate centripetal-apparatus technique.
Local Unit 4: Work enters a system; friction does not destroy energy [energy-ledger]
Given inputs, level tasks, solutions and fresh transfer. Official topic connections: 3.1, 3.2, 3.3, 3.4, 3.5. Science criteria: 1, 2, 3, 4.
materials: Use the supplied energy-inputs records, assigned source sections, calculator and an accessible graph or diagram. No physical apparatus is required.
record: Retain energy-ledger, the selected level, original inputs and first attempt, source/date, system and axes, units, assumptions, calculation/graph, uncertainty, correction and fresh transfer. Link only to the stated science criteria.
review: An educator reviews prerequisites, source interpretation, reasoning and accessibility before assessment. A paper response does not certify unobserved practical technique; approved hands-on work requires a separate record.
question: How do two legitimate system boundaries account for the same pulling work and frictional heating?
prerequisites: Use cosine of a supplied angle, kinetic energy, constant-force work, energy units and average power.
variables: Compare the cart-only and cart-track ledgers, then change only the opposing force in the transfer.
controls: Retain mass, displacement and force angle; do not import the separate lift/spring energies into the horizontal path.
replication: Reconcile two analytical ledgers and dimensional checks; these do not constitute repeat measured heat or speed records.
analysis: Compute each work term, require nonnegative final kinetic energy, solve speed and label average versus instantaneous power.
Uncertainty: Angle, force, displacement and heat-transfer uncertainties would affect an actual experiment. The supplied friction force is not inferred from an unstated coefficient or normal force. A negative predicted final kinetic energy signals an inconsistent assumed trajectory from rest, not an imaginary physical speed.
Assessment evidence: Science criteria 1-4: signed work, system boundaries, kinetic/thermal ledger, separately defined potential energies and power. These calculations do not document a measured ramp or spring experiment.
Local Unit 5: Impulse, sticking and vector conservation [impulse-collisions]
Given inputs, level tasks, solutions and fresh transfer. Official topic connections: 4.1, 4.2, 4.3, 4.4. Science criteria: 1, 2, 3, 4.
materials: Use the supplied collision-bodies, collision-force-time, collision-vectors records, assigned source sections, calculator and an accessible graph or diagram. No physical apparatus is required.
record: Retain impulse-collisions, the selected level, original inputs and first attempt, source/date, system and axes, units, assumptions, calculation/graph, uncertainty, correction and fresh transfer. Link only to the stated science criteria.
review: An educator reviews prerequisites, source interpretation, reasoning and accessibility before assessment. A paper response does not certify unobserved practical technique; approved hands-on work requires a separate record.
question: Which quantities remain conserved in sticking and elastic models, and which force-pulse features determine impulse?
prerequisites: Use signed momentum, triangle areas, kinetic energy and perpendicular vector components.
variables: Compare collision type, component direction and pulse duration while preserving explicit system boundaries.
controls: Keep negligible external impulse, the supplied masses and the piecewise-linear pulse assumption distinct from measured evidence.
replication: Compare pulse area with the independent before/after momentum calculation; these are consistency checks, not independent collision trials.
analysis: Compute system momentum first, recover each body impulse, calculate kinetic-energy change and solve vector components before speed.
Uncertainty: Three force samples would not establish a real pulse shape without the stated interpolation. Alignment, timing, sensor bandwidth and unmeasured external forces matter experimentally. Kinetic-energy loss is not lost total energy, and conserving only scalar speed or only one component is insufficient for a vector collision.
Assessment evidence: Science criteria 1-4: signed impulse, system boundary, separate momentum/energy ledgers and two-dimensional components. No collision-cart timing or apparatus skill follows from these supplied values.
Local Unit 6: Restoring force, phase, energy and period estimates [oscillation-models]
Given inputs, level tasks, solutions and fresh transfer. Official topic connections: 2.8, 7.1, 7.2, 7.3, 7.4. Science criteria: 1, 2, 3, 4.
materials: Use the supplied oscillator-inputs, oscillator-timings, oscillator-phase records, assigned source sections, calculator and an accessible graph or diagram. No physical apparatus is required.
record: Retain oscillation-models, the selected level, original inputs and first attempt, source/date, system and axes, units, assumptions, calculation/graph, uncertainty, correction and fresh transfer. Link only to the stated science criteria.
review: An educator reviews prerequisites, source interpretation, reasoning and accessibility before assessment. A paper response does not certify unobserved practical technique; approved hands-on work requires a separate record.
question: How do phase and energy constrain a spring oscillator, and what can repeated timing actually test?
prerequisites: Use square roots, fractions of a cycle, mean/sample SD, spring potential energy and the meaning of a model approximation.
variables: Compare phase, amplitude and timing estimates while separating spring parameters from the pendulum model.
controls: Keep the equilibrium origin, starting phase, linear spring regime, count definition and small-angle assumption explicit.
replication: Three synthetic timing attempts describe repeat variability for one setup; do not relabel them as independent springs or real observations.
analysis: Relate all three phase graphs, calculate period/energy, divide total time by cycles, compute sample spread and test the changed amplitude.
Uncertainty: Repeated timing of the same setup estimates timing variability, not variation across an entire population of springs. Reaction-time bias and model errors can remain even when repeat spread is small. The synthetic values do not prove the model or document practical period measurement; large pendulum amplitude can invalidate the approximation.
Assessment evidence: Science criteria 1-4: restoring-force sign, phase graph, period dependencies, energy exchange and a bounded timing-data comparison. Timing arithmetic does not earn the separate observed-technique criterion.
Local Unit 7: Axis-specific inertia, net torque and rotational motion [torque-dynamics]
Given inputs, level tasks, solutions and fresh transfer. Official topic connections: 5.1, 5.2, 5.3, 5.4, 5.5, 5.6, 6.2. Science criteria: 1, 2, 3.
materials: Use the supplied rotor-masses, torque-inputs, torque-balance records, assigned source sections, calculator and an accessible graph or diagram. No physical apparatus is required.
record: Retain torque-dynamics, the selected level, original inputs and first attempt, source/date, system and axes, units, assumptions, calculation/graph, uncertainty, correction and fresh transfer. Link only to the stated science criteria.
review: An educator reviews prerequisites, source interpretation, reasoning and accessibility before assessment. A paper response does not certify unobserved practical technique; approved hands-on work requires a separate record.
question: How do torque direction, axis choice and mass distribution determine angular motion and equilibrium?
prerequisites: Use sine of a supplied angle, signed forces, radians, squared radii and constant-acceleration algebra.
variables: Compare the original and doubled-radius fixed configurations; analyze the stationary bar as a separate system.
controls: Keep the specified axis, negligible spoke mass and constant applied/drag torque; do not confuse mass radius with force radius.
replication: Check torque dynamics against a separate rotational work-energy calculation; this is model consistency, not repeated apparatus measurement.
analysis: Sum inertia about the chosen axis, calculate signed net torque, integrate constant angular acceleration algebraically and check energy and support balance.
Uncertainty: A real extended object has its own inertia and may need additional terms; radius and angle uncertainty can strongly affect torque/inertia. The point-mass model does not measure bearing losses or certify a spinning apparatus. Zero net torque alone does not establish static equilibrium unless translational force balance and appropriate initial motion are also addressed.
Assessment evidence: Science criteria 1-3: axis and signed moment arm, simultaneous force/torque equilibrium, fixed-axis dynamics and radius-dependent inertia with a work-energy check. No balance or rotor-handling technique is inferred.
Local Unit 7: Rolling energy requires a physically possible no-slip condition [rolling-energy]
Given inputs, level tasks, solutions and fresh transfer. Official topic connections: 6.1, 6.5. Science criteria: 2, 3.
materials: Use the supplied rolling-inputs, rolling-shapes records, assigned source sections, calculator and an accessible graph or diagram. No physical apparatus is required.
record: Retain rolling-energy, the selected level, original inputs and first attempt, source/date, system and axes, units, assumptions, calculation/graph, uncertainty, correction and fresh transfer. Link only to the stated science criteria.
review: An educator reviews prerequisites, source interpretation, reasoning and accessibility before assessment. A paper response does not certify unobserved practical technique; approved hands-on work requires a separate record.
question: When is an energy-based rolling prediction consistent with the force required to prevent slipping?
prerequisites: Use I = beta m R^2, energy conservation, radians, force components and a static-friction inequality.
variables: Compare beta, available static friction and release height while keeping the selected body model and slope explicit.
controls: Retain release from rest, fixed surface, rigid-body geometry and negligible dissipative losses only where the no-slip test succeeds.
replication: Reconcile energy and force/torque predictions independently; no repeated releases or observed contact motion are supplied.
analysis: Compute energy-based speed, derive acceleration and required friction, check the normal-force bound, then accept or reject the assumed regime.
Uncertainty: Real body shape and mass distribution can differ from ideal beta values. A low-friction surface can violate the assumed rolling constraint; then v = R omega and the loss-free rolling answer cannot simply be retained. Friction direction depends on the driving situation; the upslope result here is for a body released on a fixed incline, not every powered wheel.
Assessment evidence: Science criteria 2 and 3: force/torque consistency, ideal inertia factor, translational/rotational energy and an explicit no-slip inequality. These records do not establish a performed rolling experiment.
Local Unit 7: External angular impulse and internal energy during contraction [angular-impulse]
Given inputs, level tasks, solutions and fresh transfer. Official topic connections: 6.1, 6.3, 6.4. Science criteria: 3, 4.
materials: Use the supplied angular-inputs, angular-torque-time records, assigned source sections, calculator and an accessible graph or diagram. No physical apparatus is required.
record: Retain angular-impulse, the selected level, original inputs and first attempt, source/date, system and axes, units, assumptions, calculation/graph, uncertainty, correction and fresh transfer. Link only to the stated science criteria.
review: An educator reviews prerequisites, source interpretation, reasoning and accessibility before assessment. A paper response does not certify unobserved practical technique; approved hands-on work requires a separate record.
question: How can angular momentum change in one stage and be conserved while kinetic energy changes in the next?
prerequisites: Use torque-time areas, signed L = I omega, fixed-axis assumptions and rotational kinetic energy.
variables: Compare torque-pulse sign and inertia rearrangement as separate stages with distinct conservation conditions.
controls: Keep the axis and I0 fixed during the pulse; keep net external torque zero only during the stated contraction stage.
replication: Use independent momentum and energy ledgers to check the synthetic model; no measured torque trace or repeated contraction is supplied.
analysis: Find signed pulse area, update L, apply the later conservation condition and calculate the internal work from endpoint energies.
Uncertainty: Actual bearing torque, changing-axis motion, radial kinetic energy and dissipative losses could change the accounting. The three synthetic pulse samples specify straight interpolation, not instrument bandwidth. A reduction in inertia cannot be assumed energetically free merely because external torque is zero.
Assessment evidence: Science criteria 3 and 4: same-axis inertia and energy, signed angular impulse, conditional conservation and internal-work accounting. No observed rotor technique is implied by the model.
Local Unit 8: Pressure references and the hydraulic force-distance trade [pressure-hydraulics]
Given inputs, level tasks, solutions and fresh transfer. Official topic connections: 8.1, 8.2. Science criteria: 1, 3.
materials: Use the supplied pressure-inputs, hydraulic-inputs records, assigned source sections, calculator and an accessible graph or diagram. No physical apparatus is required.
record: Retain pressure-hydraulics, the selected level, original inputs and first attempt, source/date, system and axes, units, assumptions, calculation/graph, uncertainty, correction and fresh transfer. Link only to the stated science criteria.
review: An educator reviews prerequisites, source interpretation, reasoning and accessibility before assessment. A paper response does not certify unobserved practical technique; approved hands-on work requires a separate record.
question: Which pressure differences change force, and how does ideal hydraulic volume conservation constrain output work?
prerequisites: Use pressure as force per area, density, depth, area/volume conversion and simple work calculations.
variables: Change the ambient pressure only in the tank transfer; compare piston areas in the separate ideal hydraulic model.
controls: Retain constant density, surface reference, same-height pistons and matched ambient pressures within their respective models.
replication: Check pressure and work through independent ledgers; there are no repeated pressure or piston measurements in these inputs.
analysis: Compute hydrostatic difference before adding the reference; calculate hydraulic force and travel from pressure and volume, then verify energy.
Uncertainty: Real fluids can vary in density; real pistons have seal friction, weight, leakage and height differences. Pressure-force calculations require the loaded area and the correct reference. Ideal equal work does not establish real efficiency, structural strength or safe operation at the modeled pressures.
Assessment evidence: Science criteria 1 and 3: pressure/density units, explicit gauge/absolute references and a hydraulic volume/work ledger. No pressure handling or equipment certification follows.
Local Unit 8: Displaced volume, floating equilibrium and apparent weight [buoyancy-models]
Given inputs, level tasks, solutions and fresh transfer. Official topic connections: 8.1, 8.3. Science criteria: 1, 2.
materials: Use the supplied buoyancy-fluid, buoyancy-bodies records, assigned source sections, calculator and an accessible graph or diagram. No physical apparatus is required.
record: Retain buoyancy-models, the selected level, original inputs and first attempt, source/date, system and axes, units, assumptions, calculation/graph, uncertainty, correction and fresh transfer. Link only to the stated science criteria.
review: An educator reviews prerequisites, source interpretation, reasoning and accessibility before assessment. A paper response does not certify unobserved practical technique; approved hands-on work requires a separate record.
question: Which volume belongs in a buoyancy equation for a freely floating versus a fully immersed supported object?
prerequisites: Use mass/volume density, force balance, displaced volume, fractions and Newton’s second law.
variables: Compare the stated support and immersion conditions; the transfer changes both body mass and fluid density explicitly.
controls: Retain rigid volume, constant-density fluid and the neglect of drag only for the initial acceleration calculation.
replication: Reconcile density and force approaches for the same synthetic inputs; no real displaced-fluid volumes or repeat weighings are supplied.
analysis: Choose the displaced volume from the physical condition, calculate buoyancy, close the force ledger and distinguish initial acceleration from a terminal-speed problem.
Uncertainty: Volume, fluid density, trapped air, body deformation and contact forces matter in real measurements. Greater depth increases both top and bottom pressures; for a fixed-volume fully immersed body in constant-density fluid, the net buoyancy does not grow simply because absolute pressure is larger. This model does not assess floating stability or a real vessel.
Assessment evidence: Science criteria 1 and 2: density comparison, displaced-volume choice, buoyant/support force balances and a qualified changed-fluid prediction. Supplied values do not certify the buoyancy-measurement practical.
Local Unit 8: Flow continuity, height and the limits of a pressure prediction [continuity-bernoulli]
Given inputs, level tasks, solutions and fresh transfer. Official topic connections: 8.4. Science criteria: 4.
materials: Use the supplied flow-inputs records, assigned source sections, calculator and an accessible graph or diagram. No physical apparatus is required.
record: Retain continuity-bernoulli, the selected level, original inputs and first attempt, source/date, system and axes, units, assumptions, calculation/graph, uncertainty, correction and fresh transfer. Link only to the stated science criteria.
review: An educator reviews prerequisites, source interpretation, reasoning and accessibility before assessment. A paper response does not certify unobserved practical technique; approved hands-on work requires a separate record.
question: How much pressure change follows from the stated flow and height, and what remains unknown when a comparison value differs?
prerequisites: Use area times velocity, squared speed, energy per volume, signed height change and common pressure references.
variables: Compare section area, elevation and the fictional reported pressure; the transfer changes downstream area only.
controls: Retain one streamline, steady incompressible flow, common pressure datum and no pump for the ideal prediction.
replication: The model plus one fictional comparison is not a set of independent measurements or a validated loss correlation.
analysis: Apply continuity first, calculate both kinetic and elevation pressure terms, compare the residual and state which missing evidence would discriminate explanations.
Uncertainty: A pressure residual can reflect losses, sensor offsets, wrong flow/elevation inputs or failure of the assumed steady model. It does not identify viscosity or a unique mechanism by itself. Flow-dependent losses need their own model; do not carry an old residual unchanged into a different geometry. No pipe, pump or pressure-device construction is assigned.
Assessment evidence: Science criterion 4: continuity with compatible units, a quantitative Bernoulli height/pressure ledger and a non-unique residual critique. This supplied-data work does not certify pressure or flow apparatus technique.