弹簧‑质量系统的动量守恒与周期运动

Overview

A concise visual demonstration of a three‑mass system connected by a spring. First, an elastic collision between a moving mass C and stationary mass B shows momentum conservation. Then the coupled masses A and B oscillate for two periods while the spring length varies sinusoidally. A synchronized velocity‑time plot in the lower‑right corner highlights the moment when B reaches its minimum speed (0.5 m/s) and A reaches its maximum speed (1.5 m/s).

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Phases

#	Phase Name	Duration	Description
1	Intro	~1 s	Title fades in at the top. Axes appear with origin at A (x=0). Three squares A (blue), B (green) and C (red) are placed at their initial positions; the yellow spring connects A and B. All three masses are rendered as filled squares of equal size. The smooth horizontal ground line runs just below the bottom edges of the squares, and the squares rest on top of it.
2	Collision	~3 s	C moves leftward at $v_0=2\,\text{m/s}$ and must physically travel until its left surface makes full contact with the right surface of square B before the collision is triggered — the two squares must be visually touching (zero gap) at the exact moment of impact. At impact a brief overlay shows the momentum‑conservation equation $m_C v_0 = m_B v_B' + m_C v_C'$. Velocities after collision are displayed ($v_B'=-1\,\text{m/s}$, $v_C'=+1\,\text{m/s}$). A small "elastic" label appears, then the squares separate. Immediately after the collision, B moves leftward and begins compressing the spring against A; the spring visually shortens/compresses right away. The spring's right endpoint is always anchored to the left surface of square B, and the left endpoint is always anchored to the right surface of square A, so the spring length equals the gap between the two square surfaces at every frame. Under no circumstances should square B overlap with or penetrate into the spring's interior while the spring remains at natural length.
3	Coupled Oscillation – First Period	~5 s	After the collision, B compresses the spring immediately, driving A into motion. A and B then oscillate together. Their positions follow the given sinusoidal formulas. Because $m_B = 3m_A$, the two masses do not exchange velocities symmetrically. At three key instants ($t=T/4$, $t=T/2$, $t=3T/4$) the animation pauses for 0.5 s each to highlight the spring state and the masses' speeds. At $t=T/2$ (spring at natural length), B's speed is at its minimum (0.5 m/s) and A's speed is at its maximum (1.5 m/s); both values are highlighted with red labels. Spring colour coding: orange at $T/4$, red at $T/2$, light‑yellow at $3T/4$. Throughout this phase, the camera (viewport) continuously tracks the centre of mass of the A–B system so that both squares and the spring remain fully visible on screen at all times. The coordinate axis and scale bar move with the viewport to preserve spatial reference.
4	Coupled Oscillation – Second Period	~5 s	The same motion repeats for the second period without extra pauses, reinforcing the periodicity. Camera tracking of the A–B centre of mass continues from Phase 3. If translation tracking alone is insufficient to keep both squares and the full spring within the left area (i.e., the system drifts too far from the initial viewport after t ≈ 11 s), the camera smoothly zooms out to reduce the scale so that the entire A–B system remains fully visible. The zoom is applied gradually and continuously — never as a sudden jump — and is only triggered when the system would otherwise leave the frame. The floating scale bar and axis labels update in real time to reflect the current zoom level, so the viewer always retains a clear spatial reference.
5	Outro	~2 s	The motion freezes, showing the full trajectories of A, B, and C (faded trails). The complete velocity‑time plot remains on screen. No concluding caption or text overlay is shown at the end of the video.

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Layout

┌─────────────────────────────────────────────┐
│                TOP AREA (Title)            │
├───────────────────────┬─────────────────────┤
│        LEFT AREA       │   RIGHT AREA (v‑t) │
│   (Main animation)    │   (graph)           │
├───────────────────────┴─────────────────────┤
│                BOTTOM AREA (Notes)        │
└─────────────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Top	Animation title "弹簧‑质量系统的动量守恒与周期运动"	Fades in at the start of Phase 1
Left	Main visual: a smooth horizontal ground line runs across the left area at a fixed vertical position. The three masses A (blue), B (green), and C (red) are rendered as filled squares that rest on top of this ground line — their bottom edges are flush with the ground surface at all times. The spring connects A and B with its endpoints anchored to the right surface of square A and the left surface of square B respectively. There is no axis passing through the centres of the squares; the ground line replaces the previous centre axis and sits just below the squares. A position scale (tick marks and numeric labels) is shown along the ground line or just below it to provide spatial reference; tick marks and their numeric labels must never overlap — labels are offset sufficiently below or beside each tick so that every label is fully legible and no two labels crowd each other. Position trails are drawn during the oscillation phases. From Phase 3 onward, the viewport follows the centre of mass of the A–B system (camera tracking), keeping both squares and the full spring within the left area at all times. If translation tracking alone cannot keep the system in frame after t ≈ 11 s, the viewport additionally zooms out smoothly and continuously so that the A–B system always remains fully visible. The zoom factor is adjusted gradually (no jump cuts) and only engages when necessary. A floating scale bar and shifted position labels along the ground update in real time to reflect both the current pan position and zoom level, so that the viewer always has an accurate spatial reference. The spring's endpoints are rigidly attached to the square surfaces at every frame: the left endpoint tracks the right edge of square A, and the right endpoint tracks the left edge of square B. The rendered spring length is always exactly equal to the current surface‑to‑surface distance between A and B, so square B can never appear to penetrate or overlap the spring.	Primary focus; all collisions and spring dynamics occur here
Right	Velocity‑time (v‑t) plot occupying the full right panel, covering the entire animation duration. Specific requirements: Axes: The horizontal time axis runs from $t=0$ to $t \approx 13\,\text{s}$ (matching the total animation length), labelled "t (s)". The vertical velocity axis runs from $-1.5\,\text{m/s}$ to $+1.5\,\text{m/s}$, labelled "v (m/s)". Both axes are drawn in white with evenly spaced tick marks and numeric labels. Tick marks and their numeric labels must not overlap: each label is placed with sufficient clearance from its tick mark and from adjacent labels so that every value is clearly readable. Axis labels ("t (s)" and "v (m/s)") are positioned at the far end of each axis with a comfortable margin so they never collide with tick labels. A horizontal dashed grey line is drawn at $v=0$ for reference. Curves: Three curves are drawn and updated in real time as the animation plays — A in blue, B in green, C in red — each labelled with a small legend in the top‑right corner of the graph panel. Phase 2 (Collision): Before impact, C's curve shows $v_C = -2\,\text{m/s}$ (leftward), while A and B are flat at 0. At the moment of collision, C's curve jumps to $+1\,\text{m/s}$ and B's curve jumps to $-1\,\text{m/s}$; A remains at 0 until the spring begins to act. Phase 3–4 (Oscillation): A's velocity curve follows a sinusoidal shape with amplitude $1.5\,\text{m/s}$, starting from 0 and rising. B's velocity curve follows a sinusoidal shape with amplitude $0.5\,\text{m/s}$ offset from its post‑collision value of $-1\,\text{m/s}$, reflecting the $m_B = 3m_A$ mass ratio. At $t = T/2$ and $t = 3T/2$, a red filled dot is drawn on both the A curve (at $v = +1.5\,\text{m/s}$) and the B curve (at $v = -0.5\,\text{m/s}$ or $+0.5\,\text{m/s}$ depending on direction), with small red numeric labels "1.5" and "0.5" respectively. A vertical dashed red line is drawn at these instants to visually link the two highlighted points. C's curve remains flat at $+1\,\text{m/s}$ throughout Phases 3–4 (C moves away freely after the collision). Real‑time update: A moving vertical cursor line (thin white or yellow) sweeps from left to right across the time axis in sync with the animation clock, so the viewer can always identify the current moment on the graph. The curves are drawn progressively (trace‑style) as time advances rather than being shown all at once.	Synchronized with the left‑side motion; updates in real time. The v‑t panel is fixed and does not move with the camera tracking or zoom in the left area.
Bottom	Small caption area for brief equations or labels that appear during pauses (e.g., momentum equation, spring‑length formula). No concluding summary text or analysis caption is displayed here at any point during the outro.	Text appears only when needed during active phases, otherwise stays empty

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Notes

• All durations are approximate; the total runtime is about 15 seconds, well within the 30‑second guideline.
• No additional text is shown except the essential equations and labels required for understanding the physics. In particular, no concluding analysis caption is shown at the end of the video (the outro freezes on the final frame with trails only).
• Mass shape: All three masses A, B, and C are rendered as filled squares (not circles or discs) throughout the entire animation. Their bottom edges rest flush on the smooth horizontal ground line at all times.
• Ground line: A smooth horizontal line represents the ground surface and runs across the full width of the left area. The squares slide along this ground without friction. No axis passes through the centres of the squares. Position scale markings (tick marks and numeric labels) are placed along or just below the ground line to provide spatial reference; tick marks and numeric labels must never visually overlap — labels are spaced and offset so every value is clearly legible at all times. These markings update in real time during camera pan and zoom.
• Axis and graph readability (critical): All axes — both the position scale along the ground line in the left area and the v‑t plot axes in the right area — must be rendered so that tick marks and their associated numeric labels are clearly separated. The minimum clearance between a tick label and any adjacent tick label or axis label must be large enough that no two text elements touch or overlap at any zoom level. If the viewport zooms out in Phase 4, the position scale labels along the ground must be thinned out or re‑spaced automatically to prevent crowding.
• Collision timing (critical): The collision between C and B must occur at the moment their surfaces are in full physical contact — i.e., the gap between the right surface of B and the left surface of C is exactly zero. C must travel the full distance to reach B before the collision event fires. Under no circumstances should the collision be triggered while C and B are still visually separated.
• Key physics constraint: $m_B = 3m_A$. This means the coupled oscillation does NOT produce a simple velocity exchange. At the moment the spring returns to its natural length ($t = T/2$), B's speed is at its minimum (0.5 m/s) and A's speed is at its maximum (1.5 m/s). The v‑t curves must be drawn to reflect this asymmetry.
• v‑t plot correctness: The velocity curves for A and B during oscillation must be derived from the correct sinusoidal position formulas that account for the $m_B = 3m_A$ mass ratio and the conserved centre‑of‑mass velocity. The centre‑of‑mass velocity of the A–B system after collision is $v_{cm} = \frac{m_B \cdot (-1)}{m_A + m_B} = -0.75\,\text{m/s}$ (taking leftward as negative). The oscillatory components are superimposed on this drift. The amplitude of B's velocity oscillation is 0.25 m/s and A's is 0.75 m/s in the centre‑of‑mass frame, giving peak lab‑frame speeds of 0.5 m/s (B minimum) and 1.5 m/s (A maximum) at $t = T/2$. These values must be reflected exactly in the plotted curves.
• Spring–square coupling (critical for correctness): The spring is rendered as a geometric object whose two endpoints are computed every frame from the current positions of square A's right surface and square B's left surface. The spring's visual length is therefore always equal to the physical gap between the two squares. This strictly prevents the visually incorrect situation where square B appears to sit inside or overlap the spring while the spring still shows its natural length. This constraint must be enforced from the moment of collision onward.
• Spring compression must begin immediately after the collision: as soon as B acquires leftward velocity $v_B' = -1\,\text{m/s}$, it closes the gap with A and the spring starts compressing visually without delay.
• Camera tracking and zoom from Phase 3 onward: Starting at the beginning of the coupled oscillation (approximately t = 3 s into the video), the viewport continuously follows the centre of mass of the A–B system via smooth translation. After t ≈ 11 s, if translation tracking alone is no longer sufficient to keep both squares and the full spring within the left area (because the centre of mass has drifted too far from the initial frame), the camera additionally zooms out gradually and continuously. The zoom engages only when needed and is applied at a smooth, constant rate — never as a sudden jump — so the audience can follow the motion without disorientation. The zoom level is chosen to be the minimum required to keep the entire A–B system (both square surfaces and the full spring) within the left area at all times. The floating reference scale and position labels along the ground line are re‑scaled in real time to match the current zoom level, maintaining a clear and accurate spatial reference for the viewer throughout. The velocity‑time plot in the right area is fixed and is never affected by the camera pan or zoom.
• The spring colour changes are instantaneous at the highlighted instants and revert to yellow after each pause.
• Velocity‑time plot axes remain static; only the curves and the real‑time cursor animate.
• The final frozen frame includes faint trails for the full trajectories. No concluding caption, summary text, or analysis overlay appears at any point during or after the outro.