Differentiate a vector trajectory and distinguish its directions
Compare times 0,1 and 2 at A=3; derive every component and normalize only nonzero vectors. Compare scalar speed change with acceleration magnitude.
AP Physics C: Mechanics — calculus-based modeling
Sixty-one original destinations provide the finite authored calculus-mechanics scope across seven content units and cumulative synthesis. Authored coverage and retained evidence workflows do not certify actual learner mastery, performed laboratory time or AP authorization. Independent scope review and technical/external gates are recorded separately.
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, trigonometry and vector components; derivatives and definite integrals are introduced in context but concurrent single-variable calculus is recommended. AP Physics 1 concepts may support intuition, but these investigations require new calculus derivations rather than reuse its algebraic completion evidence.
Course outcome: Derive and test motion using derivatives, definite integrals, initial conditions, piecewise/event boundaries and explicit uncertainty; integrate Newtonian dynamics, distinguish exact and numerical ODE behavior, and test force constraints; integrate rotational work and angular impulse, distinguish changing-inertia energy budgets, and test rolling and orbital constraints; derive harmonic initial-value and pendulum models, challenge shared calibration, and defend a full-course evidence portfolio.
All lessons are directly accessible. Reused foundation explanations are identified where relevant; completing a different course does not award this course's evidence. The 61 registered lessons keep separate predictions, captures, practice responses and reflections.
Calculus kinematics and evidence
Connect vector, graphical, analytic and sampled motion while distinguishing exact rates from finite estimates and model boundaries.
Recover an instantaneous rate from finite secants
At t=2 compare h=1,.5 and.1. Derive forward and centered errors and explain which shrinks linearly versus quadratically at leading order.
Reconstruct motion with two integration constants
Compare j=−1,0,+1 at v0=2; derive both functions, check their derivatives and initial values, and contrast a constant-acceleration shortcut.
Integrate distance across a velocity reversal
Compare T=1,2,3 and 5 at tau=2. Locate the velocity zero, integrate with signs and verify distance≥|displacement| including endpoint reversal.
Match piecewise motion without inventing jumps
Derive switch values for a=2; compare braking 1,2 and 4. Check continuity at the switch and first rest, then show why extrapolation through rest violates the stated hold.
Differentiate a moving-origin transformation
At observer acceleration 0 compare U=−2,0,2; then hold U=1 and compare accelerations −1,0,1. Differentiate the transformation and explicitly label which cases are noninertial.
Resolve independent components and an event boundary
At vertical launch velocity 2 compare horizontal jerk 0,1,3. Derive the same positive vertical root and different horizontal ranges; then change vertical velocity and explain both consequences.
Design a sampling interval with competing error sources
At epsilon=.04 compare h=.1,.2,.3,.5,1. Retain endpoint readings, the analytic teacher value and both error terms; choose an interval by the declared bound, not visual smoothness.
Forces, differential equations and constraints
Derive recipient-specific Newtonian models, integrate continuous distributions and initial-value dynamics, and test numerical and physical feasibility.
Locate a center by integrating a nonuniform mass distribution
Compare beta 0,2 and 4 at length 3. Derive mass and first moment separately, check units, and explain why multiplying all density values by the same factor leaves the center unchanged.
Derive a vector trajectory from recipient-specific forces
Compare initial horizontal force −4,0,4 at ramp 2. Draw the recipient diagram, integrate both components and distinguish force zero, velocity zero and displacement zero.
Solve a signed linear-resistance initial-value problem
At b1 compare v0=0,4,8; derive velocity, acceleration and position, retain the equilibrium case separately, and check each derivative and initial value.
Solve a quadratic-drag branch with explicit sign limits
Compare c=.25,.5 and 1 at mass 2. Derive terminal speed, response scale, velocity and displacement; compare early-time slope and late-time behavior.
Diagnose numerical overshoot in a stable physical model
Compare step ratios.5,1,1.5 and 2.5; retain recurrence multiplier and final time. Independently halve dt and double step count at one fixed final time.
Match a friction threshold to a moving-stage integral
At ramp 2 compare static coefficients.2,.4,.6. Derive threshold time and initial postrelease acceleration, then integrate one second of motion without resetting absolute force to zero.
Integrate a coupled system without canceling the wrong forces
Compare force ramps 0,2,4 with F0=8. Derive combined acceleration, each recipient net force, tension and the integrated common velocity/displacement.
Infer a field cancellation from signed source contributions
Compare right masses 1,5,9 at positions 1,2,3. Compute signed fields and derive the between-source root; explain why adding magnitudes would miss it.
Compare a local force derivative with a secant stiffness
Compare x=−.5,0,.5 at beta 4, then compare beta 0 and 8. Derive signed force, tangent stiffness and the nonzero-displacement secant; explain the omitted 0/0 value.
Resolve circular acceleration and test force constraints
Compare angular accelerations 0,1,2,3 and times 0,1,2. Derive tangent/normal components, required forces and the capacity screen; distinguish instantaneous angular speed from a true period.
Work, potential and evidence integrals
Derive work and power integrals, recover forces and equilibrium from potentials, and enforce accessibility and uncertainty boundaries.
Derive kinetic-energy change from Newtonian dynamics
Compare A=2,b=0 and A=2,b=2. Derive the complete K(x), locate any force zero, and explain why endpoint work does not describe every local change. Retain both captures.
Parameterize work along competing planar paths
Compare A=1,B=2 with A=B=2. Independently parameterize both legs, then reverse one path to build a closed-loop argument. Retain a unit audit and explain the role of unspecified forces.
Recover force and classify equilibrium from an energy landscape
Compare B=2 and B=6 at A=1. Derive all roots, classify with nearby forces, and independently integrate force between the center and a minimum. Explain how adding 100 J to U changes the plot but not motion.
Distinguish allowed positions from reachable motion
Compare E=.5,1 and 1.5 at the same starting well. Sketch allowed components, state whether finite-time crossing is possible, and identify the missing initial-direction information. Retain the threshold case rather than rounding it into either side.
Integrate gravitational work with a defensible reference
Compare final radii 3 and 8 for m=2. Reconstruct the definite integral and both work signs, then express the relative change of g. Propose how a small-height comparison would test the approximation without claiming a launch experiment.
Integrate dissipation only along an accessible branch
Compare K 0=4 and 8 at c=1. Independently locate the first stop, retain the target flag, and reconcile friction work with thermal energy at the physical endpoint. Explain why a new contact branch is needed after rest.
Integrate power without confusing one interaction with net work
Compare a=1 and−1 at T=2. Derive each power curve, find any sign change and integrate signed areas. Explain what extra evidence would be needed to infer net kinetic-energy change.
Separate quadrature error from an uncertain force record
Compare n=2,4,8 at epsilon=.2. Reconstruct the endpoint weights, separate both error terms, and propose a calibration check that more intervals cannot replace. Retain failed and revised estimates.
Momentum integrals, systems and collisions
Integrate distributed momentum and force histories, audit system boundaries and distinguish collision constraints from independent evidence.
Integrate momentum and energy over a moving distribution
Compare q=0 and−1 at beta=0, then beta=2. Reconstruct M,P and K from integrals; retain the center-relative energy and explain why this instantaneous packet cannot predict later density without further dynamics.
Integrate a vector force and differentiate the resulting momentum
Compare A=1 and 4 at B=1. Integrate before opening feedback; sketch momentum slopes and force areas. Then change B and explain how final momentum direction differs from the horizontal pulse direction.
Distinguish impulse from a force-pulse maximum
Compare all three n values at T=2, then double T from 1 to 2 for one shape. Reconstruct the normalization and peak, retain common impulse, and name a measurement limitation that could hide a peak.
Recover center motion from an internal-force audit
Compare A=0,b=0 with A=2,b=1. Derive individual and center trajectories, retain both initial center conditions, and identify why internal forces cannot supply the external impulse. Analyze the two conveyor inflow velocities as a written boundary check.
Derive collision endpoints from momentum and relative motion
At second mass 3 compare e=0,.5,1. Independently solve both equations, calculate K before/after and retain the changed constraint. Identify an external impulse that would invalidate the isolated-event approximation.
Conserve planar momentum and test an inferred external impulse
Compare b=2 and−2 at a=3. Reconstruct vector and energy results, then suppose the measured final velocity is(1,1): calculate the required residual impulse and list two non-force alternatives that could explain it.
Derive center and relative energy for an internal release
Compare w=0,3,5 at V=2, then reverse V. Reconstruct both fragment velocities and the center-frame momenta, retain the internal-energy decrease and explain why the event is not an isolated creation of energy.
Audit an impulse window before accepting endpoint agreement
Compare h=.5,1,1.5 at B=.5, then set B=0. Retain all error terms separately and propose a baseline observation outside the pulse. Explain how opposite errors could produce misleading agreement.
Torque and distributed rotational inertia
Differentiate fixed-axis motion, integrate distributed loads and mass, and reconcile rotational dynamics with geometric and unilateral constraints.
Differentiate an unwrapped angular record
Compare omega 0=−1,c=1 with omega 0=1,c=1. Locate any reversal, integrate absolute angular speed by intervals and retain both signed and total angle. Check the derivative at a chosen intermediate time.
Differentiate the radial basis of a rigid rotation
Compare alpha=0 and 2 at radius 1, then halve radius. Reconstruct radial/tangential and Cartesian components independently and verify squared magnitudes agree. Explain which components are kinematic requirements rather than named forces.
Integrate signed torque across a distributed load
Compare a=1,b=1 with a=−1,b=1. Integrate force and torque about two origins; explain why the effective-force coordinate disappears for the second case. Retain signs of the load rather than converting all terms to magnitudes.
Integrate rotational inertia of a nonuniform rod
Compare beta=0,1,3. Independently integrate M,Q and Iend, derive Icm and compare normalized shape factors. Then double alpha and predict required torque without reintegrating the unchanged distribution.
Normalize a radial density before integrating disk inertia
At R=2 compare n=0,2,4. Derive sigma 0 before I, retain normalized shape factors and compare the thin-rim limit. Explain which extra geometric factor appears for a spherical shell.
Infer inertia from parallel-axis evidence without hiding bias
Compare d 1=.5,1,1.5 at second offset+.1. Derive the error amplification and propose a third-axis prediction before looking at its result. Retain both fitted and proposed held-out evidence.
Test support feasibility after integrating a distributed load
Compare point positions 2 and 4 at a=0, then add distributed load a=4. Integrate its moment, solve both reactions and retain the feasibility flag. Sketch the failed assumed contact separately from any proposed new support model.
Integrate a rotational torque law with its initial condition
At b=1 compare initial omega 0=0,6,8. Derive the rate and angle, retain net torque and explain the different approach directions. Then double b and predict the steady value and response time before evaluating.
Couple translation and rotation through a massive pulley
At m 2=4 compare I=.5,1,2. Solve acceleration, both tensions and torque independently; retain the no-slip conversion and compare the massless limit. Explain which observations would challenge the assumed constraint.
Rotational work, angular momentum and contact events
Integrate rotational energy and angular impulse, select valid angular-momentum systems, and test rolling or orbital constraints with explicit branch boundaries.
Integrate rotational energy without double counting
Compare beta=0,1,2 at omega=2. Integrate the normalized density independently, calculate the two energy decompositions and verify their sums. Reverse omega and retain the changed angular-momentum sign alongside unchanged energy. Explain why holding lambda0 fixed would answer a different question.
Integrate torque over an accessible angular path
Compare A=0,B=4 with A=4,B=0. Derive work and accessible endpoint before using the workbench. Sketch signed torque-angle areas and mark where instantaneous power vanishes. Retain a rejected target and explain why clipping negative energy is not a valid continuation.
Reconstruct angular momentum from a torque history
Use A=3,B=1 and then A=3,B=2. Predict signed areas and final momenta. Differentiate your reconstructed L at t=0,1,2 and compare with the torque function. Integrate omega independently and explain why zero net impulse does not imply zero torque at every instant.
Differentiate particle angular momentum about a fixed origin
Compare two origins for the same assigned r,p,F. Compute both cross products directly and through the translation identities. Find a supported case with zero torque and explain why its angular momentum need not be zero. Identify which change would invalidate the fixed-origin derivative proof.
Conserve angular momentum while inertia changes
Compare final radii 2,1,.5 m at initial omega=2. Derive L, final speed and energy change before checking. Then audit the two-rotor coupling example with separately selected subsystems and show their angular impulses cancel. Identify where assuming conserved kinetic energy would contradict the stated interaction.
Derive and test a no-slip rolling constraint
Compare beta=.5 and 1 at 30 and 40 degrees. Draw the force diagram and derive the required friction, then test the coefficient before interpreting acceleration. For a feasible case, independently obtain the speed after a specified vertical drop from energy and from the constant-acceleration geometry.
Integrate sliding motion to its first no-slip event
Analyze forward slip, backward slip and an already rolling case. Predict the friction direction, solve both motion equations and locate the first event before checking. Reconcile kinetic-energy loss with the contact-slip integral, and explain why center kinetic energy alone is not a dissipation measurement.
Reconcile orbital energy with angular momentum
Compare f=.5,1,1.25,1.5 at r0=4. Compute E and L independently, classify the orbit and test whether a finite second turning radius exists. Explain why changing K around a bound eccentric orbit does not imply energy loss. For the circular case, derive a period ratio for a changed radius without using a full numerical trajectory.
Calculus oscillations and parameter evidence
Derive harmonic initial-value and energy relations, evaluate approximation boundaries, and identify distributed or torsional pendulum parameters without conflating fitted records with calibration.
Solve and verify a harmonic initial-value equation
Choose two supported initial velocities at the same initial displacement. Derive both functions and verify their initial conditions, amplitudes, phases and second derivatives. Compare their identical periods despite different amplitudes. Explain why a periodic triangle-wave position would not satisfy the same harmonic differential equation.
Derive spring topology and shift the equilibrium coordinate
Derive all three stiffnesses for two different k2 values. Compute each equilibrium first, then compare periods at the same mass. Double mass in a written changed case and predict the period factor before calculating. Draw which quantity is shared in a series connection and which in a parallel connection.
Use the chain rule to audit an oscillator clock
Compare phase indices0 and1 at gain1, then hold phase fixed while changing gain. Derive first and second derivatives in both time coordinates. Retain a same-physical-instant comparison and propose an independent clock or force/mass check that could distinguish the alternatives.
Recover harmonic timing from an energy integral
Compare two amplitudes and the positions x/A=0,.5,1. Retain E,U,K and both velocity signs, then choose a phase branch explicitly. Re-derive the quarter-period integral and compare energy frequency with displacement frequency. Explain why equal time-average energies do not mean equal instantaneous energies.
Test a local harmonic approximation against a nonlinear force
Compare lambda=0,4,8 at the same quadratic coefficient. Calculate the force-term ratio and exact center speed, then halve amplitude in a written changed case. State which conclusions are exact under the supplied potential and which require a small-displacement approximation.
Audit a pendulum small-angle approximation
Compare amplitudes5,15,30 degrees at one length, then double the length. Convert to radians, calculate restoring ratios and both bottom-speed predictions. Explain why changing length affects the linear period while the restoring ratio at the same angle does not. Identify an independent timing observation required to study the actual finite-amplitude period.
Integrate a physical pendulum and test the best pivot distance
Integrate the beta=0 and beta=1 mass distributions independently. For each, compare at least three pivot offsets, derive the continuous optimum and test the sign of the period-squared derivative on either side. Explain why moving the pivot closer to the center does not always shorten the period.
Identify torsional stiffness with a known inertia increment
Compare two added inertias at unbiased settings, then bias only the second period. Derive both inferred parameters and explain the sensitivity of a difference. Contrast this with a written common-gain case. Propose an independent timing or stiffness check rather than treating the known generator values as measured evidence.
Full-course calculus mechanics evidence defense
Choose systems and intervals, reconcile calculus representations with conservation, test constraints and distinguish independent evidence from agreement caused by shared assumptions.
Defend calculus and conservation across a changing system
Compare tau=1 and2 at captured mass2, then change the captured mass. Retain pulse integrals, capture energy loss and post-capture phase for each. Explain why the station is placed at x(tau) in each virtual scenario rather than treating the changing endpoint as the same fixed apparatus. Complete a new written force-function case without using the generator formula as independent evidence.
Defend independent evidence rather than shared model agreement
Retain three method estimates and both conditional reading intervals for an unbiased and a biased packet. Write at least two alternative explanations, then specify an observation that would discriminate them without reusing the disputed reference. Complete the cumulative mixed tasks and keep reviewer critique, answer-exposure disclosures and revised reasoning in the versioned portfolio.
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.
Analytic and limiting representation audit
Evidence to retain: Derive component velocity/acceleration and speed change, omit undefined directions, then compare forward/centered secants with analytic rates and changed spacing.
Boundary: Prescribed analytic functions, not measured trajectories or a force identification.
Initial-value, reversal and matching study
Evidence to retain: Retain both initial constants, sign-partitioned distance integrals and matched segment values; challenge an extrapolation beyond first rest.
Boundary: Synthetic piecewise motion; ideal derivative discontinuities and explicit hold constraint, no actual actuator.
Frame and event independence study
Evidence to retain: Differentiate observer transformations, label inertial versus accelerating cases, and test whether one component changes the other event equation.
Boundary: Nonrotating frames, separable planar model, no drag or physical launches.
Sampling and deterministic uncertainty study
Evidence to retain: Retain raw endpoint readings, several spacings, exact cubic truncation and reading bounds; justify a choice and propose held-out and clock-calibration checks.
Boundary: Teacher-known generator and assigned bounds, not measured instrument accuracy or a confidence interval.
Continuous mass and recipient-force audit
Evidence to retain: Integrate mass and first moment with units; construct separate recipient-specific free-body equations, recover the shared constraint and retain both initial conditions.
Boundary: Original prescribed densities and force histories; no actual balances or force sensors.
Exact and numerical resistive dynamics
Evidence to retain: Separate and integrate the linear ODE, differentiate the result, contrast quadratic rest-release and compare Euler step refinements at the same final time. Retain failure as well as convergent captures.
Boundary: Linear and quadratic laws are stipulated models. Terminal behavior does not identify a real drag law or validate a numerical step.
Contact transition and spatial force inference
Evidence to retain: Locate marginal static release and integrate the kinetic branch; compare a gravitational null with a mass center and local spring stiffness with a secant ratio.
Boundary: Ideal contact transition, point/spherical gravity and assigned spring law; no physical drops, launches or apparatus.
Changing-speed circular constraint
Evidence to retain: Derive radial and tangential acceleration from the moving basis, reconstruct connector/drive forces, and reject a capture above the declared capacity. Compare banked and conical components in written work.
Boundary: Prescribed circular path is conditional on sufficient forces; an infeasible path is not a simulated broken connector.
Work integral and path audit
Evidence to retain: Derive net work-energy from Newton2, then compare three selected-field paths and a reversed loop. Retain a case with negative local force and a positive accumulated K.
Boundary: Constant mass, inertial frame, prescribed fields/paths; no hidden claim that selected work equals net work.
Potential geometry and accessible states
Evidence to retain: Differentiate a potential, classify equilibria with nearby forces and test disconnected wells, the exact barrier and above-barrier energy. Retain missing initial-direction information.
Boundary: Conservative one-dimensional models; energy geometry does not certify a finite-time trajectory or settling.
Conservative and dissipative budget
Evidence to retain: Integrate exact gravity and compare a local approximation; then locate first frictional stopping and reconcile thermal inventory after changing the system boundary.
Boundary: Fixed spherical source, ideal contact branch, no actual launches or calorimetry.
Power and uncertain work reconstruction
Evidence to retain: Integrate a signed power history and reconstruct trapezoid weights at three resolutions. Separate generator bias, coherent calibration allowance and unmeasured model discrepancy.
Boundary: Synthetic exact-distance generator, not measured force accuracy or a probabilistic confidence interval.
Distributed momentum and center audit
Evidence to retain: Integrate momentum and kinetic energy over a shared mass measure, compare center and relative motion, and reconcile individual/center trajectories with only external impulse. Analyze changed inflow momentum in a written conveyor boundary case.
Boundary: Instantaneous deformable distribution and prescribed point-particle interactions; no real material motion or conveyor.
Impulse shape and captured-window investigation
Evidence to retain: Derive vector impulse and momentum slope, normalize contrasting pulse shapes, then separate missing tails from a coherent sensor offset. Retain an initially misleading agreement and propose independent baseline/timing checks.
Boundary: Assigned pulses and exact times, not injury thresholds or calibrated sensors.
Collision endpoint and residual-impulse study
Evidence to retain: Solve momentum with elastic, sticking and stated restitution conditions; reconstruct planar components, compare kinetic inventories and infer a residual external impulse from an altered observed endpoint.
Boundary: Ideal short events and known masses, not performed impacts or secure assessment.
Internal release and frame comparison
Evidence to retain: Derive fragment velocities from center and relative constraints, audit the internal store and compare laboratory/center-frame momenta and energies.
Boundary: Virtual unspecified energy conversion, no explosive or mechanical release instructions.
Angular representation and rotating-basis audit
Evidence to retain: Derive angular functions with initial conditions, split swept angle at a reversal and independently transform inward/tangential acceleration into Cartesian components.
Boundary: Fixed-axis planar rigid rotation; no simultaneous multi-plane rotation or apparatus.
Load and mass integrals across axes
Evidence to retain: Integrate force/torque and differential mass/inertia with explicit units; test a pure couple, normalize radial density and compare end/center or disk/rim geometries.
Boundary: Assigned load and positive mass distributions; equivalent resultants do not certify local stresses.
Parallel-axis inference and unilateral equilibrium
Evidence to retain: Infer mass/inertia from two axes, propagate a controlled reading fault and propose a third-axis check; independently solve distributed-load support reactions and reject lift-off branches.
Boundary: Synthetic readings and ideal unilateral contacts; no actual sensor calibration or load rating.
Rotational initial-value and coupled-constraint study
Evidence to retain: Solve a fixed-inertia torque ODE with two angular initial conditions, then derive two pulley tensions and match translational and rotational equations through no-slip.
Boundary: Optional viscous model and virtual suspended masses, with fixed-axis/no-slip assumptions; no physical experiment.
Energy density and torque-angle branch audit
Evidence to retain: Normalize distributed mass, integrate local kinetic energy and reconcile center/end-axis accounts; integrate signed torque over angle and identify a first turning point before an inaccessible target.
Boundary: Fixed-axis rigid bodies and assigned functions; no apparatus or physical rotation.
Time impulse, reference point and subsystem audit
Evidence to retain: Reconstruct L and angle from a torque history, differentiate L as a graph check, compare particle cross products about two fixed origins and reconcile equal/opposite angular impulses in a two-rotor coupling.
Boundary: Fixed inertial origins and stipulated negligible external torque; self-reported derivations are not measured isolation.
No-slip feasibility and first slip event
Evidence to retain: Derive required static friction and test its limit, then independently integrate translational and rotational sliding equations until the first zero of relative contact velocity; reconcile energy loss with contact-slip work.
Boundary: Ideal stationary surfaces, stated Coulomb coefficients and no rolling resistance; no experimental releases.
Circular and eccentric invariant comparison
Evidence to retain: Derive circular energy/period relations, classify a changed tangential-speed packet using E and L, and compare a second turning radius under the optional effective-potential model.
Boundary: Synthetic dominant-source point field, not an actual trajectory, central-body interior model or launch plan.
Harmonic initial-value and spring-constraint derivations
Evidence to retain: Verify both initial conditions and second derivatives, derive parallel/series stiffness from constraints and translate a vertical equilibrium before comparing periods.
Boundary: Ideal massless linear springs, supplied equations and no physical assembly; amplitude invariance applies only to the linear model.
Clock-chain-rule and energy-time representation audit
Evidence to retain: Retain phase-aware first/second derivatives in true and reported time, identify calibration alternatives, derive both speed branches and evaluate the integrable quarter-period energy integral.
Boundary: Synthetic clock intervention; a waveform does not independently calibrate time and energy alone does not select velocity sign.
Exact versus local and small-angle approximations
Evidence to retain: Derive a nonlinear force from its supplied potential, quantify a neglected term and compare exact energy speed with a local prediction; audit sine-torque and bottom-speed approximations for a simple pendulum.
Boundary: Local-curvature treatment is optional enrichment; neither a force error nor a speed difference is asserted to be an exact period-error bound.
Distributed and torsional pendulum identification
Evidence to retain: Integrate a normalized rod distribution, apply the parallel-axis theorem and derive/optimize its small-angle period; use a known inertia increment to identify torsional stiffness and challenge a biased second period.
Boundary: Stipulated rigid pivots and known inertia increments, not measured apparatus calibration; finite-angle and mounting effects remain external.
Multi-stage calculus and independent-evidence capstone
Evidence to retain: Retain force integrals, new collision-system boundaries, internal energy conversion and post-capture oscillator initial conditions; compare three inferred stiffnesses and conditional reading bounds, then design a discriminating calibration check.
Boundary: Supplied virtual packets and proposed checks, not apparatus construction, measured calibration, performed laboratory time or independent mastery.
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 42 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. Two constants and an absolute-value partition
An original trajectory obeys a(t)=6t m/s², v(0)=−3 m/s and x(0)=2 m. Derive v and x. On 0≤t≤2 find displacement and distance; explain why inserting endpoint acceleration into a constant-acceleration formula fails.
Course-specific review criteria
- Integrate with both initial conditions.
- Find the interior velocity zero and split speed integral.
- Check units and distinguish x from displacement.
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. Directions and an observer derivative
r(t)=2t i+3t² j in SI units. At t=1 derive velocity, acceleration, speed and scalar speed derivative. A nonrotating observer has R(t)=(1+t)i+t² j. Find relative velocity/acceleration and explain the frame qualification.
Course-specific review criteria
- Differentiate fixed-basis components.
- Distinguish |a| from d|v|/dt.
- Differentiate observer motion rather than subtract position from velocity.
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. Event-limited component transfer
x=3t+t³/3,y=4−2t−4t² in SI units. Derive component accelerations, choose the first positive ground time and find horizontal position there. Would doubling only the t³ coefficient change that time?
Course-specific review criteria
- Use derivatives and the vertical event equation.
- Reject the negative root.
- Apply the same time to the horizontal integral and state component assumptions.
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 sharper interval can worsen a derivative
For supplied x=t³−2t at t=2, each position reading has bound±.02 m. Compare centered half-widths.1,.2 and.5 s. Give analytic velocity, exact centered bias and total conservative error bound at each. Does the smallest bound validate the clock?
Course-specific review criteria
- Derive cubic bias and endpoint-difference bound.
- Compare errors rather than graphical smoothness.
- Distinguish deterministic bounds from probabilities and calibration evidence.
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. A weighted mass is not a force law
A rod spans 0≤x≤2 m with lambda(x)=(3 kg/m²)x. Find total mass and center. Separately, a 3 kg point mass has Fx=6t N and v(0)=−1 m/s. Find v(2) and displacement on 0–2 s. Explain why the rod center does not supply this point mass’s force history.
Course-specific review criteria
- Derive the relevant force, integral or differential equation before substituting.
- Retain initial conditions, signs and units.
- Test the stated physical or numerical boundary and explain what the evidence does not establish.
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. Initial condition versus numerical stability
For 2 dv/dt=4−v, v(0)=6, derive v(t) and x(t) with x(0)=0. Compare exact v(4) with Euler using two 2 s steps. Does terminal value 4 forbid starting at 6? If a separate rest-release quadratic model has m=3 kg, g=12 m/s² and c=1 kg/m, find its terminal speed.
Course-specific review criteria
- Derive the relevant force, integral or differential equation before substituting.
- Retain initial conditions, signs and units.
- Test the stated physical or numerical boundary and explain what the evidence does not establish.
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. A change of contact model before integration
A 2 kg object on a horizontal surface has mu_s=.4, mu_k=.15 and g=10; applied force is 2t N. Find marginal release time and velocity/displacement one second after release. Separately two frictionless masses 2 and 4 kg are joined and pulled by 12 N on the 4 kg mass. Find acceleration and connector force.
Course-specific review criteria
- Derive the relevant force, integral or differential equation before substituting.
- Retain initial conditions, signs and units.
- Test the stated physical or numerical boundary and explain what the evidence does not establish.
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. Local response and feasible curvature
Equal point masses at x=0 and 6 have where-zero field on the connecting line? A spring has F=−5x−2x³ in SI units: find restoring force and tangent stiffness at x=1. A 3 kg object moves on radius 2 m at speed 4 m/s with tangential acceleration 2 m/s²: find inward and tangential force components and their resultant. Can a purely radial connector alone provide both?
Course-specific review criteria
- Derive the relevant force, integral or differential equation before substituting.
- Retain initial conditions, signs and units.
- Test the stated physical or numerical boundary and explain what the evidence does not establish.
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. Circular time scales and force directions
In a circular orbit around a fixed central mass, derive the period-radius relation from Newtonian gravity and circular acceleration; compare radii r and 4r. For a frictionless banked turn with v=6 m/s, r=12 m and g=10, find bank angle. A 2 kg conical pendulum makes 30 degrees from vertical; find tension and v²/r.
Course-specific review criteria
- Derive the relevant force, integral or differential equation before substituting.
- Retain initial conditions, signs and units.
- Test the stated physical or numerical boundary and explain what the evidence does not establish.
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. One field, several paths
A selected planar field is F=(2y, x) in SI units. Compare its work along r(u)=(u,u²), 0≤u≤1, with horizontal-then-vertical and vertical-then-horizontal legs between the same endpoints. Can selected work alone establish a change in K?
Course-specific review criteria
- Derive the integral or derivative and retain units and signs.
- Declare the system, selected versus net interaction and initial conditions.
- Reject an inaccessible continuation and separate uncertainty from model assumptions.
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. Gradient, reference and threshold
For U=2(x²−1)² J and mass 2 kg, locate and classify equilibria. A trajectory starts at x=−1 with total energy E=2 J: does it cross x=0 in finite time? What changes at E=3? Would adding 7 J to both U and E change the conclusion?
Course-specific review criteria
- Derive the integral or derivative and retain units and signs.
- Declare the system, selected versus net interaction and initial conditions.
- Reject an inaccessible continuation and separate uncertainty from model assumptions.
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. Gravity versus a dissipative branch
For fixed GM=12, a 2 kg object moves quasistatically from r=3 to6 m. Compute ΔU and gravity/external work. Separately a right-moving object has K0=9 J and resistance2+x N; find first stop and explain why evaluating K at x=4 is invalid.
Course-specific review criteria
- Derive the integral or derivative and retain units and signs.
- Declare the system, selected versus net interaction and initial conditions.
- Reject an inaccessible continuation and separate uncertainty from model assumptions.
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. Signed power is not endpoint power
A selected interaction has F=(t,−2) N along r=(t²,t), 0≤t≤2 in SI units. Derive P, total work, average power and endpoint power. What claim would additionally require the force to be net?
Course-specific review criteria
- Derive the integral or derivative and retain units and signs.
- Declare the system, selected versus net interaction and initial conditions.
- Reject an inaccessible continuation and separate uncertainty from model assumptions.
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.
14. A second work method and its uncertainty
Synthetic net force F=1+2x² N acts over0–3 m. Compute exact work and the composite trapezoid bias at n=6. Each force reading has bound±.2 N and distances are exact. Give a conservative total error bound. An independent2 kg speed record changes from1 to sqrt(22) m/s: compare ΔK and state what agreement does not prove.
Course-specific review criteria
- Derive the integral or derivative and retain units and signs.
- Declare the system, selected versus net interaction and initial conditions.
- Reject an inaccessible continuation and separate uncertainty from model assumptions.
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.
15. An integral and an unseen velocity profile
A uniform 2 kg/m distribution spans0≤x≤2 m with v(x)=2−x m/s. Find M,P,vCM and total K, then KCM and relative K. Separately a net force F(t)=3t² N acts on a 2 kg object from0 to2 s with initial velocity−1 m/s. Find final momentum and velocity.
Course-specific review criteria
- Derive vector momentum and relevant integrals with units.
- State fixed-material versus open boundaries and the extra event constraint.
- Audit energy separately and distinguish model compatibility from independent evidence.
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.
16. Equal impulse and a biased observation window
A triangular force pulse rises from0 to a peak over1 s and falls to0 over the next1 s. Required impulse is18 N s. Find peak and mean. A sensor includes a constant+.5 N offset and integrates a3 s window containing the entire pulse: find indicated impulse. An independent 3 kg velocity record changes from−1 to5 m/s: compare momentum change and explain the discrepancy.
Course-specific review criteria
- Derive vector momentum and relevant integrals with units.
- State fixed-material versus open boundaries and the extra event constraint.
- Audit energy separately and distinguish model compatibility from independent evidence.
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.
17. Internal forces versus incoming momentum
A fixed 5 kg material system initially has P=−5 kg m/s and receives net external force2t N over0–3 s. Find final P and center velocity. Separately a conveyor at4 m/s collects mass at3 kg/s with zero incoming horizontal velocity. Find drive force needed to maintain speed, then repeat if incoming material already has horizontal velocity4 m/s.
Course-specific review criteria
- Derive vector momentum and relevant integrals with units.
- State fixed-material versus open boundaries and the extra event constraint.
- Audit energy separately and distinguish model compatibility from independent evidence.
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.
18. An endpoint can conserve momentum but violate elasticity
Two masses2 and3 kg approach with velocities4 and−1 m/s. A proposed common endpoint velocity1 m/s satisfies which conservation statement and loses how much translational K? Give the elastic endpoints instead. For a separate planar sticking packet, initial momenta(6,0) and(0,−6) combine into5 kg: find final velocity and explain why adding magnitudes fails.
Course-specific review criteria
- Derive vector momentum and relevant integrals with units.
- State fixed-material versus open boundaries and the extra event constraint.
- Audit energy separately and distinguish model compatibility from independent evidence.
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.
19. The minimum energy at fixed momentum
A5 kg virtual system has total momentum−10 kg m/s and splits into masses2 and3 with relative velocity v1−v2=4 m/s, without external impulse. Find both velocities, center and relative K, total K and required internal-store decrease from an initially co-moving state. Does reversing the center velocity change the relative term?
Course-specific review criteria
- Derive vector momentum and relevant integrals with units.
- State fixed-material versus open boundaries and the extra event constraint.
- Audit energy separately and distinguish model compatibility from independent evidence.
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.
20. A reversal and a point acceleration
A fixed-axis record has omega(t)=−2+t² rad/s from 0 to 2 s and theta(0)=0. Find signed angular displacement and total swept angle. Separately a point at radius .5 m has omega=4 rad/s and alpha=3 rad/s²: find inward, tangential and total acceleration magnitudes.
Course-specific review criteria
- Derive the appropriate angular derivative, load integral or inertia integral with units.
- Declare axis, reference point and fixed-distribution assumptions.
- Test contact/constraint feasibility and distinguish inferred from independently measured quantities.
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.
21. Two differential elements with different geometry
A line from 0 to 3 m has lambda(x)=(2 kg/m²)x. Find mass, center and inertia about a perpendicular end axis, then central inertia. Separately a thin disk has mass 3 kg,radius 2 m and normalized surface density proportional to r²: find its central inertia.
Course-specific review criteria
- Derive the appropriate angular derivative, load integral or inertia integral with units.
- Declare axis, reference point and fixed-distribution assumptions.
- Test contact/constraint feasibility and distinguish inferred from independently measured quantities.
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.
22. An origin shift is not the parallel-axis theorem
A signed vertical force density f(x)=−1+x N/m acts on 0–2 m. Find net force and torque about x=0 and x=1. Separately a rigid body has indicated I1=5 and I2=14.3 kg m² at parallel offsets 1 and 2 m from its center axis. Infer mass and central inertia and compare with true M=3, Icm=2 if only I2 was biased.
Course-specific review criteria
- Derive the appropriate angular derivative, load integral or inertia integral with units.
- Declare axis, reference point and fixed-distribution assumptions.
- Test contact/constraint feasibility and distinguish inferred from independently measured quantities.
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.
23. Equations plus contact directions
For the taught beam with supports x=1 and 3, own weight 20 at 2, added density a(1+x/4) and point weight 40 at b, use a=0,b=4. Compute both required reactions and say whether two-contact static equilibrium is possible. Would merely replacing a negative reaction by zero preserve both balances?
Course-specific review criteria
- Derive the appropriate angular derivative, load integral or inertia integral with units.
- Declare axis, reference point and fixed-distribution assumptions.
- Test contact/constraint feasibility and distinguish inferred from independently measured quantities.
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.
24. Torque history and a massive pulley
A fixed I=2 kg m² body obeys 2 dω/dt=8−2ω with omega0=0 and theta0=0. Derive omega(t),theta(t),then evaluate at 1 s. Separately the taught pulley has m1=2,m2=4,R=.5,I=1,g10: derive acceleration,tensions and net torque.
Course-specific review criteria
- Derive the appropriate angular derivative, load integral or inertia integral with units.
- Declare axis, reference point and fixed-distribution assumptions.
- Test contact/constraint feasibility and distinguish inferred from independently measured quantities.
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.
25. Two energy accounts and an inaccessible angle
A uniform rod has M=3 kg,L=2 m and fixed endpoint omega=3 rad/s. Derive endpoint inertia and total energy, then split that energy into center translation and center-axis rotation. Separately a fixed I=2 body begins with K=4 J and net torque tau=−4theta. Can its positive branch reach theta=2 rad? Find the first stop and work to it.
Course-specific review criteria
- Declare the selected system, fixed origin/axis and assumptions behind the conservation or work statement.
- Derive the appropriate integral, constraint or invariant with units and initial conditions.
- Check turning/contact branches and reconcile total versus component energy or angular momentum.
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.
26. Time integral, angle integral and an origin change
A fixed I=2 body has omega0=−1 and theta0=0. Net torque is 3t(2−t)−1 over 0–2 s in SI units. Find angular impulse, final L and angle. Separately r=(1,2),p=(3,2),F=(1,−2): calculate Lz and tauz about O, then about fixed Oprime=(1,0).
Course-specific review criteria
- Declare the selected system, fixed origin/axis and assumptions behind the conservation or work statement.
- Derive the appropriate integral, constraint or invariant with units and initial conditions.
- Check turning/contact branches and reconcile total versus component energy or angular momentum.
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.
27. Redistribution versus dissipative coupling
For the taught inertia-change system, omega_i=2 and final radius 1 m. Find initial/final inertia, angular momentum, speed and kinetic energy. Then independently couple two rotors I1=2,omega1=3 and I2=1,omega2=−2 without external torque. Find common final speed, energy loss and angular impulse received by each rotor.
Course-specific review criteria
- Declare the selected system, fixed origin/axis and assumptions behind the conservation or work statement.
- Derive the appropriate integral, constraint or invariant with units and initial conditions.
- Check turning/contact branches and reconcile total versus component energy or angular momentum.
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.
28. Static feasibility and reverse slip
A beta=1 rolling body is on a 40 degree incline with available mu_s=.30 and g=10. Is the no-slip solution supported? Separately use the taught horizontal cylinder with v0=1,omega0=8. Find contact-slip direction, friction direction, first no-slip time, final speeds and energy loss.
Course-specific review criteria
- Declare the selected system, fixed origin/axis and assumptions behind the conservation or work statement.
- Derive the appropriate integral, constraint or invariant with units and initial conditions.
- Check turning/contact branches and reconcile total versus component energy or angular momentum.
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.
29. Invariant constraints beyond the circular case
For GM=12, m=2,r0=4 and initial tangential factor f=.5 with zero radial speed, find E,L and the second turning radius. Identify whether r0 is the larger or smaller turning point. Separately change a circular orbit radius from 4 to 16 around the same source: find the period factor and energy-magnitude factor.
Course-specific review criteria
- Declare the selected system, fixed origin/axis and assumptions behind the conservation or work statement.
- Derive the appropriate integral, constraint or invariant with units and initial conditions.
- Check turning/contact branches and reconcile total versus component energy or angular momentum.
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.
30. Two initial conditions and a spring connection
For m=2 kg,k=8 N/m,x0=.3 m,v0=.4 m/s, derive x(t), amplitude and cosine phase, then x,v,a at t=pi/4 s. Separately place 6 and 3 N/m ideal springs in series under m=2 kg with g=10. Derive effective stiffness, equilibrium extension and period.
Course-specific review criteria
- Derive the restoring differential equation or relevant energy integral with units.
- Retain both initial conditions, the equilibrium reference and any small-amplitude or linear-response assumptions.
- Separate a mathematical fit from independent calibration and identify what the observations do not determine.
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.
31. A reported clock and an unchanged energy budget
The true record is x=.4cos(2t), but u=1.2t is reported. Derive indicated angular frequency and first/second derivative scale factors. Separately an ideal m=2,k=8 oscillator has A=.5 m and x=.25 m: find E,U,K, both velocities and endpoint-to-center time. Can its period be inferred from one energy-position pair without the model parameters?
Course-specific review criteria
- Derive the restoring differential equation or relevant energy integral with units.
- Retain both initial conditions, the equilibrium reference and any small-amplitude or linear-response assumptions.
- Separate a mathematical fit from independent calibration and identify what the observations do not determine.
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.
32. A force approximation is not a period measurement
For U=4x²/2+4x⁴/4,m=2 and release amplitude .5 m, find local frequency, cubic/linear force ratio at the endpoint and exact center speed. For a separate simple pendulum l=1 m,g=10,theta0=30 degrees, compute sine/linear restoring ratio and exact versus small-angle bottom speed. State whether either comparison gives an exact finite-amplitude period correction.
Course-specific review criteria
- Derive the restoring differential equation or relevant energy integral with units.
- Retain both initial conditions, the equilibrium reference and any small-amplitude or linear-response assumptions.
- Separate a mathematical fit from independent calibration and identify what the observations do not determine.
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.
33. Inertia normalization and an optimal gravity lever arm
For the normalized M=3,L=2 rod with beta=1, derive center coordinate and central inertia. At center-to-pivot distance .5 m, find pivot inertia and small-angle omega squared. Derive the continuous period-minimizing offset and explain the d→0 limit.
Course-specific review criteria
- Derive the restoring differential equation or relevant energy integral with units.
- Retain both initial conditions, the equilibrium reference and any small-amplitude or linear-response assumptions.
- Separate a mathematical fit from independent calibration and identify what the observations do not determine.
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.
34. One parameter ratio and two competing clock faults
A synthetic torsional body has I=2,kappa=4. Add independently known DeltaI=.5. First period is unbiased, second is multiplied by .90. Infer kappa and I from the two indicated periods. Then consider a different case in which both periods share gain1.1. Explain which inferred quantity remains correct and why one period alone cannot identify both parameters.
Course-specific review criteria
- Derive the restoring differential equation or relevant energy integral with units.
- Retain both initial conditions, the equilibrium reference and any small-amplitude or linear-response assumptions.
- Separate a mathematical fit from independent calibration and identify what the observations do not determine.
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.
35. Vector derivatives and a moving origin
A supplied planar trajectory is r(t)=(t²,2t) in SI units. At t=1 find velocity, acceleration, speed and rate of change of speed. A nonrotating observer has origin R(t)=(t,0). Find relative velocity/acceleration and explain whether this observer accelerates relative to the stipulated laboratory.
Course-specific review criteria
- Declare the system, interval, frame/axis, initial conditions and physical assumptions before selecting equations.
- Show the calculus or conservation derivation with units and independently recompute a changed case.
- Check branch/constraint feasibility, separate generated from observed evidence and retain a justified correction.
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.
36. A continuous center and a resistive initial-value law
A line from0 to2 m has lambda(x)=(3 kg/m²)x. Derive mass and center coordinate. Separately a2 kg body starts at rest under net force F=4−2v in SI units. Solve its velocity and position from x0=0, then evaluate at1 s. State why initial acceleration is not valid as a constant over the interval.
Course-specific review criteria
- Declare the system, interval, frame/axis, initial conditions and physical assumptions before selecting equations.
- Show the calculus or conservation derivation with units and independently recompute a changed case.
- Check branch/constraint feasibility, separate generated from observed evidence and retain a justified correction.
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.
37. A spatial force integral and instantaneous power
A2 kg particle begins at x=0 with kinetic energy1 J and positive velocity. Net force F(x)=6x−2x² N acts over0≤x≤2 m with consistent SI coefficients. Find work, final speed and instantaneous power at2. Construct a compatible potential and test whether the selected positive branch can reach2.
Course-specific review criteria
- Declare the system, interval, frame/axis, initial conditions and physical assumptions before selecting equations.
- Show the calculus or conservation derivation with units and independently recompute a changed case.
- Check branch/constraint feasibility, separate generated from observed evidence and retain a justified correction.
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.
38. An inelastic interval is not energy-conserving
A2 kg cart with velocity+3 m/s sticks to a1 kg cart with velocity−1 m/s. External impulse during collision is negligible. Find common velocity, kinetic-energy conversion and the impulse on each cart. Explain which quantities depend on the chosen two-cart boundary.
Course-specific review criteria
- Declare the system, interval, frame/axis, initial conditions and physical assumptions before selecting equations.
- Show the calculus or conservation derivation with units and independently recompute a changed case.
- Check branch/constraint feasibility, separate generated from observed evidence and retain a justified correction.
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.
39. Mass distribution and a time-dependent torque
A line over0–2 m has lambda(x)=(2 kg/m²)x. Compute M,xCM,Iend and Icm for perpendicular axes. It rotates about its fixed endpoint from rest under net torque tau(t)=t² N m with the appropriate SI coefficient over0–2 s. Derive final angular momentum, angular speed, angle and kinetic energy.
Course-specific review criteria
- Declare the system, interval, frame/axis, initial conditions and physical assumptions before selecting equations.
- Show the calculus or conservation derivation with units and independently recompute a changed case.
- Check branch/constraint feasibility, separate generated from observed evidence and retain a justified correction.
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.
40. Reject a contact shortcut and an orbital shortcut
An ideal beta=2/5 rolling body is on a30 degree incline with available mu_s=.15. Is no-slip motion supported? Separately a point-source orbit has GM=18,r=6 and tangential speed sqrt5 with zero radial speed. Find specific energy, compare circular and escape speeds and classify the ideal orbit. Optionally derive its other turning radius.
Course-specific review criteria
- Declare the system, interval, frame/axis, initial conditions and physical assumptions before selecting equations.
- Show the calculus or conservation derivation with units and independently recompute a changed case.
- Check branch/constraint feasibility, separate generated from observed evidence and retain a justified correction.
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.
41. Torsional initial conditions and a shared clock
A torsional oscillator has I=3 kg m²,kappa=12 N m/rad,theta0=.2 rad and initial angular velocity .4 rad/s. Find theta(t), amplitude, period and total energy. If known added inertia3 kg m² produces a second period, but both periods share gain1.1, what I and kappa would the two-period method infer?
Course-specific review criteria
- Declare the system, interval, frame/axis, initial conditions and physical assumptions before selecting equations.
- Show the calculus or conservation derivation with units and independently recompute a changed case.
- Check branch/constraint feasibility, separate generated from observed evidence and retain a justified correction.
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.
42. A shared-error alternative survives method agreement
For the taught stiffness packet, true k=8 and clock gain1.05 while static-force gain1. Compare both dynamic estimates with the static estimate. Give an independent test that separates timing and force calibration, and explain why the conditional reading intervals do not automatically cover the hidden gains. Then describe one retained unfamiliar transfer and one practical-evidence gate that remain after the authored61 destinations are available.
Course-specific review criteria
- Declare the system, interval, frame/axis, initial conditions and physical assumptions before selecting equations.
- Show the calculus or conservation derivation with units and independently recompute a changed case.
- Check branch/constraint feasibility, separate generated from observed evidence and retain a justified correction.
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.
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.
Calculus mechanics full-course evidence portfolio
Retain original predictions, calculus derivations, replayable virtual evidence and unfamiliar transfer across all seven units. Authored content and submitted self-reports do not establish independent mastery, performed laboratory time or AP authorization.
Kinematics and frame evidence
Derive a vector or piecewise trajectory with all initial conditions; compare derivatives, integrals and an observer transformation, including an event or uncertainty boundary.
Evidence to retain: First prediction, units, representation and changed-case calculation; distinguish generator truth from observations.
Forces and accessible motion
Create recipient-specific force diagrams, integrate a variable or resistive force law and test a static/contact or circular-motion constraint before accepting the trajectory.
Evidence to retain: System diagram, differential equation, supported versus rejected case and independently recomputed endpoint.
Energy and linear momentum intervals
Use a spatial work integral and a short collision separately; account for internal conversion, choose the system and carry the new initial state forward.
Evidence to retain: Time/angle/position axes labeled, momentum and energy budgets with changed assumptions and a retained critique.
Distributed rotation and angular systems
Integrate a mass distribution, select a fixed axis and reconcile angular impulse/work with changing-inertia or rolling/orbital constraints.
Evidence to retain: Mass normalization, inertia or cross-product derivation, origin/system choice and a feasibility check.
Oscillation and calibration challenge
Derive a harmonic initial state and a physical or torsional period; compare an approximation with an exact constraint and propose an independent calibration test.
Evidence to retain: Phase/energy derivation, parameter-identification limits, uncertainty assumptions and independent-reference plan.
Unfamiliar cumulative defense
Complete the mixed tasks across all seven units before reviewing their public solutions where possible; disclose exposure and preserve critique and revised reasoning.
Evidence to retain: Versioned task attempts, answer-exposure records, reviewer comments and a genuinely changed transfer case, not copied model output.
Supervised practical and external gates
Prepare a proposal with variables, instrument calibration, repeats and a risk assessment for qualified local educator review. Record execution only if it actually occurs; distinguish proposed, simulated and empirical evidence.
Evidence to retain: Externally supplied supervision, actual learner records and documented time where applicable. Never generate fictional raw measurements or claim approval from page availability.
Project review criteria
- All61 original destinations and the authored evidence workflow may be available without any particular learner having completed them.
- A qualified independent reviewer must evaluate unfamiliar learner reasoning; public solutions and arithmetic checks are not a secure examination.
- Current AP guidance requires minimum25% laboratory instructional time. Where onsite access is prevented, qualifying virtual/simulation student work may meet the requirement, but science-educator supervision and retained student records remain necessary. This app does not certify performed time or supervision.
- AP Course Audit authorization, college credit, actual laboratory completion, independent mastery and browser/mobile/keyboard validation are separate gates.
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
Defend a new multi-stage calculus mechanism and a new inference with changed geometry or shared bias. Identify what each method actually measures and what an independent observation could falsify.
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:ap-physics-c-mechanics.
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
- College Board AP Physics C Mechanics CED: Current Units 1–7 topic/LO/EK actions verified through official indexed extracts and Fall 2026 clarifications, checked 2026-10-30. Full-PDF materialization remained unavailable; original teaching and cases, no authorization claim.
- OpenStax University Physics — integral kinematics: Derivative/integral and initial-condition verification; supplied original polynomial cases.
- OpenStax University Physics — vector kinematics: Fixed-basis vector derivatives and direction definitions.
- All 61 finite authored destinations are present and independently reviewed; actual learner performance and external practical/technical gates remain open.
- No AP Course Audit approval, official credit or completed supervised laboratory time claimed. The minimum 25% laboratory-time requirement needs qualified supervision and retained learner records; conditional qualifying virtual work does not make page availability evidence of participation.
- Public solutions and locally retained self-reports do not establish independent learner mastery.
- Browser/mobile/keyboard and durable evidence export/import require technical verification separately from model tests.
Original LearnAI explanations and tasks use the references for scientific scope. Any attributed dataset or image has its own declared provenance. Laboratory or field activity requires an appropriate educator-approved protocol; no hazardous real-world operation is required by the guided path.
Save a science backup or import one
Science work is stored locally in this browser, separately from mathematics and AI. It does not sync automatically. Use this science-only backup; this checkout's older all-subject export does not include science.
A legacy file is available only when the complete snapshot fits the 36 MB limit. For larger snapshots, prepare a complete file set below; never discard notebooks to make a smaller file.
Matching imported notebook drafts replace their current drafts. Omitted lessons and known first attempts remain. Export your current work first if you want to retain both drafts.
Complete file sets and course backups
A file set can hold more work than one science file. Keep its manifest and every part together. Each file is limited to 36 MB. This format needs this version of LearnAI or a compatible newer version; older versions can use only the legacy single-file backup above.
The set includes every saved science course and global continuation at this moment. Later edits need a new export. A conflicting tab exports its own visible work, which may differ from saved work in another tab.
0 file(s) selected. File order and filenames do not determine identity. Missing, changed or conflicting parts block the entire restore; no good part is applied by itself.
Science storage: loading. No account or remote computation is required.