Shadow Length of a Walking Boy

Overview

A boy of height $0.90\,\text{m}$ walks away from a lamppost that is $3.6\,\text{m}$ tall at $1.2\,\text{m/s}$. The animation visualizes the geometry after $4$ seconds and shows the computed shadow length using similar triangles.

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Phases

#	Phase Name	Duration	Description
1	Intro	~3s	The lamppost and boy appear; a brief title "Shadow Length – Related Rates" fades in.
2	Setup	~4s	The lamppost height is labeled, the boy’s height is shown, and a light ray from the lamp top to the ground is drawn, forming the initial shadow line.
3	Motion	~6s	The boy moves horizontally away from the lamppost for 4 seconds (distance $1.2\times4 = 4.8\,\text{m}$). The light ray updates continuously, extending the shadow.
4	Calculation	~4s	At the 4‑second mark the scene pauses; a transparent overlay displays the similar‑triangle relation $\frac{3.6}{x+0.9}=\frac{3.6}{\text{shadow}+\text{boy}}$ and solves for the shadow length, revealing the numeric result.
5	Result	~3s	The final shadow length $\approx 3.6\,\text{m}$ is highlighted, and the boy, lamppost, and shadow remain on screen for a moment before fading out.

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Layout

┌─────────────────────────────────────────────┐
│                                             │
│                MAIN (visual scene)          │
│                                             │
├─────────────────────────────────────────────┤
│  Caption (optional) – step title or formula │
└─────────────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Main	Lamppost, boy figure, moving shadow line, light ray, distance markers	Central visual focus; occupies most of the frame
Caption	Short step titles (e.g., "Setup", "Calculation") and the final formula $\frac{3.6}{x}=\frac{3.6}{x+0.9}$	Appears at the bottom band only when needed

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Notes

• Assumptions: Boy’s height is taken as $0.90\,\text{m}$; lamppost height $3.6\,\text{m}$; walking speed $1.2\,\text{m/s}$; time interval exactly $4$ s. No textual narration is used; all information is conveyed visually.
• The shadow length after 4 s is computed via similar triangles: $$\frac{3.6}{x+0.9}=\frac{3.6}{x}$$ leading to $x = 3.6\,\text{m}$.
• Total animation time is kept under 20 seconds to maintain brevity while fully illustrating the problem.
• Only one Manim Scene class is required; all phases are sequenced within that scene.
• No persistent on‑screen text beyond the caption band; equations appear only during the calculation phase.