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Open Model Lab
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Pick a bounded task, then open the exact concept bench that can solve it.

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Start with Short-period matchBrowse starter paths
Best first challengeTo tryCore

Short-period match

Starting from Calm start, make the oscillator complete a shorter cycle without turning it into a wider swing. Keep the displacement graph open so the timing change stays visible.

Simple Harmonic Motion

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Starter challenge paths

Open these only when you want the next few challenge moves narrowed into one authored path.

Starter track0/10 solved

Motion and Circular Motion

Start with vector components, move into projectile paths, and then use circular motion to understand how velocity can keep changing direction.

Short-period force band is the next best challenge from Uniform Circular Motion.

Starter track0/15 solved

Rotational Mechanics

Start with torque as the turning effect of force, use centre of mass and support region for static balance, then carry the same rotational language into moment of inertia, rolling motion, and angular momentum.

Zero turn at the handle is the next best challenge from Torque.

Starter track0/12 solved

Gravity and Orbits

Start with one source mass creating a field and potential well, then use that same gravity model to explain circular speed, orbital periods, and the escape threshold.

Keep the heavier source circular is the next best challenge from Circular Orbits and Orbital Speed.

Show 16 more starter paths
Starter track0/6 solved

Oscillations and Energy

Build from one clean oscillator to energy exchange and then to driven resonance, so the same system grows without changing its core ideas.

Starter track0/10 solved

Fluid and Pressure

Start with pressure in a resting fluid, then carry that same branch through continuity, Bernoulli, buoyancy, and drag-limited motion.

Starter track0/19 solved

Waves

Use oscillation as the entry point, lock down wave speed and wavelength, carry that into longitudinal sound and pitch-versus-loudness cues, add beats as the nearby-frequency superposition bridge, then move into Doppler shifts, interference, standing-wave patterns, and open-vs-closed air-column resonance without losing the live connection between motion and graph.

Starter track0/8 solved

Thermodynamics and Kinetic Theory

Start with temperature-versus-internal-energy bookkeeping, reuse that particle story for gas pressure, then follow energy transfer into heating curves and phase-change shelves.

Starter track0/7 solved

Electricity

Start with source charges and voltage, then carry that same circuit story into current, power, branch behavior, and equivalent resistance.

Starter track0/6 solved

Magnetism

Start with current-made magnetic fields, turn changing flux into induced emf with Faraday and Lenz, and then reuse that same field direction story to explain magnetic force on charges and currents.

Starter track0/11 solved

Sound and Acoustics

Stay on the sound branch long enough that longitudinal motion, pitch-versus-loudness cues, beats, Doppler shifts, and open-vs-closed air-column resonance feel like one acoustics path instead of isolated pages.

Starter track0/10 solved

Wave Optics

Follow the bounded wave-optics branch from polarization into diffraction, double-slit interference, color-dependent refraction, and imaging limits so the newer optics pages read like one compact path instead of isolated stops.

Starter track0/10 solved

Modern Physics

Follow the bounded modern-physics branch from threshold emission into line spectra, matter waves, the Bohr hydrogen model, and half-life so the new concept set reads like one path instead of five isolated pages.

Starter track0/6 solved

Functions and Change

Keep the first math path compact: read parent-curve moves first, then rational asymptotes and domain breaks, then exponential growth and decay, local slope, visible limit behavior, and finally accumulation so change stays graph-first all the way through.

Starter track0/6 solved

Complex and Parametric Motion

Start with complex numbers as points on one plane, turn that plane into unit-circle and polar-coordinate geometry, deepen that same bench into trig identities and inverse-angle reasoning, then carry the coordinate language into motion traced from x(t) and y(t).

Starter track0/4 solved

Vectors and Motion Bridge

Start with vectors as geometric objects on a 2D plane, then carry the same component language into the existing motion-facing vectors bench.

Starter track0/4 solved

Rates and Equilibrium

Start with successful collisions setting reaction rate, then reuse the same chemistry language inside a reversible system that re-balances after a disturbance.

Starter track0/3 solved

Stoichiometry and Yield

Start with one visible reaction recipe, use the lower batch cap to identify the limiting reagent, and then compare actual output with the same theoretical marker.

Starter track0/6 solved

Solutions and pH

Start with concentration in one beaker, add solubility limits and saturation, then reuse that same solution language to read pH, buffers, and neutralization.

Starter track0/5 solved

Algorithms and Search Foundations

Start with visible list work, reuse that search language for binary search, and then carry the branch into one live graph bench for adjacency, BFS, DFS, and visited-state behavior.

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Showing 172 of 172 challenge entries.

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Core

121 prompts in this depth

OscillationsTo tryCore

Short-period match

Starting from Calm start, make the oscillator complete a shorter cycle without turning it into a wider swing. Keep the displacement graph open so the timing change stays visible.

Simple Harmonic Motion

Path: Oscillations and Energy, Waves

Match3 checks
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OscillationsTo tryCore

Build five joules

From the mixed-energy baseline, raise the stored energy to about 5 J without making the oscillator heavier than about 1.2 kg.

Oscillation Energy

Path: Oscillations and Energy

Target3 checks
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WavesTo tryCore

One-cycle probe lag

Starting from Baseline, move the probe until it sits one wavelength downstream: same phase shape, one full cycle later.

Wave Speed and Wavelength

Path: Waves

Condition4 checks
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WavesTo tryCore

Faster wave, same spacing

From Baseline, keep the crest spacing at 1.6 m and raise the wave speed until the source cycles faster and the probe delay shrinks.

Wave Speed and Wavelength

Path: Waves

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Show 117 more core challenges
SoundTo tryCore

Tune slow pulses

Starting near unison, tune the source pair until the envelope pulses at about 0.2 Hz while the source amplitude stays near the baseline.

Beats

Path: Waves, Sound and Acoustics

Match5 checks
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SoundTo tryCore

Remove the beat without muting

Starting near unison, make the loud-soft beat disappear while keeping the source amplitude near the baseline.

Beats

Path: Waves, Sound and Acoustics

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WavesTo tryCore

Higher pitch ahead

Keep the emitted tone near 1.1 Hz and tune the live setup so the observer clearly hears a higher pitch on the moving-source bench.

Doppler Effect

Path: Waves, Sound and Acoustics

Target6 checks
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SoundTo tryCore

Find a strong compression

Starting from Baseline, move the probe until it sits inside a strong compression while the probe-pressure graph and compression overlay stay visible.

Sound Waves and Longitudinal Motion

Path: Waves, Sound and Acoustics

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WavesTo tryCore

Find a dark band

Starting from Center bright, move the probe onto a dark region where the screen intensity almost vanishes.

Wave Interference

Path: Waves

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WavesTo tryCore

Probe on a node

Starting from the third harmonic, move the probe onto a node so the local oscillation envelope collapses almost to zero.

Standing Waves

Path: Waves

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SoundTo tryCore

Closed-end stillness

Starting from the closed-pipe third harmonic, move the probe onto the closed wall so parcel motion nearly disappears while the pressure cue stays strong.

Resonance in Air Columns / Open and Closed Pipes

Path: Waves, Sound and Acoustics

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MechanicsTo tryCore

Short-period force band

Starting from the reference orbit, keep the radius close to the original circle but shorten the period to about 2.2 s. Land the motion in the speed and centripetal-acceleration bands that go with that stronger centripetal-force requirement.

Uniform Circular Motion

Path: Motion and Circular Motion

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MechanicsTo tryCore

Tight inward pull

Starting from the reference orbit, tune the motion until the inward acceleration sits in the 7 to 8.5 m/s² band while the tangential speed stays between 2.5 and 3.1 m/s.

Uniform Circular Motion

Path: Motion and Circular Motion

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OscillationsTo tryCore

Lock near resonance

Starting from Free swing, switch into the response view and tune the driver until it sits very close to resonance with a strong steady-state response.

Damping / Resonance

Path: Oscillations and Energy

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MechanicsTo tryCore

Equal components

Build a vector whose horizontal and vertical components are nearly the same size. Keep the component graph open so the match is visible in the real readout.

Vectors and Components

Path: Motion and Circular Motion, Vectors and Motion Bridge

Match6 checks
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MechanicsTo tryCore

Downward, same length

From Balanced split, keep the vector length near 8 m/s but rotate it into a downward-right direction. Prove the negative vertical component means direction, not a smaller vector.

Vectors and Components

Path: Motion and Circular Motion, Vectors and Motion Bridge

Condition8 checks
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MechanicsTo tryCore

Zero turn at the handle

Keep the push point near the handle but make the bar feel almost no turning effect.

Torque

Path: Rotational Mechanics

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MechanicsTo tryCore

Compact and quick

Keep the motor near the baseline torque, then make the rotor spin up sharply by changing only the mass layout.

Rotational Inertia / Moment of Inertia

Path: Rotational Mechanics

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MechanicsTo tryCore

Balance the heavy right load

Starting from Tips right, move the support centre until the heavy right load is back in static equilibrium.

Static Equilibrium / Centre of Mass

Path: Rotational Mechanics

Target12 checks
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MechanicsTo tryCore

Barely stable, not tipping

Starting from Support under load, shift the support region left until the plank is only just stable but not yet tipping.

Static Equilibrium / Centre of Mass

Path: Rotational Mechanics

Target10 checks
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MechanicsTo tryCore

Wide and still same-L

Start from the lab baseline, then build a wide layout that still carries nearly the same angular momentum.

Angular Momentum

Path: Rotational Mechanics

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MechanicsTo tryCore

Match momentum with a gentler pulse

Starting from the short hard push, build a lower, longer rightward pulse that still delivers nearly the same impulse and final momentum.

Momentum and Impulse

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MechanicsTo tryCore

Brake with negative impulse

Make a right-moving cart receive a leftward pulse, then explain how the negative force-time area reduces the final momentum.

Momentum and Impulse

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MechanicsTo tryCore

Fast run, same ramp

Keep the ramp near its baseline angle and tune the roller so it reaches the bottom in under about 1.85 s.

Rolling Motion

Path: Rotational Mechanics

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MechanicsTo tryCore

Reverse A, keep total positive

Make cart A end up moving left while the isolated two-cart system still has positive total momentum and a rightward center-of-mass drift.

Conservation of Momentum

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MechanicsTo tryCore

Sticky but still conserved

Build a nearly sticky equal-mass collision, keep the energy graph open, and explain why momentum can stay conserved while kinetic energy drops.

Collisions

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MechanicsTo tryCore

Light cart rebounds

Make a light cart A rebound from a heavier cart B while the velocity graph and momentum bars stay visible.

Collisions

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MechanicsTo tryCore

Flat long shot

Starting from Earth shot, stretch the landing point into the 35 to 38 m range while keeping the apex below about 10 m.

Projectile Motion

Path: Motion and Circular Motion

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FluidsTo tryCore

Build the 27 kPa throat

Start from Level venturi and adjust only the throat width until the throat pressure is about 27.1 kPa while the entry state stays near baseline.

Bernoulli's Principle

Path: Fluid and Pressure

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FluidsTo tryCore

Half-submerged balance

Adjust the block so it could stay about half submerged without extra support.

Buoyancy and Archimedes' Principle

Path: Fluid and Pressure

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Gravity and OrbitsTo tryCore

Keep the heavier source circular

Starting from Reference orbit, make the source heavier while keeping the chosen radius near 1.6 m and the orbit circular.

Circular Orbits and Orbital Speed

Path: Gravity and Orbits

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Gravity and OrbitsTo tryCore

Too slow falls inward

Starting from Reference orbit, lower the speed just enough that gravity is clearly stronger than the turning requirement and the path bends inside the dashed circle.

Circular Orbits and Orbital Speed

Path: Gravity and Orbits

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FluidsTo tryCore

Make section B twice as fast

Start from Uniform pipe and adjust only section B until the middle speed is about twice the section A speed while the same baseline flow rate is kept.

Continuity Equation

Path: Fluid and Pressure

Match6 checks
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FluidsTo tryCore

Build a slower terminal-speed case

Starting from Baseline drop, keep the mass near 2 kg and lower the terminal speed by increasing both area and drag strength.

Drag and Terminal Velocity

Path: Fluid and Pressure

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ElectricityTo tryCore

Build the upward field

Starting from Dipole reference, turn the source pair into an equal positive arch so the horizontal field cancels while the net field still points upward.

Electric Fields

Path: Electricity

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ElectricityTo tryCore

Flip the force, keep the field

Starting from Like-charge arch, reverse the force on the test charge while keeping the same upward field symmetry at the probe.

Electric Fields

Path: Electricity

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Gravity and OrbitsTo tryCore

Quarter the field by distance

Starting from Axis near, move the probe to the doubled-distance case on the same horizontal line so the field magnitude falls to about one quarter of the 1 m reference.

Gravitational Fields

Path: Gravity and Orbits

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Gravity and OrbitsTo tryCore

Double the force, not the field

Starting from Baseline diagonal, change only the probe mass so the force magnitude doubles while the gravitational field stays the same.

Gravitational Fields

Path: Gravity and Orbits

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Gravity and OrbitsTo tryCore

Half the potential, quarter the field

Starting from Axis 1 m, move the probe to the doubled-distance case on the same horizontal line so phi is about half as deep and the field magnitude is about one quarter as large.

Gravitational Potential and Potential Energy

Path: Gravity and Orbits

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Gravity and OrbitsTo tryCore

Double the energy, not the potential

Starting from Baseline diagonal, change only the probe mass so the potential energy doubles in magnitude while the potential and field stay fixed.

Gravitational Potential and Potential Energy

Path: Gravity and Orbits

Target8 checks
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ThermodynamicsTo tryCore

Make radiation largest

Tune the live setup until radiation is the largest pathway while the block is still clearly hotter than the room.

Heat Transfer

Path: Thermodynamics and Kinetic Theory

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ThermodynamicsTo tryCore

Compress to double the pressure

Start from Room baseline and lower only the volume until the pressure is about double while the temperature and particle count stay near the baseline values.

Ideal Gas Law and Kinetic Theory

Path: Thermodynamics and Kinetic Theory

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FluidsTo tryCore

Hit 24 kPa by depth alone

Start from Water baseline and adjust only the probe depth until the total pressure is about 24 kPa while the piston load, area, density, and gravity stay near baseline.

Pressure and Hydrostatic Pressure

Path: Fluid and Pressure

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ElectricityTo tryCore

Double the storage with area

Starting from Compact starter, change only the plate area until the stored charge and stored energy are both doubled while the battery voltage and gap stay fixed.

Capacitance and Stored Electric Energy

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ElectricityTo tryCore

Quadruple the energy with voltage

Starting from Low-voltage reference, change only the battery voltage until the stored energy is about four times larger while the geometry stays fixed.

Capacitance and Stored Electric Energy

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ElectricityTo tryCore

Positive midpoint plateau

Starting from Dipole reference, tune the setup until the midpoint has almost zero field but still sits on a clearly positive potential hill.

Electric Potential

Path: Electricity

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Gravity and OrbitsTo tryCore

Heavier source, shorter year

Starting from Baseline year, raise the source mass while keeping the same radius circular so the period becomes clearly shorter.

Kepler's Third Law and Orbital Periods

Path: Gravity and Orbits

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Gravity and OrbitsTo tryCore

Break the circular-year case

Starting from Baseline year, lower the speed enough that the path is no longer the circular orbit Kepler's law is describing.

Kepler's Third Law and Orbital Periods

Path: Gravity and Orbits

Target9 checks
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CircuitsTo tryCore

Full-voltage parallel pair

Starting from the matched series pair, change only the circuit structure needed to give each branch the full battery voltage and make the total current land near 4 A.

Basic Circuits

Path: Electricity

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Gravity and OrbitsTo tryCore

Remove the finite turnaround

Starting from High but bound, raise the launch just to the threshold case at the same source mass and launch radius so the total specific energy is about zero and the finite turnaround disappears.

Escape Velocity

Path: Gravity and Orbits

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Gravity and OrbitsTo tryCore

High climb, still bound

Starting from Threshold launch, tune a launch that still begins from 1.6 m around a 4 kg source, climbs high, but remains bound with a finite turnaround near 10.4 m.

Escape Velocity

Path: Gravity and Orbits

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OpticsTo tryCore

Leave visible, keep the wave

Move the source just beyond visible red into infrared while keeping the field pair visible and the visible-window overlay on.

Light as an Electromagnetic Wave

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OpticsTo tryCore

Set a half-power case

Starting from Aligned pass, tune the bench until the detector reads about one half of the incoming intensity for a linear input.

Polarization

Path: Wave Optics

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OpticsTo tryCore

Create a half-bright first polarizer

Starting from Crossed axes, switch the bench to an unpolarized first-pass case that still leaves the detector near half brightness.

Polarization

Path: Wave Optics

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OpticsTo tryCore

Set a clear spread angle

Starting from Wide slit, tune the controls until the first minimum lands between 22 ° and 28 °.

Diffraction

Path: Wave Optics

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OpticsTo tryCore

Find the first dark band

Starting from Center bright, move the probe onto the first dark band without changing the slit or wavelength.

Diffraction

Path: Wave Optics

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OpticsTo tryCore

Find the first dark fringe

Starting from Center bright, move the probe onto the first dark fringe without changing wavelength, slit separation, or screen distance.

Double-Slit Interference

Path: Wave Optics

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OpticsTo tryCore

Set wide fringes

Starting from Tight fringes, tune the geometry until the bright-fringe spacing lands between 1.8 m and 2.1 m.

Double-Slit Interference

Path: Wave Optics

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CircuitsTo tryCore

Steady 18-watt load

Starting from Gentle glow, keep the 8 ohm load and raise the source until the stage power bar settles near 18 W.

Power and Energy in Circuits

Path: Electricity

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OpticsTo tryCore

Hit the transmitted-angle target

Starting from Air to glass, tune the setup until the refracted angle lands between 25 and 28 degrees while the speed ratio v2/v1 stays between 0.62 and 0.69.

Refraction / Snell's Law

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OpticsTo tryCore

Hit the index-and-bend target

Starting from Crown green, tune the current wavelength and prism so the selected refractive index lands between 1.53 and 1.55 while the selected deviation lands between 11.0 and 12.0 degrees.

Dispersion / Refractive Index and Color

Path: Wave Optics

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CircuitsTo tryCore

Blue branch at full voltage

Starting from Unequal series loads, rewire the setup so Load B keeps the full battery voltage while its branch current stays around 1 A.

Series and Parallel Circuits

Path: Electricity

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OpticsTo tryCore

Land just below the threshold

Starting from Glass to air near critical, tune the setup until the incident angle stays just below theta_c: keep theta_1 - theta_c between -2 and -0.4 degrees while theta_2 remains between 74 and 89 degrees.

Total Internal Reflection

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OpticsTo tryCore

Cross into TIR cleanly

Starting from Glass to air below critical, raise the setup until the boundary is clearly above threshold while staying on the same media pair.

Total Internal Reflection

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CircuitsTo tryCore

Build the clean current split

Starting from the balanced parallel pair, change only R3 until the split reads about I_total = 1.5 A, I_2 = 1.0 A, and I_3 = 0.5 A.

Kirchhoff Loop and Junction Rules

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CircuitsTo tryCore

Build the clean loop balance

Starting from the balanced series group, change only R3 until the full loop reads about V_1 = 3 V, V_2 = 3 V, and V_3 = 6 V with one consistent loop equation.

Kirchhoff Loop and Junction Rules

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Mirrors and LensesTo tryCore

Real-image target

Starting from the concave real-image preset, tune the setup until the image distance lands between 1.0 and 1.2 m and the magnification lands between -1.4 and -1.1.

Mirrors

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Mirrors and LensesTo tryCore

Virtual-image target

Starting from the inside-focus preset, make a virtual upright image with d_i between -0.90 and -0.75 m and magnification between 2.2 and 2.6.

Mirrors

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ElectromagnetismTo tryCore

One wavelength downstream

Starting from the baseline wave, place the probe exactly one wavelength downstream so the source and probe have the same phase one full cycle later.

Electromagnetic Waves

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CircuitsTo tryCore

Collapse the parallel group

Starting from the balanced series group, switch the highlighted pair into the parallel case until the reduction card reads about 3 ohm for the grouped pair and about 7 ohm for the full circuit.

Equivalent Resistance

Path: Electricity

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ElectromagnetismTo tryCore

High flux, zero emf

Starting from Approach and pass, hold the magnet near the coil center so the coil still links strong flux while the induced emf collapses nearly to zero.

Faraday's Law and Lenz's Law

Path: Magnetism

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Mirrors and LensesTo tryCore

Real-image target

Starting from the converging reference, tune the setup until the image distance lands between 1.0 and 1.2 m and the magnification lands between -1.4 and -1.1.

Lens Imaging

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Mirrors and LensesTo tryCore

Virtual-image target

Starting from the diverging reference, make a virtual image with d_i between -0.65 and -0.45 m and magnification between 0.3 and 0.5.

Lens Imaging

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MagnetismTo tryCore

Build the upward magnetic field

Starting from Same-current sweep, reverse Wire B so the sideways contributions nearly cancel while the net magnetic field points strongly upward above the midpoint.

Magnetic Fields

Path: Magnetism

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MagnetismTo tryCore

Charge down, wire up

Starting from Positive bends down, change the setup so the moving charge force points downward while the wire-segment force points upward for the same rightward direction.

Magnetic Force on Moving Charges and Currents

Path: Magnetism

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Mirrors and LensesTo tryCore

Hit the Rayleigh threshold

Starting from Blurred pair, tune the aperture or wavelength until the point spacing sits right on the Rayleigh limit.

Optical Resolution / Imaging Limits

Path: Wave Optics

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Mirrors and LensesTo tryCore

Open a clear central dip

Starting from Near threshold, make the split clearly visible without changing the detector sample control.

Optical Resolution / Imaging Limits

Path: Wave Optics

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Modern PhysicsTo tryCore

Find the stopping point

Starting from Violet above threshold, make the collected current almost vanish without changing the frequency or work function.

Photoelectric Effect

Path: Modern Physics

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Modern PhysicsTo tryCore

Brightness is not enough

Starting from Bright but still below threshold, keep the beam bright while proving the collector current can stay essentially zero.

Photoelectric Effect

Path: Modern Physics

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CircuitsTo tryCore

Build the slower response

Starting from Fast charge, change only R and C until the time constant is about 4 s while the source voltage stays fixed.

RC Charging and Discharging

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Modern PhysicsTo tryCore

Keep only two visible emission lines

Starting from Hydrogen-like emission, tune the gaps so only two visible lines remain while the spectrum still stretches from blue-visible to red-visible wavelengths.

Atomic Spectra

Path: Modern Physics

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Modern PhysicsTo tryCore

Match the same lines in absorption

Starting from Wide upper gap, switch into absorption and tune the ladder until you have three visible notches with a clear red-to-blue spread.

Atomic Spectra

Path: Modern Physics

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CircuitsTo tryCore

Heavy-load terminal drop

Starting from Lossy source, change only the load until the source is driving about 2 A and the terminal voltage lands near 6 V.

Internal Resistance and Terminal Voltage

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CircuitsTo tryCore

Recover the near-ideal source

Starting from Lossy source, reduce only the internal resistance until the internal loss is below 1 W and the terminal voltage is above 10 V.

Internal Resistance and Terminal Voltage

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Modern PhysicsTo tryCore

Find the one-fit electron

Starting from Slow electron, tune the speed until the fixed loop is close to one wavelength long without changing the particle mass.

de Broglie Matter Waves

Path: Modern Physics

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Modern PhysicsTo tryCore

Make a heavier same-speed particle

Starting from Near one-fit electron, keep the speed near the same value but make the particle heavy enough that roughly two wavelengths fit around the loop.

de Broglie Matter Waves

Path: Modern Physics

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Modern PhysicsTo tryCore

Find the red Balmer line

Starting from Balmer beta, tune the live state until the active transition is the classic red Balmer line while the page stays in emission.

Bohr Model

Path: Modern Physics

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Modern PhysicsTo tryCore

Match the reverse UV excitation

Starting from Lyman alpha emission, switch to the matching reverse excitation from the ground level while keeping the same ultraviolet wavelength.

Bohr Model

Path: Modern Physics

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Modern PhysicsTo tryCore

Land on the one-half-life checkpoint

Starting from Class-lab sample, scrub to about one half-life so the expectation is halved while the live tray stays slightly below it.

Radioactivity and Half-Life

Path: Modern Physics

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Modern PhysicsTo tryCore

Catch a noisy small sample

Starting from Small noisy sample, scrub to about one half-life so the live tray sits well below the smooth expectation and the spread is obvious.

Radioactivity and Half-Life

Path: Modern Physics

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FunctionsTo tryCore

Reflect and land the vertex

Build a reflected graph whose transformed vertex lands near (-1, 2.4). Keep the reference curve and shift guide on so the move stays tied to the base landmark.

Graph Transformations

Path: Functions and Change

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FunctionsTo tryCore

Quarter-target checkpoint

Build a decay case where the target is about one quarter of the start, so the curve reaches it in about two half-lives.

Exponential Change / Growth, Decay, and Logarithms

Path: Functions and Change

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FunctionsTo tryCore

Domain-break checkpoint

Build a reciprocal family where the true vertical asymptote sits near x=-1, the removable hole sits on positive x, and the right branch stays below the horizontal asymptote.

Rational Functions / Asymptotes and Behavior

Path: Functions and Change

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CalculusTo tryCore

Catch the flat tangent

Move to the right-hand turning point so the tangent is almost flat, then shrink delta x until the secant slope is almost flat there too.

Derivative as Slope / Local Rate of Change

Path: Functions and Change

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CalculusTo tryCore

Continuity classification checkpoint

Switch to the case where both one-sided values nearly agree, but the graph still is not continuous at x=0.

Limits and Continuity / Approaching a Value

Path: Functions and Change

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CalculusTo tryCore

Find the square maximum

Move the width until the fixed-perimeter rectangle reaches the maximum area and the local area slope is essentially zero.

Optimization / Maxima, Minima, and Constraints

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CalculusTo tryCore

Negative height, positive total

Move the bound into a region where the source height is already negative, but the running total is still positive overall.

Integral as Accumulation / Area

Path: Functions and Change

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VectorsTo tryCore

Near-zero resultant

Adjust the vectors until the resultant lands very close to the origin, but keep the scaled first vector clearly nontrivial so the cancellation has to be earned.

Vectors in 2D

Path: Vectors and Motion Bridge

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Right-shear checkpoint

Start from the identity matrix and build a right shear that keeps the first basis vector near its original x-axis direction while the unit square becomes a right-leaning parallelogram. Keep the basis and square visible so the column story stays on the plane.

Matrix Transformations / Stretch, Shear, Reflection

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Complex Numbers and Parametric MotionTo tryCore

Rotate onto the positive imaginary axis

Build a multiplication case where z · w lands almost on the positive imaginary axis while the multiplier magnitude stays close to one.

Complex Numbers on the Plane

Path: Complex and Parametric Motion

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VectorsTo tryCore

Orthogonal projection checkpoint

Adjust \vecB until the amber projection nearly collapses while both arrows stay clearly nonzero. Keep the angle marker and projection guide visible so the right-angle story stays geometric.

Dot Product / Angle and Projection

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Complex Numbers and Parametric MotionTo tryCore

Tall, fast, and near the axis

Build a curve that is clearly taller than it is wide, then pause when the point is near the y-axis and still moving quickly.

Parametric Curves / Motion from Equations

Path: Complex and Parametric Motion

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Radius-angle to x-y checkpoint

Build a point in Quadrant II where the leftward x projection is noticeably larger in magnitude than the upward y projection. Keep the coordinate guides on so r, θ, x, and y stay tied to the same point.

Polar Coordinates / Radius and Angle

Path: Complex and Parametric Motion

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Quadrant II sign checkpoint

Start from the axis-crossing view, then push the point just into Quadrant II so cosine has flipped negative while sine is still strongly positive. Keep the projection guides and sign map on so the crossing stays visible.

Unit Circle / Sine and Cosine from Rotation

Path: Complex and Parametric Motion

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Rates and EquilibriumTo tryCore

More success, not just more hits

Build a setup where the reaction is clearly active even though concentration stays modest. Keep the threshold cue and collision pulses on so the rate story stays visible.

Reaction Rate / Collision Theory

Path: Rates and Equilibrium

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Catalyst, not heat

Use the catalyst to raise the success story while keeping temperature near baseline. Prove the barrier changed instead of pretending the particles got hotter.

Reaction Rate / Collision Theory

Path: Rates and Equilibrium

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Complex Numbers and Parametric MotionTo tryCore

Three-four-five identity checkpoint

Start in Quadrant I and set the point so cosine is near 0.6 while sine is near 0.8. Keep the squared-projection graph open so the identity line stays in view while you tune the angle.

Trig Identities from Unit-Circle Geometry

Path: Complex and Parametric Motion

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Quadrant II angle-from-ratio checkpoint

Build a point whose ratio y / x is negative but whose full angle is clearly in Quadrant II, not Quadrant IV. Keep the angle-recovery graph open so the principal-angle output and the actual angle disagree visibly.

Inverse Trig / Angle from Ratio

Path: Complex and Parametric Motion

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Stoichiometry and YieldTo tryCore

Build the matched 3:2 run

Set the bench to a 3 A + 2 B recipe with supplies that still finish four full batches together.

Stoichiometric Ratios and Recipe Batches

Path: Stoichiometry and Yield

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Stoichiometry and YieldTo tryCore

Make B limit first

Keep the 2 A + 3 B recipe, but set the supplies so B is limiting and 3 A packets remain after the full-yield run.

Limiting Reagent and Leftover Reactants

Path: Stoichiometry and Yield

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Stoichiometry and YieldTo tryCore

Hit 75% yield on a matched run

Use the matched 2 A + 3 B run and set percent yield to 75% so the actual output lands at 3.75 batches.

Percent Yield and Reaction Extent

Path: Stoichiometry and Yield

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Solutions and pHTo tryCore

Dilute without losing solute

Starting from a fairly crowded beaker, lower the concentration clearly while keeping the solute amount near the same starting value.

Concentration and Dilution

Path: Solutions and pH

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Dissolve the excess without removing solute

Starting from a saturated beaker, make the excess pile disappear while keeping the same total solute amount in the system.

Solubility and Saturation

Path: Solutions and pH

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Extra solute after the capacity line

Keep the at-the-limit capacity fixed, then add much more solute. Prove the dissolved amount stays capped while excess solid grows.

Solubility and Saturation

Path: Solutions and pH

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Land near neutral

Adjust the mixture until the pH sits near neutral while acid and base character stay visibly close together.

Acid-Base / pH Intuition

Path: Solutions and pH

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Algorithms and SearchTo tryCore

Use insertion where it pays off

Build a case where insertion sort finishes a nearly sorted list with very few writes.

Sorting and Algorithmic Trade-offs

Path: Algorithms and Search Foundations

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Solutions and pHTo tryCore

Hold near neutral under an acid pulse

Starting from the unbuffered acid pulse, add enough buffer reserve to bring pH near neutral while keeping the acid and base amounts in place and without flooding the beaker with extra water.

Buffers and Neutralization

Path: Solutions and pH

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Algorithms and SearchTo tryCore

Find a far-right target fast

Build a large ordered list where the target sits near the far-right edge, but binary search still finds it in five checks or fewer.

Binary Search / Halving the Search Space

Path: Algorithms and Search Foundations

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Build a wider first frontier

Start from the hub-and-detours graph at C, keep the adjacency cue visible, and explain which direct neighbors make the first frontier wider than the edge start.

Graph Representation and Adjacency Intuition

Path: Algorithms and Search Foundations

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Build a wide layered frontier

Start from the hub-and-detours graph at C, keep BFS mode on, and explain why the frontier widens before the target is reached without breaking layer order.

Breadth-First Search and Layered Frontiers

Path: Algorithms and Search Foundations

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Build a deep backtracking branch

Switch to the bridge-cycle DFS scene, keep DFS mode on, and explain why the current depth can rise before the search backs up instead of looping through the cycle.

Depth-First Search and Backtracking Paths

Path: Algorithms and Search Foundations

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Challenge depth

Stretch

51 prompts in this depth

OscillationsTo tryStretch

Compare the timing

Use compare mode to keep a calm baseline in Setup A and make Setup B cycle faster while both setups keep about the same swing size.

Simple Harmonic Motion

Path: Oscillations and Energy, Waves

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OscillationsTo tryStretch

Equal split

Starting from Mixed energy, pause at a moment when kinetic and potential energy are nearly equal. Keep the energy graph visible so the balance is honest.

Oscillation Energy

Path: Oscillations and Energy

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SoundTo tryStretch

Same beat, lower carrier

Enter compare mode and make Setup B keep the same beat frequency as Setup A while clearly lowering the average carrier frequency.

Beats

Path: Waves, Sound and Acoustics

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WavesTo tryStretch

Lower behind, higher ahead

Enter compare mode and make Setup A hear a lower pitch than emitted while Setup B hears a higher pitch, with both sources keeping the same emitted frequency.

Doppler Effect

Path: Waves, Sound and Acoustics

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SoundTo tryStretch

Louder, same pitch

Enter compare mode and make Setup B louder than Setup A while keeping the pitch the same.

Pitch, Frequency, and Loudness / Intensity

Path: Waves, Sound and Acoustics

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Higher, same loudness cue

Enter compare mode and make Setup B higher in pitch than Setup A while keeping the amplitude and intensity cue matched.

Pitch, Frequency, and Loudness / Intensity

Path: Waves, Sound and Acoustics

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SoundTo tryStretch

One wavelength later

Enter compare mode and keep Setup B one full wavelength farther downstream than Setup A so both probes share the same phase relation but Setup B arrives one cycle later.

Sound Waves and Longitudinal Motion

Path: Waves, Sound and Acoustics

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WavesTo tryStretch

Bright zero crossing

From Center bright, pause at a moment when the combined amplitude is still large but the instantaneous probe displacement has crossed through zero.

Wave Interference

Path: Waves

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WavesTo tryStretch

Center zero crossing

From the fundamental mode, keep the probe at the center antinode and pause right as that antinode crosses through zero displacement.

Standing Waves

Path: Waves

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SoundTo tryStretch

Same slider, different harmonic family

Enter compare mode, keep both setups at the same tube length and the same resonance-order slider setting of 2, then make Setup B the closed-open tube so Setup A lands on the 2nd harmonic while Setup B lands on the 3rd harmonic at a lower frequency.

Resonance in Air Columns / Open and Closed Pipes

Path: Waves, Sound and Acoustics

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MechanicsTo tryStretch

Same period, bigger inward pull

Open compare mode and keep Setup A and Setup B on nearly the same period, but make Setup B need the larger centripetal pull by giving it the wider orbit.

Uniform Circular Motion

Path: Motion and Circular Motion

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MechanicsTo tryStretch

Compare the spin

Open compare mode from the reference orbit and make Setup B complete turns noticeably faster than Setup A while both setups keep nearly the same radius.

Uniform Circular Motion

Path: Motion and Circular Motion

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OscillationsTo tryStretch

Late small motion

From Free swing, make the transient decay quickly enough that a late inspected sample shows only a very small displacement.

Damping / Resonance

Path: Oscillations and Energy

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MechanicsTo tryStretch

Hit the endpoint

Pause at the end of the 4 s walk and make the point land near (16 m, 12 m).

Vectors and Components

Path: Motion and Circular Motion, Vectors and Motion Bridge

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MechanicsTo tryStretch

Same torque, shorter lever arm

Open compare mode and make Setup B twist just as hard as Setup A even though Setup B pushes much closer to the pivot.

Torque

Path: Rotational Mechanics

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MechanicsTo tryStretch

Clockwise finish

Pause at the end of the clip and tune a clean clockwise twist.

Torque

Path: Rotational Mechanics

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MechanicsTo tryStretch

Same torque, very different response

Open compare mode and make Setup B much harder to spin than Setup A without changing the torque in either setup.

Rotational Inertia / Moment of Inertia

Path: Rotational Mechanics

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MechanicsTo tryStretch

Strong motor, same inertia

Pause at the end of the clip and tune a wide-rim rotor that still reaches a moderate final angular speed.

Rotational Inertia / Moment of Inertia

Path: Rotational Mechanics

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MechanicsTo tryStretch

Same centre of mass, different cargo

Open compare mode. Keep Setup A on Support under load, then tune Setup B so a heavier cargo placed closer in lands on the same combined centre of mass.

Static Equilibrium / Centre of Mass

Path: Rotational Mechanics

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MechanicsTo tryStretch

Same L, different spin

Open compare mode and make Setup A compact and Setup B wide while keeping their angular momenta nearly matched.

Angular Momentum

Path: Rotational Mechanics

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MechanicsTo tryStretch

Read the end-of-clip consequence

Pause at the end of the clip and tune a wide same-L rotor whose slow spin makes the accumulated angle stay small.

Angular Momentum

Path: Rotational Mechanics

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MechanicsTo tryStretch

Same ramp, different finish

Open compare mode and make Setup B finish much later than Setup A while keeping both setups on the same slope and radius.

Rolling Motion

Path: Rotational Mechanics

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MechanicsTo tryStretch

Same roll, faster spin

Inspect the run near the end and tune a small sphere that keeps the same rolling logic but reaches a high angular speed.

Rolling Motion

Path: Rotational Mechanics

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MechanicsTo tryStretch

Same impulse, different force shape

Enter compare mode: keep Setup A as a short strong split and make Setup B a longer gentler split with the same total internal impulse.

Conservation of Momentum

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MechanicsTo tryStretch

Same start, different energy story

Open compare mode with the same equal-mass starting collision, then make Setup A elastic and Setup B sticky so only elasticity changes the energy story.

Collisions

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MechanicsTo tryStretch

Freeze the apex

From Earth shot, pause exactly at the top of the arc where the vertical velocity has dropped to zero but the projectile is still high above the ground.

Projectile Motion

Path: Motion and Circular Motion

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MechanicsTo tryStretch

Same range, different height

Open compare mode and build two Earth-gravity launches with the same speed: Setup A near 30 degrees and Setup B near 60 degrees. Make their ranges match while the apex heights and airtimes separate clearly.

Projectile Motion

Path: Motion and Circular Motion

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FluidsTo tryStretch

Same entry state, wider B recovers pressure

Start from Baseline venturi, switch to compare mode, leave Setup A alone, and tune Setup B until it keeps the same entry pressure and flow rate but recovers the throat pressure by widening only the throat.

Bernoulli's Principle

Path: Fluid and Pressure

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FluidsTo tryStretch

Same block, less submersion in brine

Start from Wood in water, switch to compare mode, leave Setup A alone, and tune only Setup B until the same block balances with a noticeably smaller submerged height in denser fluid.

Buoyancy and Archimedes' Principle

Path: Fluid and Pressure

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Gravity and OrbitsTo tryStretch

Inner orbit, faster compare

Open compare mode and keep both setups circular with the same source mass, but make Setup B the smaller-radius orbit so it moves faster and finishes sooner.

Circular Orbits and Orbital Speed

Path: Gravity and Orbits

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FluidsTo tryStretch

Same flow, slower wide section

Start from Baseline stream, switch to compare mode, leave Setup A alone, and tune Setup B until it keeps the same flow rate but slows section B down by widening that middle section.

Continuity Equation

Path: Fluid and Pressure

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FluidsTo tryStretch

Catch the near-terminal moment

Starting from Draggy disk, pause when drag is almost equal to weight and the remaining net downward force is tiny.

Drag and Terminal Velocity

Path: Fluid and Pressure

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ThermodynamicsTo tryStretch

Same contrast, slower loss

Start from Metal on cool bench, switch to compare mode, and edit only Setup B until it keeps nearly the same temperature contrast as Setup A but loses energy at less than half the rate.

Heat Transfer

Path: Thermodynamics and Kinetic Theory

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ThermodynamicsTo tryStretch

Match pressure with a different microscopic story

Start from Hotter same box, switch to compare mode, and edit only Setup B until it reaches about the same pressure while staying cooler and using more particles instead of more temperature.

Ideal Gas Law and Kinetic Theory

Path: Thermodynamics and Kinetic Theory

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FluidsTo tryStretch

Match total pressure with less surface load

Start from Water baseline, switch to compare mode, leave Setup A alone, and tune only Setup B until it reaches the same total pressure with a smaller surface-pressure part and a denser fluid.

Pressure and Hydrostatic Pressure

Path: Fluid and Pressure

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ThermodynamicsTo tryStretch

Same pulse, smaller delta T

Start from Low-c warm sample, switch to compare mode, and edit only Setup B until both setups use the same 4 minute pulse but Setup B warms much less because its specific heat is larger.

Specific Heat and Phase Change

Path: Thermodynamics and Kinetic Theory

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ThermodynamicsTo tryStretch

Catch the real shelf

Starting from Warming toward the shelf, pause on a real shelf moment where temperature is near 0 degC but the phase fraction is still between fully solid and fully liquid.

Specific Heat and Phase Change

Path: Thermodynamics and Kinetic Theory

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ThermodynamicsTo tryStretch

Same temperature, bigger store

Start from the small warm sample, switch to compare mode, and edit only Setup B until it keeps about the same temperature as Setup A but clearly stores much more internal energy.

Temperature and Internal Energy

Path: Thermodynamics and Kinetic Theory

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ThermodynamicsTo tryStretch

Catch the true shelf

Starting from Warming toward a shelf, pause the run on a real shelf moment where temperature is nearly flat even though the sample is still taking in energy.

Temperature and Internal Energy

Path: Thermodynamics and Kinetic Theory

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Gravity and OrbitsTo tryStretch

Inner year vs outer year

Open compare mode, keep the same source mass in both setups, set Setup A to the inner circular orbit, and set Setup B to the much wider circular orbit so Setup B has the longer year.

Kepler's Third Law and Orbital Periods

Path: Gravity and Orbits

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OpticsTo tryStretch

Green enters glass

Enter compare mode with Setup A as green light in air and Setup B as the same green source in glass. Keep source frequency matched while speed and in-medium wavelength drop in B.

Light as an Electromagnetic Wave

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OpticsTo tryStretch

Compare a denser variant

Open compare mode from Air to glass. Leave Setup A near the baseline, then edit Setup B so the lower medium is noticeably denser and the incident angle is steeper.

Refraction / Snell's Law

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Compare a weak and strong disperser

Open compare mode from Crown green. Keep Setup A weakly dispersive, then edit Setup B until the same prism angle produces a much larger red-violet spread.

Dispersion / Refractive Index and Color

Path: Wave Optics

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ElectromagnetismTo tryStretch

Slower wave, same spacing

Enter compare mode and make Setup B use the same electric amplitude, wavelength, and probe position as Setup A, but a slower wave speed.

Electromagnetic Waves

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Oppose the rising flux

Pause during the left-side approach so the magnet is still outside the coil, the linked flux is increasing, and the induced current runs in the clockwise Lenz response.

Faraday's Law and Lenz's Law

Path: Magnetism

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MagnetismTo tryStretch

Lift versus cancel

Open compare mode from Opposite-current lift. Keep Setup A on the upward above-midpoint bridge, but turn Setup B into the midpoint-cancel case where the net field nearly vanishes even though the current magnitudes still match.

Magnetic Fields

Path: Magnetism

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MagnetismTo tryStretch

Faster force, wider arc

Open compare mode from Positive bends down. Keep Setup A as the baseline, but make Setup B show the bigger moving-charge force and the wider orbit that go with a faster charge in the same field.

Magnetic Force on Moving Charges and Currents

Path: Magnetism

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CircuitsTo tryStretch

Catch the 1tau charge point

Starting from the 1tau reference preset, inspect the loop at 1tau and keep the charging setup where the capacitor voltage is about 63% of the source and the current fraction is about 37%.

RC Charging and Discharging

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Rates and EquilibriumTo tryStretch

Disturb, then rebalance

Build a product-favored equilibrium from a reactant-heavy start, then let the forward and reverse rates come back together while the product share stays clearly above one-half.

Dynamic Equilibrium / Le Chatelier's Principle

Path: Rates and Equilibrium

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Rates and EquilibriumTo tryStretch

Rebalance the other way

Move the same reversible bench toward reactants and wait until the rates nearly match again. The product share should fall because the equilibrium target moved, not because the reaction stopped.

Dynamic Equilibrium / Le Chatelier's Principle

Path: Rates and Equilibrium

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Solutions and pHTo tryStretch

Same concentration, different amounts

Open compare mode. Keep Setup A as the diluted same-solute beaker, then tune Setup B to use less solute and less liquid while landing on the same concentration.

Concentration and Dilution

Path: Solutions and pH

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