🔬 Science & Mathematics

Compare a secant through two points with the instantaneous tangent slope, and see exactly where the limit does — and doesn't — exist.

Build a signed total from rectangles or trapezoids and compare each approximation with the exact accumulated value.

Track a rotating radius and its projections to see how one rotation produces two connected sine and cosine waves.

Transform a wireframe cube with scale, shear and rotation, and see how operation order changes signed volume and orientation.

Send two carts along a frictionless track and compare elastic and inelastic impacts through momentum and energy loss.

Release a spring-mass oscillator and compare ideal, underdamped, critically damped and overdamped return.

Compare a nonlinear pendulum's measured period against the small-angle approximation while tracking energy exchange.

Launch a projectile and resolve horizontal and vertical motion independently under constant gravity.

Build a head-to-tail chain and its parallelogram to add or subtract two free vectors in 3D.

Move a local origin and its coordinate planes around a fixed point to explore octants and projections.

Scan waves, saddle, bell and cone surfaces with perpendicular slicing planes to read cross-sections.

Probe a scalar field, test directional slopes and follow a gradient-ascent trail across four terrains.

Dial in x/y/z components and watch the component staircase, projection, rotation and normalization.

Build square, sawtooth and custom waveforms from harmonics and study convergence, Gibbs ringing and aliasing.

Morph continuously between sphere, cylinder, cone and hyperboloids by editing the defining equation.

Place two material layers between temperature-controlled plates and trace series thermal resistance, transient storage and a contact-resistance temperature jump.

Cool or heat an aluminum sphere in an airflow chamber and compare Newton's-law cooling against heat-transfer coefficient, area and heater power.

Heat a gray specimen inside an isothermal enclosure and separate emitted, absorbed and net radiative heat exchange across the infrared spectrum.

Inspect a concentric-tube counterflow exchanger and adjust flow rates, fouling and a cold-side bypass while checking outlet temperatures and energy balance.

Heat or cool a sealed water charge under a constant-pressure piston and watch sensible heating give way to a boiling plateau and back through condensation.

Compare two matched exchangers side by side and see why counterflow delivers more heat transfer than parallel flow at the same UA.

Trace a superheated steam power plant through pump, boiler, turbine and condenser, and see how condenser pressure and turbine losses affect output.

Run a cutaway gas turbine through compression, heat addition and expansion, and balance pressure ratio and firing temperature against compressor demand.

Follow an ideal-gas piston through four reversible isothermal and adiabatic steps and explore the thermodynamic efficiency limit as a heat engine or refrigerator.

Heat a bar on a precision bench and compare free expansion against a restrained bar that converts thermal strain into elastic stress.

Steer a charged beam through crossed electric and magnetic fields and find the velocity where the forces balance.

Turn a coil between magnetic poles and follow flux linkage, induced emf and load current through a generator circuit.

Spin a puck on a frictionless air table and release the tether to compare centripetal force against free-flight tangent motion.

Launch a satellite around Earth and compare circular, elliptical, impact and escape trajectories under inverse-square gravity.

Push a carriage against static and kinetic friction and separate applied force, friction and the actuator reaction.

Run a controlled hoist through a rest-to-rest lift and account for cable work, kinetic energy, power and drive loss.

Aim a light ray at a mirror and confirm the angle of incidence always equals the angle of reflection.

Send a ray across a boundary between two media and watch it bend according to Snell's law as refractive index changes.

Increase the incidence angle inside a denser medium and find the critical angle where refraction gives way to total internal reflection.

Trace principal rays through convex and concave lenses and watch real and virtual images form as object distance and focal length change.

Trace reflected rays off curved mirrors and compare real, virtual, upright and inverted images as object position and curvature change.

Adjust the eye's lens curvature to focus light from near and far objects onto the retina, and see how near- and far-sightedness form blurred images.

Send coherent light through two narrow slits and read the bright and dark fringe pattern Young's experiment produces on a screen.

Pass light through a multi-slit grating and watch sharp spectral orders spread out as slit spacing and wavelength change.

Rotate a polarizing filter in front of polarized light and watch transmitted intensity follow Malus's law.

Send white light through a glass prism and watch wavelength-dependent refraction spread it into a visible spectrum.

Compare side-to-side rope-wave motion against slinky-style compression and rarefaction driven by the same source.

Adjust frequency and speed and watch wavelength update live through the wave equation v = fλ.

See how wave energy and power scale with the square of amplitude on a live energy bar and pulsing power indicator.

Overlay two waves and watch constructive and destructive interference build the summed waveform in real time.

Drive a string fixed at both ends and find the harmonic modes where nodes and antinodes lock into place.

Drive a damped oscillator near its natural frequency and watch amplitude build sharply on a live response curve.

Move a wave source past a stationary observer and see wavefronts compress and stretch, shifting observed frequency.

Combine two close frequencies and watch the amplitude envelope pulse at the beat frequency.

Watch air molecules bunch into compressions and spread into rarefactions as a pressure wave propagates.

Send a pulse toward a boundary and compare fixed-end, free-end and medium-change reflection and transmission.

Compare all eight planets on circular orbits and switch between compressed and linear AU spacing to see period and speed against distance.

Spin an enlarged Earth beside its orbit and see why a sidereal day differs from a solar day, with solar elevation and local solar time.

Tilt a globe through a circular year and read solar declination, day length, noon elevation and daily mean sunlight at any latitude.

Move the Moon around a plan-view Sun-Earth-Moon model and read the illuminated fraction, elongation and phase angle on an Earth-facing disk.

Vary lunar distance, inclination and node angle and watch umbra, antumbra and penumbra intersect or miss the target body.

Set eccentricity, semimajor axis and central mass on a Kepler ellipse and compare speed, swept area and orbital energy.

Scrub a five-stage tour from protostar to compact remnant for different initial masses beside a temperature-luminosity diagram.

Compare a supported clock near a nonrotating black hole with a distant clock and follow gravitational redshift of emitted light.

Separate the bulge, disk and halo and compare their contributions to a rotation curve with a movable radial probe.

Launch a two-stage rocket with a pitch program and check whether the integrated trajectory has a perigee above Earth.

Flip a classical bit between 0 and 1, then dial a qubit's superposition angle and phase to see how much richer its state space is.

Drag a state vector around the Bloch sphere or jump to named presets and read live measurement probabilities and amplitudes.

Apply the Hadamard gate to |0> or |1> and watch it create — then undo — an equal superposition on the Bloch sphere.

Build a starting qubit state and apply X, Y and Z gates to trace their 180-degree Bloch-sphere rotations.

Dial a qubit's relative phase around the equator, then apply a second Hadamard to turn invisible phase into visible probability.

Apply the controlled-NOT gate to definite and superposed control qubits and see classical logic give way to entanglement.

Build a gate sequence from H, X, Y, Z, S and T and step through it while tracking the Bloch vector and probabilities.

Shift the phase between two amplitude paths and watch them reinforce or cancel as phasors and combining waves.

Step through Hadamard-then-CNOT to build an entangled Bell pair, then measure it to see the correlated outcome.

Walk through the full protocol that moves an unknown qubit state to Bob using entanglement and two classical bits.

Send single quanta through a Mach–Zehnder interferometer and watch individual clicks build phase-dependent probabilities, then remove the second splitter to erase the interference.

Accumulate discrete detections on a screen and compare the histogram with the predicted diffraction envelope and fringes as you change wavelength, slit geometry and coherence.

Prepare a qubit on the Bloch sphere, precess its relative phase, and see why Z probabilities stay fixed while X and Y probabilities oscillate.

Send spin-½ atoms through sequential Stern–Gerlach analyzers and see why a repeated measurement repeats while a rotated one adds uncertainty.

Scatter an electron from a rectangular barrier and vary energy, height and width to see penetration, exponentially small transmission and above-barrier resonances.

Collect independent position measurements of hydrogen 1s and compare a movable radius gate with the exact radial probability.

Compare hydrogen 1s, 2pz and 3dz² signed-amplitude shapes, angular nodes and animated cross-sections with an adjustable density threshold.

Pick upper and lower hydrogen levels and read the transition energy, photon wavelength and frequency for emission or absorption.

Tune a probe across a Lorentzian gas line on a spectroscopy bench and compare incoming, absorbed and transmitted intensity in absorption and emission modes.

Rotate Alice and Bob polarization analyzers and watch joint correlations change while each local outcome stays 50:50.

Emit two light pulses in opposite directions and transform the events into a moving observer's frame to see both frames measure exactly c.

Follow a photon bouncing between spacecraft mirrors and compare laboratory and spacecraft clock readings as velocity rises.

Compare a rod's rest length with its laboratory measurement while it moves, using synchronized endpoint gates.

Flash both ends of a moving train at once in the station frame and see why the train frame disagrees.

Send one twin to a distant marker and back and compare the ages accumulated along each worldline.

Release two opposed radiation packets from a resting source and track energy exported against rest mass lost.

Combine a carrier velocity with a probe launched from it and compare the relativistic and classical sums.

Place an event on a Minkowski diagram, tilt the moving-frame axes and classify the interval as timelike, lightlike or spacelike.

Explore Schwarzschild spatial geometry with an orbiting clock and compare coordinate radius with proper radial distance.

Place clocks at two radii near a compact object and compare tick rates and the redshift of light sent upward.

Build an atom or ion by counting protons, neutrons and electrons and read mass number, charge and shell filling.

Step through the first eighteen elements and watch electrons fill orbital boxes with spin arrows.

Transfer a sodium valence electron to chlorine and compare the ion pair with a rock-salt lattice fragment.

Stretch a hydrogen molecule and compare shared electron density with its bond-energy well.

Switch among seven molecular shapes, reveal lone-pair domains and inspect bond angles in three dimensions.

Track every hydrogen and oxygen atom as reactants reorganize into water and identify the limiting reactant.

Add NaOH to HCl in a stirred vessel and follow pH, excess ions and the equivalence point.

Compress, heat and add gas in a piston cylinder and watch pressure respond with PV = nRT.

Connect zinc and copper half-cells with a wire and salt bridge and track emf, current and electrode mass change.

Compare uncatalyzed and catalyzed energy paths and see how temperature changes the rate constant.

Draw repeatable samples from normal, binomial and uniform models and compare the growing histogram with the theoretical mean and variance.

Move one observation and watch the mean balance point, the sorted median pair and squared-deviation columns respond differently.

Average repeated samples from skewed or binary populations and watch the distribution of sample means tighten and become bell shaped.

Link X and Y through a hidden common cause, then intervene on X to see that association and true effect can disagree.

Fit a line to thirty noisy points, inspect every residual square, and compare your line with the least-squares optimum.

Sample a two-group population with fair and unequal selection rules and see how bias moves the sample mean away from the truth.

Shade two-sided p-value tails on a normal null model, pick a significance level, and simulate repeated true-null tests.

Follow a screening probability tree from prevalence, sensitivity and specificity to the chance a positive result is a real defect.

Run real k-means iterations on a point cloud and watch memberships flip and centroids converge as k and separation change.

Fit polynomials of rising degree to noisy data and compare training and held-out error with and without ridge regularization.

Explore a cutaway from crust to inner core and slide a depth probe to compare layer thickness and volume with the planet radius.

Compare divergent, convergent and transform boundaries in a cross-section and track plate speed and displacement over geologic time.

Load a locked strike-slip fault until friction fails and watch strain release, surface offset and a seismograph trace.

Launch P and S wave packets along particle chains and time their arrival, then see why S waves stop in a liquid.

Cut away a volcano and balance magma recharge against vent resistance to see reservoir pressure and eruption outflow evolve.

Move water between ocean, atmosphere, catchment and groundwater with adjustable transfer rates while total water is conserved.

Explore idealized Hadley, Ferrel and polar cells on a rotating Earth and read the Coriolis parameter at any latitude.

Compare wind-driven surface flow with density-driven overturning and vary polar temperature and salinity to change its strength.

Follow a specimen through cooling, weathering, compaction, metamorphism and melting and replay each transformation.

Flow water through a saturated aquifer with adjustable conductivity and heads and compare Darcy flux with pore velocity.

A cutaway animal cell follows labeled cargo from rough ER through Golgi stacks into transport vesicles and across the plasma membrane. Select the nucleus, mitochondria, lysosome or cytoskeleton to inspect compartment functions.

Inspect a rectangular plant-cell cutaway with a cellulose wall, membrane, central vacuole, chloroplast grana and nucleus. Change external osmolarity to compare water uptake, turgor and plasmolysis.

Explore an antiparallel double helix with two sugar-phosphate backbones, complementary base pairs and grooves. Change GC content and helical twist, then open the base pairs to inspect hydrogen bonding.

A moving fork separates parental DNA while leading synthesis follows continuously and lagging synthesis forms Okazaki fragments. Compare helicase progress, polymerase speed and the effect of inhibiting ligase.

A nucleus-to-ribosome pathway transcribes a short DNA coding example, exports the mRNA and advances a ribosome through AUG GCU UUU GGA UAA. tRNAs deliver amino acids and the chain stops at UAA.

A model diploid cell with 2n=4 enters mitosis after S phase. Condensed chromosomes attach to spindle fibers, align at metaphase, separate at anaphase and form daughter nuclei during cytokinesis.

A chloroplast cutaway shows stacked thylakoids, light absorption, water oxidation, proton movement and ATP synthase. Follow ATP and NADPH into a separate Calvin-cycle ring in the stroma.

Separate glycolysis in the cytosol, pyruvate oxidation and the citric-acid cycle in the matrix, and electron transport on the mitochondrial inner membrane. Follow carriers, proton pumping and ATP synthase.