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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.
簡諧運動
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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.
簡諧運動
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Guided challenge paths
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Start with vector components, move into projectile paths, and then use circular motion to understand how velocity can keep changing direction.
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Short-period force band is the next best challenge from 勻速圓周運動.
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.
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Zero turn at the handle is the next best challenge from 力矩.
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.
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Keep the heavier source circular is the next best challenge from 圓形軌道與軌道速度.
Build from one clean oscillator to energy exchange and then to driven resonance, so the same system grows without changing its core ideas.
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Short-period match is the next best challenge from 簡諧運動.
Start with pressure in a resting fluid, then carry that same branch through continuity, Bernoulli, buoyancy, and drag-limited motion.
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Build the 27 kPa throat is the next best challenge from 伯努利原理.
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.
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Short-period match is the next best challenge from 簡諧運動.
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.
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Same contrast, slower loss is the next best challenge from 熱傳遞.
Start with source charges and voltage, then carry that same circuit story into current, power, branch behavior, and equivalent resistance.
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Build the upward field is the next best challenge from 電場.
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.
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High flux, zero emf is the next best challenge from 法拉第定律與楞次定律.
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.
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Tune slow pulses is the next best challenge from 拍頻.
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.
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Set a half-power case is the next best challenge from 偏振.
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.
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Find the stopping point is the next best challenge from 光電效應.
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.
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Reflect and land the vertex is the next best challenge from 圖像變換.
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).
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Rotate onto the positive imaginary axis is the next best challenge from 複平面上的複數.
Start with vectors as geometric objects on a 2D plane, then carry the same component language into the existing motion-facing vectors bench.
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Equal components is the next best challenge from 向量與分量.
Start with successful collisions setting reaction rate, then reuse the same chemistry language inside a reversible system that re-balances after a disturbance.
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More success, not just more hits is the next best challenge from 反應速率／碰撞理論.
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.
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Build the matched 3:2 run is the next best challenge from 化學計量比與反應配方批次.
Start with concentration in one beaker, add solubility limits and saturation, then reuse that same solution language to read pH, buffers, and neutralization.
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Dilute without losing solute is the next best challenge from 濃度與稀釋.
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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Use insertion where it pays off is the next best challenge from 排序與演算法取捨.
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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.
簡諧運動
在同一個反覆系統中，從位移到加速度再回到位移，直接看懂週期性運動。
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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.
簡諧運動
在同一個反覆系統中，從位移到加速度再回到位移，直接看懂週期性運動。
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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.
振盪能量
觀看簡諧運動中的動能與勢能如何互相轉換，而總能量仍由振幅與彈簧剛度決定。
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From the mixed-energy baseline, raise the stored energy to about $5\,\mathrm{J}$ without making the oscillator heavier than about $1.2\,\mathrm{kg}$.
振盪能量
觀看簡諧運動中的動能與勢能如何互相轉換，而總能量仍由振幅與彈簧剛度決定。
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Starting near unison, tune the source pair until the envelope pulses at about $0.2\,\mathrm{Hz}$ while the source amplitude stays near the baseline.
拍頻
把兩個相近頻率的聲音疊加，看到快速載波包在較慢包絡線內，並把拍頻率連到頻率差。
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Enter compare mode and make Setup B keep the same beat frequency as Setup A while clearly lowering the average carrier frequency.
拍頻
把兩個相近頻率的聲音疊加，看到快速載波包在較慢包絡線內，並把拍頻率連到頻率差。
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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.
都卜勒效應
看著移動聲源在前方壓縮、後方拉開波前，並理解聲源與觀察者運動如何共同改變聽到的音高。
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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.
都卜勒效應
看著移動聲源在前方壓縮、後方拉開波前，並理解聲源與觀察者運動如何共同改變聽到的音高。
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Enter compare mode and make Setup B louder than Setup A while keeping the pitch the same.
音高、頻率與響度／強度
在同一個小型聲學實驗台上分清音高與頻率、響度與振幅／強度，以及探針延遲與訊源聲音本身。
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Starting from Baseline, move the probe until it sits inside a strong compression while the probe-pressure graph and compression overlay stay visible.
聲波與縱波運動
把聲音視為縱波，讓介質微粒運動、疏密變化、探針讀值與能量傳遞都留在同一個介質優先的實驗台上。
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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.
聲波與縱波運動
把聲音視為縱波，讓介質微粒運動、疏密變化、探針讀值與能量傳遞都留在同一個介質優先的實驗台上。
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Starting from Center bright, move the probe onto a dark region where the screen intensity almost vanishes.
波的干涉
把兩個相干波源疊加，從路徑差追到相位差，並在同一個即時畫面上看亮暗區如何形成。
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From Center bright, pause at a moment when the resultant amplitude is still large but the instantaneous probe displacement has crossed through zero.
波的干涉
把兩個相干波源疊加，從路徑差追到相位差，並在同一個即時畫面上看亮暗區如何形成。
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Starting from the third harmonic, move the probe onto a node so the local oscillation envelope collapses almost to zero.
駐波
在同一條即時弦上追蹤固定波節、振幅最大的波腹與諧波模態，同時用探針讀出時間上的振動。
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From the fundamental mode, keep the probe at the center antinode and pause right as that antinode crosses through zero displacement.
駐波
在同一條即時弦上追蹤固定波節、振幅最大的波腹與諧波模態，同時用探針讀出時間上的振動。
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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.
空氣柱共鳴／開管與閉管
在同一個空氣柱中比較開管與閉管邊界條件，讓駐波形狀、偶次諧波缺失、探針運動與壓力線索都綁在同一個共鳴狀態。
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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.
空氣柱共鳴／開管與閉管
在同一個空氣柱中比較開管與閉管邊界條件，讓駐波形狀、偶次諧波缺失、探針運動與壓力線索都綁在同一個共鳴狀態。
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Starting from the reference orbit, keep the radius close to the original circle but shorten the period to about $2.2\,\mathrm{s}$. Land the motion in the speed and centripetal-acceleration bands that go with that stronger centripetal-force requirement.
勻速圓周運動
追蹤粒子以恆定速率繞圓運動，並把半徑、角速度、切向速度、向心加速度與向內力需求連到同一個即時狀態。
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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.
勻速圓周運動
追蹤粒子以恆定速率繞圓運動，並把半徑、角速度、切向速度、向心加速度與向內力需求連到同一個即時狀態。
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Starting from the reference orbit, tune the motion until the inward acceleration sits in the $7$ to $8.5\,\mathrm{m/s^2}$ band while the tangential speed stays between $2.5$ and $3.1\,\mathrm{m/s}$.
勻速圓周運動
追蹤粒子以恆定速率繞圓運動，並把半徑、角速度、切向速度、向心加速度與向內力需求連到同一個即時狀態。
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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.
勻速圓周運動
追蹤粒子以恆定速率繞圓運動，並把半徑、角速度、切向速度、向心加速度與向內力需求連到同一個即時狀態。
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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.
阻尼／共振
探索阻尼如何帶走能量、驅動頻率如何改變振幅，以及為何在固有頻率附近共振會特別明顯。
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From Free swing, make the transient decay quickly enough that a late inspected sample shows only a very small displacement.
阻尼／共振
探索阻尼如何帶走能量、驅動頻率如何改變振幅，以及為何在固有頻率附近共振會特別明顯。
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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.
向量與分量
旋轉並縮放一個即時向量，把它拆成水平與垂直分量，並看這些分量如何驅動同一條直線運動與幾何。
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Pause at the end of the $4\,\mathrm{s}$ walk and make the point land near $(16\,\mathrm{m}, 12\,\mathrm{m})$.
向量與分量
旋轉並縮放一個即時向量，把它拆成水平與垂直分量，並看這些分量如何驅動同一條直線運動與幾何。
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Keep the push point near the handle but make the bar feel almost no turning effect.
力矩
在同一根可繞支點轉動的桿上施力，看看力臂長度、施力方向與轉動效果如何保持對應。
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Open compare mode and make Setup B twist just as hard as Setup A even though Setup B pushes much closer to the pivot.
力矩
在同一根可繞支點轉動的桿上施力，看看力臂長度、施力方向與轉動效果如何保持對應。
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Pause at the end of the clip and tune a clean clockwise twist.
力矩
在同一根可繞支點轉動的桿上施力，看看力臂長度、施力方向與轉動效果如何保持對應。
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Keep the motor near the baseline torque, then make the rotor spin up sharply by changing only the mass layout.
轉動慣量／慣性矩
保持總質量與力矩不變，再把等質量往內或往外移，直接看出為何有些轉子比其他轉子更難加速轉動。
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Open compare mode and make Setup B much harder to spin than Setup A without changing the torque in either setup.
轉動慣量／慣性矩
保持總質量與力矩不變，再把等質量往內或往外移，直接看出為何有些轉子比其他轉子更難加速轉動。
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Pause at the end of the clip and tune a wide-rim rotor that still reaches a moderate final angular speed.
轉動慣量／慣性矩
保持總質量與力矩不變，再把等質量往內或往外移，直接看出為何有些轉子比其他轉子更難加速轉動。
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Starting from Tips right, move the support center until the heavy right load is back in static equilibrium.
靜力平衡／重心
在同一塊受力木板下移動支撐範圍，看看重心、支撐反力與力矩平衡如何決定物體是穩定還是翻倒。
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Starting from Support under load, shift the support region left until the plank is only just stable but not yet tipping.
靜力平衡／重心
在同一塊受力木板下移動支撐範圍，看看重心、支撐反力與力矩平衡如何決定物體是穩定還是翻倒。
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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.
靜力平衡／重心
在同一塊受力木板下移動支撐範圍，看看重心、支撐反力與力矩平衡如何決定物體是穩定還是翻倒。
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Start from the lab baseline, then build a wide layout that still carries nearly the same angular momentum.
角動量
在同一個小型轉子上把角動量當作旋轉版的動量來看，讓質量分佈、轉速與守恆故事都對應同一組讀值。
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Open compare mode and make Setup A compact and Setup B wide while keeping their angular momenta nearly matched.
角動量
在同一個小型轉子上把角動量當作旋轉版的動量來看，讓質量分佈、轉速與守恆故事都對應同一組讀值。
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Pause at the end of the clip and tune a wide same-L rotor whose slow spin makes the accumulated angle stay small.
角動量
在同一個小型轉子上把角動量當作旋轉版的動量來看，讓質量分佈、轉速與守恆故事都對應同一組讀值。
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Keep the ramp near its baseline angle and tune the roller so it reaches the bottom in under about $1.85\,\mathrm{s}$.
滾動運動
讓球體、圓柱、圓環或自訂質量分佈沿斜面滾下，看看無滑動滾動如何把平移、轉動與轉動慣量綁在同一次運動裡。
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Open compare mode and make Setup B finish much later than Setup A while keeping both setups on the same slope and radius.
滾動運動
讓球體、圓柱、圓環或自訂質量分佈沿斜面滾下，看看無滑動滾動如何把平移、轉動與轉動慣量綁在同一次運動裡。
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Inspect the run near the end and tune a small sphere that keeps the same rolling logic but reaches a high angular speed.
滾動運動
讓球體、圓柱、圓環或自訂質量分佈沿斜面滾下，看看無滑動滾動如何把平移、轉動與轉動慣量綁在同一次運動裡。
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Starting from Earth shot, stretch the landing point into the $35$ to $38\,\mathrm{m}$ range while keeping the apex below about $10\,\mathrm{m}$.
拋體運動
發射一個拋體，觀察軌跡如何形成，並把射程、最高點與分量運動直接連到發射設定。
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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.
拋體運動
發射一個拋體，觀察軌跡如何形成，並把射程、最高點與分量運動直接連到發射設定。
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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.
伯努利原理
沿著同一條穩定理想流管追蹤，看看壓力、速度與高度如何在同一個伯努利預算中互相轉換，同時由連續條件保持流量故事誠實。
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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.
伯努利原理
沿著同一條穩定理想流管追蹤，看看壓力、速度與高度如何在同一個伯努利預算中互相轉換，同時由連續條件保持流量故事誠實。
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Adjust the block so it could stay about half submerged without extra support.
浮力與阿基米德原理
用同一個浸沒方塊實驗台，把壓差、排開流體與浮沉／中性浮力背後的密度平衡連起來。
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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.
浮力與阿基米德原理
用同一個浸沒方塊實驗台，把壓差、排開流體與浮沉／中性浮力背後的密度平衡連起來。
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Starting from Reference orbit, make the source heavier while keeping the chosen radius near 1.6 m and the orbit circular.
圓形軌道與軌道速度
理解圓形軌道為何需要剛好的橫向速度、重力如何提供向心加速度，以及來源質量與半徑如何一起決定軌道速度與週期。
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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.
圓形軌道與軌道速度
理解圓形軌道為何需要剛好的橫向速度、重力如何提供向心加速度，以及來源質量與半徑如何一起決定軌道速度與週期。
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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.
圓形軌道與軌道速度
理解圓形軌道為何需要剛好的橫向速度、重力如何提供向心加速度，以及來源質量與半徑如何一起決定軌道速度與週期。
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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.
連續方程
保持同一條穩定流線管在畫面上，用 Q = Av 連結截面積、流速與在寬窄區域保持一致的體積流率。
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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.
連續方程
保持同一條穩定流線管在畫面上，用 Q = Av 連結截面積、流速與在寬窄區域保持一致的體積流率。
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Starting from Baseline drop, tune the setup into a much slower terminal-speed case by keeping the mass near $2\,\mathrm{kg}$ while increasing both area and drag strength.
阻力與終端速度
讓物體穿過流體下落，利用質量、截面積與阻力強度理解阻力如何隨速度增加，直到與重力平衡而達到終端速度。
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Starting from Draggy disk, pause when the object is effectively at terminal speed: drag nearly equals weight and the remaining net downward force is tiny.
阻力與終端速度
讓物體穿過流體下落，利用質量、截面積與阻力強度理解阻力如何隨速度增加，直到與重力平衡而達到終端速度。
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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.
電場
看來源電荷的正負、距離與疊加如何決定探測點的電場，再觀察測試電荷如何把該電場轉成力而不改變電場本身。
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Starting from Like-charge arch, reverse the force on the test charge while keeping the same upward field symmetry at the probe.
電場
看來源電荷的正負、距離與疊加如何決定探測點的電場，再觀察測試電荷如何把該電場轉成力而不改變電場本身。
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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.
重力場
看一個來源質量如何形成向內的重力場，以及來源質量與距離如何決定場強，並讓探測質量把場轉成受力而不改變場本身。
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Starting from Baseline diagonal, change only the probe mass so the force magnitude doubles while the gravitational field stays the same.
重力場
看一個來源質量如何形成向內的重力場，以及來源質量與距離如何決定場強，並讓探測質量把場轉成受力而不改變場本身。
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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.
重力勢與重力勢能
看單一來源質量如何形成負值的勢阱，比較勢與勢能隨距離的變化，並在同一個即時模型中把 φ 的下降趨勢連到重力場。
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Starting from Baseline diagonal, change only the probe mass so the potential energy doubles in magnitude while the potential and field stay fixed.
重力勢與重力勢能
看單一來源質量如何形成負值的勢阱，比較勢與勢能隨距離的變化，並在同一個即時模型中把 φ 的下降趨勢連到重力場。
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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.
熱傳遞
把熱看成由溫差驅動的能量轉移，同時比較傳導、對流與輻射如何在同一個小型熱學實驗台上競爭。
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Tune the live setup until radiation is the largest pathway while the block is still clearly hotter than the room.
熱傳遞
把熱看成由溫差驅動的能量轉移，同時比較傳導、對流與輻射如何在同一個小型熱學實驗台上競爭。
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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.
理想氣體定律與氣體分子動理論
在同一個受限粒子箱中連結壓力、體積、溫度與粒子數，然後把這些壓力變化再讀回粒子速度與碰壁頻率的改變。
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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.
理想氣體定律與氣體分子動理論
在同一個受限粒子箱中連結壓力、體積、溫度與粒子數，然後把這些壓力變化再讀回粒子速度與碰壁頻率的改變。
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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.
壓力與靜水壓
用同一個活塞與液槽實驗台，把力除以面積、壓力向各方向作用，以及密度、重力與深度如何造成靜水壓連在一起。
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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.
壓力與靜水壓
用同一個活塞與液槽實驗台，把力除以面積、壓力向各方向作用，以及密度、重力與深度如何造成靜水壓連在一起。
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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.
比熱與相變
看看同樣的能量脈衝為何會讓不同材料升溫幅度不同，以及為何相變平台能在不改變溫度的情況下吸收或釋放能量。
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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.
比熱與相變
看看同樣的能量脈衝為何會讓不同材料升溫幅度不同，以及為何相變平台能在不改變溫度的情況下吸收或釋放能量。
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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.
溫度與內能
比較粒子的平均運動與整體樣本的能量，改變物質量與加熱，並在同一個熱學實驗台上看相變平台如何打破只看溫度的直覺。
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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.
溫度與內能
比較粒子的平均運動與整體樣本的能量，改變物質量與加熱，並在同一個熱學實驗台上看相變平台如何打破只看溫度的直覺。
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Starting from Dipole reference, tune the setup until the midpoint has almost zero field but still sits on a clearly positive potential hill.
電勢
描繪來源電荷的正負與距離如何塑造電勢，比較同一條掃描線上的電勢差，並把 V 的下降斜率連到電場。
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Open compare mode, keep the same source mass in both setups, and make Setup B the much wider circular orbit so it has the longer year.
克卜勒第三定律與軌道週期
比較繞同一來源質量的圓形軌道，理解較大軌道為何需要更長時間：路徑更長、圓周速度更低，而同一個即時模型讓週期定律清楚可見。
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Starting from Baseline year, raise the source mass while keeping the same radius circular so the period becomes clearly shorter.
克卜勒第三定律與軌道週期
比較繞同一來源質量的圓形軌道，理解較大軌道為何需要更長時間：路徑更長、圓周速度更低，而同一個即時模型讓週期定律清楚可見。
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Starting from Baseline year, lower the speed enough that the path is no longer the circular orbit Kepler's law is describing.
克卜勒第三定律與軌道週期
比較繞同一來源質量的圓形軌道，理解較大軌道為何需要更長時間：路徑更長、圓周速度更低，而同一個即時模型讓週期定律清楚可見。
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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.
基本電路
把一個電池與兩個電阻留在視野中，讓電流、電壓、電阻、歐姆定律，以及串並聯對比都綁在同一個誠實電路上。
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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.
逃逸速度
從同一個受限重力來源向外發射，看看來源質量、發射半徑與總比能量如何決定物體是逃逸還是最終返回。
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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.
逃逸速度
從同一個受限重力來源向外發射，看看來源質量、發射半徑與總比能量如何決定物體是逃逸還是最終返回。
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Starting from Aligned pass, tune the bench until the detector reads about one half of the incoming intensity for a linear input.
偏振
用同一個小型偏振片實驗台理解偏振是橫波方向的故事，觀察角度不匹配如何決定透射光，以及理想偏振片如何把未偏振光整理成單一軸向。
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Starting from Crossed axes, switch the bench to an unpolarized first-pass case that still leaves the detector near half brightness.
偏振
用同一個小型偏振片實驗台理解偏振是橫波方向的故事，觀察角度不匹配如何決定透射光，以及理想偏振片如何把未偏振光整理成單一軸向。
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Starting from Wide slit, tune the controls until the first minimum lands between 22 deg and 28 deg.
繞射
看波在通過狹縫後如何展開，理解當波長與縫寬相近時繞射為何變得更明顯，並為雙縫干涉建立波動光學橋樑。
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Starting from Center bright, move the probe onto the first dark band without changing the slit or wavelength.
繞射
看波在通過狹縫後如何展開，理解當波長與縫寬相近時繞射為何變得更明顯，並為雙縫干涉建立波動光學橋樑。
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Starting from Center bright, move the probe onto the first dark fringe without changing wavelength, slit separation, or screen distance.
雙縫干涉
利用兩條相干狹縫與一面屏幕，把路徑差、相位差與條紋間距連到波長、縫距與屏距。
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Starting from Tight fringes, tune the geometry until the bright-fringe spacing lands between 1.8 m and 2.1 m.
雙縫干涉
利用兩條相干狹縫與一面屏幕，把路徑差、相位差與條紋間距連到波長、縫距與屏距。
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Starting from Gentle glow, keep the 8 ohm load and raise the source until the stage power bar settles near 18 W.
電路中的功率與能量
把一個電源與一個電阻負載保持在視野中，讓電流、功率與隨時間累積的能量都綁在同一個誠實電路裡。
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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.
折射／斯涅爾定律
看一道光線穿過介面，把折射率與速度變化連起來，並理解斯涅爾定律如何決定折射角、彎折方向與臨界角極限。
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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.
折射／斯涅爾定律
看一道光線穿過介面，把折射率與速度變化連起來，並理解斯涅爾定律如何決定折射角、彎折方向與臨界角極限。
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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.
色散／折射率與顏色
用一個小型薄稜鏡實驗台看折射率如何依波長改變、不同顏色為何折得不同，以及受限的稜鏡模型如何分開顏色而不變成完整光譜學系統。
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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.
色散／折射率與顏色
用一個小型薄稜鏡實驗台看折射率如何依波長改變、不同顏色為何折得不同，以及受限的稜鏡模型如何分開顏色而不變成完整光譜學系統。
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Starting from Unequal series loads, rewire the setup so Load B keeps the full battery voltage while its branch current stays around 1 A.
串聯與並聯電路
把同一對負載在單一路徑與雙支路之間切換，追蹤電流、電壓、亮度與電荷流動如何重新組織，而電池本身不變。
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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.
全內反射
把光線從高折射率介質推向低折射率邊界，觀察臨界角如何出現，並看同一張即時圖如何從普通折射切換到完全內反射。
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Starting from Glass to air below critical, raise the setup until the boundary is clearly above threshold while staying on the same media pair.
全內反射
把光線從高折射率介質推向低折射率邊界，觀察臨界角如何出現，並看同一張即時圖如何從普通折射切換到完全內反射。
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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.
等效電阻
先把高亮的電阻群化簡為等效方塊，再誠實地把整個混合電路逐步縮減，觀察總電流與分組行為如何一起改變。
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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.
鏡面成像
用平面鏡、凹面鏡與凸面鏡在同一張即時光線圖上追蹤等角反射、有號像距與放大率。
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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.
鏡面成像
用平面鏡、凹面鏡與凸面鏡在同一張即時光線圖上追蹤等角反射、有號像距與放大率。
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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.
法拉第定律與楞次定律
追蹤一塊磁鐵穿過一個線圈，觀察磁通量連結改變如何產生感應電動勢，以及楞次定律如何決定回應方向。
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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.
法拉第定律與楞次定律
追蹤一塊磁鐵穿過一個線圈，觀察磁通量連結改變如何產生感應電動勢，以及楞次定律如何決定回應方向。
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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.
透鏡成像
透過主光線追蹤會聚與發散透鏡，將有號薄透鏡方程直接對應到圖像，並看像距與放大率如何回應同一組物體設定。
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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.
透鏡成像
透過主光線追蹤會聚與發散透鏡，將有號薄透鏡方程直接對應到圖像，並看像距與放大率如何回應同一組物體設定。
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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.
磁場
看電流方向、導線間距、距離與疊加如何決定一到兩條長直導線周圍的磁場，讓舞台箭頭與掃描圖都綁在同一個即時來源配置上。
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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.
磁場
看電流方向、導線間距、距離與疊加如何決定一到兩條長直導線周圍的磁場，讓舞台箭頭與掃描圖都綁在同一個即時來源配置上。
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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.
磁場對運動電荷與電流的作用力
讓一個運動電荷穿過均勻磁場，再與同方向電流段比較，並在同一個受限即時舞台上連結力的方向、軌跡彎曲與電流受力。
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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.
磁場對運動電荷與電流的作用力
讓一個運動電荷穿過均勻磁場，再與同方向電流段比較，並在同一個受限即時舞台上連結力的方向、軌跡彎曲與電流受力。
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Starting from Blurred pair, tune the aperture or wavelength until the point spacing sits right on the Rayleigh limit.
光學解析度／成像極限
讓兩個接近的點光源通過有限孔徑成像，理解繞射、波長與孔徑直徑如何限制光學系統分辨它們的能力。
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Starting from Near threshold, make the split clearly visible without changing the detector sample control.
光學解析度／成像極限
讓兩個接近的點光源通過有限孔徑成像，理解繞射、波長與孔徑直徑如何限制光學系統分辨它們的能力。
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Starting from Violet above threshold, make the collected current almost vanish without changing the frequency or work function.
光電效應
用一個小型光源到金屬的實驗台理解為何光頻率決定電子是否逸出、為何只靠強度無法跨越閾值，以及截止電壓如何誠實讀出電子能量。
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Starting from Bright but still below threshold, keep the beam bright while proving the collector current can stay essentially zero.
光電效應
用一個小型光源到金屬的實驗台理解為何光頻率決定電子是否逸出、為何只靠強度無法跨越閾值，以及截止電壓如何誠實讀出電子能量。
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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.
原子光譜
把離散發射與吸收線連到允許的能階差，用同一個小型能階與光譜實驗台保持轉換、波長與模式切換的一致性。
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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.
原子光譜
把離散發射與吸收線連到允許的能階差，用同一個小型能階與光譜實驗台保持轉換、波長與模式切換的一致性。
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Starting from Slow electron, tune the speed until the fixed loop is close to one wavelength long without changing the particle mass.
德布羅意物質波
用一個小型物質波實驗台理解粒子動量如何決定波長、為何較重或較快的粒子有更短波長，以及整數圈適配如何成為早期量子行為的受限橋樑。
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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.
德布羅意物質波
用一個小型物質波實驗台理解粒子動量如何決定波長、為何較重或較快的粒子有更短波長，以及整數圈適配如何成為早期量子行為的受限橋樑。
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Starting from Balmer beta, tune the live state until the active transition is the classic red Balmer line while the page stays in emission.
波耳模型
用一個精簡的氫原子實驗台把量子化能階、允許轉換與命名譜線系列連起來，同時清楚知道波耳模型是有用的歷史模型而不是最終量子描述。
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Starting from Lyman alpha emission, switch to the matching reverse excitation from the ground level while keeping the same ultraviolet wavelength.
波耳模型
用一個精簡的氫原子實驗台把量子化能階、允許轉換與命名譜線系列連起來，同時清楚知道波耳模型是有用的歷史模型而不是最終量子描述。
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Starting from Class-lab sample, scrub to about one half-life so the expectation is halved while the live tray stays slightly below it.
放射性與半衰期
用一個小型衰變實驗台理解為何單個原子核衰變無法預測、為何大量樣本仍遵循規律的半衰期曲線，以及如何誠實解讀剩餘數量圖。
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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.
放射性與半衰期
用一個小型衰變實驗台理解為何單個原子核衰變無法預測、為何大量樣本仍遵循規律的半衰期曲線，以及如何誠實解讀剩餘數量圖。
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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.
圖像變換
用誠實的控制項移動同一條母函數曲線，讓平移、垂直伸縮與反射都綁在同一張疊圖與標誌點上。
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Build a decay case where the target is about one quarter of the start, so the curve reaches it in about two half-lives.
指數變化／增長、衰減與對數
改變同一條曲線的初值、變化率與目標值，讓增長、衰減、倍增／半衰期與對數求時都留在同一個即時圖上。
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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.
有理函數／漸近線與行為
改變同一個平移後的倒數函數族，讓定義域缺口、垂直與水平漸近線、截距與可去間斷都綁在同一張圖上。
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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
Slide a point along one curve, tighten a secant into a tangent, and connect local steepness to the derivative graph without leaving the same live bench.
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Switch to the case where both one-sided values nearly agree, but the graph still is not continuous at $x=0$.
極限與連續／逼近某個值
從左與右逼近同一個目標點，比較極限高度與函數實際值，並在同一張圖上對照連續、可去間斷、跳躍與發散行為。
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Move the width until the fixed-perimeter rectangle reaches the maximum area and the local area slope is essentially zero.
最佳化／最大值、最小值與限制
在固定周長下改變矩形寬度，觀察面積曲線如何達到峰值，並用局部斜率看出正方形為何是最佳受限形狀。
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Move the bound into a region where the source height is already negative, but the running total is still positive overall.
積分作為累積／面積
把上限沿著來源曲線移動，看有號面積如何累積成為運行總量，讓累積保持可視化而不是只剩符號。
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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.
二維向量
在同一平面上組合、相減與縮放向量，讓大小、方向與分量都綁在同一個即時物件上。
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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.
矩陣變換／伸縮、剪切、反射
讓同一個 2×2 矩陣同時作用在網格、基底向量與範例圖形上，使平面變換保持可視化而不只是符號。
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Build a multiplication case where z · w lands almost on the positive imaginary axis while the multiplier magnitude stays close to one.
複平面上的複數
把複數視為平面上的點與向量，再讓加法與乘法保持幾何直覺，而不只停留在符號操作。
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Adjust $\vec{B}$ 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.
內積／夾角與投影
把兩個向量、它們的夾角、其中一個投影到另一個的有號分量，以及內積一起顯示，讓對齊程度從幾何上變得清楚。
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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.
參數曲線／由方程描述運動
把 x(t)、y(t)、描出的路徑與移動點同時顯示，清楚分開形狀本身與沿著曲線前進的方式。
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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 $r$, $\theta$, $x$, and $y$ stay tied to the same point.
極座標／半徑與角度
讓同一個點同時出現在極座標與直角座標中，使半徑與角度如何變成 x、y 直接可見。
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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.
單位圓／由旋轉理解正弦與餘弦
把旋轉點、x/y 投影與正弦餘弦曲線連在一起，讓單位圓成為這兩個函數的即時來源。
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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.
反應速率／碰撞理論
把一個化學盒子留在視野中，讓溫度、濃度、活化門檻與催化劑都可被理解為成功碰撞數目的變化。
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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.
由單位圓幾何理解三角恆等式
讓旋轉點與其投影保持可見，使核心三角恆等式仍然連著幾何，而不是脫離圖像的規則。
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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.
反三角函數／由比值求角
讓一個極座標點與其座標正負保持可見，使反三角函數變成帶象限檢查的由比值求角，而不只是計算器輸出。
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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.
動態平衡／勒沙特列原理
看一個可逆反應實驗台如何在微觀上持續變化，同時在每次擾動後走向新的平衡。
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Set the bench to a 3 A + 2 B recipe with supplies that still finish four full batches together.
化學計量比與反應配方批次
用同一個反應配方實驗台讓化學計量比讀起來像完成一批批反應，而不是脫離情境的工作紙比例。
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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.
限量試劑與剩餘反應物
用同一個配方實驗台看清哪個反應物先限制產量，以及為何另一個反應物會留下過量。
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Use the matched 2 A + 3 B run and set percent yield to 75% so the actual output lands at 3.75 batches.
百分產率與反應進度
把實際產出與同一配方下的理論上限並列，讓百分產率在同一個實驗台上保持可視與誠實。
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Starting from a fairly crowded beaker, lower the concentration clearly while keeping the solute amount near the same starting value.
濃度與稀釋
用同一個燒杯分清加入溶劑時濃度如何改變，以及加入更多溶質時濃度又如何不同地改變。
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Starting from a saturated beaker, make the excess pile disappear while keeping the same total solute amount.
溶解度與飽和
把已溶解量、剩餘固體與當前容量都放在同一個燒杯裡，讓飽和讀起來像一個看得見的極限而不是口號。
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Adjust the mixture until the pH sits near neutral while acid and base character stay visibly close together.
酸鹼／pH 直覺
把酸量、鹼量、水與 pH 試紙一起顯示，讓酸性與鹼性保持直觀而不只是背誦。
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Build a case where insertion sort finishes a nearly sorted list with very few writes.
排序與演算法取捨
把排序看成對即時列表的可見工作，使輸入順序、比較次數與寫入動作都保持具體，而不是只剩最後答案。
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Starting from the unbuffered acid pulse, keep the pH near neutral by adding buffer reserve while keeping the same acid push and without flooding the beaker with extra water.
緩衝液與中和
把中和反應、緩衝儲備與 pH 試紙一起放在視野中，讓 pH 穩定不再看起來像化學沒有發生變化。
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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.
二分搜尋／把搜尋空間減半
把有序列表、low-mid-high 標記與不斷縮小的區間一起留在視野中，讓二分搜尋變得可視，而不只是流程步驟。
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