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  <title>Proof of the Work-Energy Theorem</title>
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<body>

  <div class="header-bar">
    <div class="header-title">Work-Energy Proof</div>
    <div class="nav-bar">
      <button id="stepBtn0" class="btn-step active" title="Step 1">1</button>
      <button id="stepBtn1" class="btn-step" title="Step 2">2</button>
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      <button id="stepBtn3" class="btn-step" title="Step 4">4</button>
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  </div>

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    <div id="card0" class="step-card active current" data-step="0">
      <div class="split-row">
        <div class="mini-box">
          <div class="tag"><span class="step-num-badge">1</span> Newton's 2nd Law</div>
          <div class="math-eq">
            <span class="var">F</span><span class="sub">net</span>
            <span class="op">=</span>
            <span class="var">m</span><span class="op">&middot;</span><span class="var">a</span>
          </div>
        </div>
        <div class="mini-box">
          <div class="tag">Kinematics Eq.</div>
          <div class="math-eq">
            <span class="var">v</span><span class="sub">f</span><span class="sup">2</span>
            <span class="op">&minus;</span>
            <span class="var">v</span><span class="sub">i</span><span class="sup">2</span>
            <span class="op">=</span>
            2<span class="var">a</span><span class="var">d</span>
          </div>
        </div>
      </div>
    </div>

    <div id="arrow0" class="arrow-down active">&darr;</div>

    
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      <div class="tag"><span class="step-num-badge">2</span> Solve for Acceleration (<span class="var">a</span>)</div>
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        <span class="var">a</span>
        <span class="op">=</span>
        <span class="frac">
          <span class="num"><span class="var">v</span><span class="sub">f</span><span class="sup">2</span><span class="op">&minus;</span><span class="var">v</span><span class="sub">i</span><span class="sup">2</span></span>
          <span class="den">2<span class="var">d</span></span>
        </span>
      </div>
    </div>

    <div id="arrow1" class="arrow-down">&darr;</div>

    
    <div id="card2" class="step-card" data-step="2">
      <div class="tag"><span class="step-num-badge">3</span> Substitute <span class="var">F</span><span class="sub">net</span> &amp; <span class="var">a</span> into Work</div>
      <div class="math-eq">
        <span class="var">W</span><span class="sub">net</span>
        <span class="op">=</span>
        <span class="var">F</span><span class="sub">net</span><span class="op">&middot;</span><span class="var">d</span>
        <span class="op">=</span>
        <span class="var">m</span>
        <span class="paren">[</span><span class="frac">
          <span class="num"><span class="var">v</span><span class="sub">f</span><span class="sup">2</span><span class="op">&minus;</span><span class="var">v</span><span class="sub">i</span><span class="sup">2</span></span>
          <span class="den">2<span class="var">d</span></span>
        </span><span class="paren">]</span>
        <span class="op">&middot;</span><span class="var">d</span>
      </div>
    </div>

    <div id="arrow2" class="arrow-down">&darr;</div>

    
    <div id="card3" class="step-card final-card" data-step="3">
      <div class="tag"><span class="step-num-badge">4</span> Work-Energy Theorem Result</div>
      <div class="math-eq">
        <span class="var">W</span><span class="sub">net</span>
        <span class="op">=</span>
        <span class="frac"><span class="num">1</span><span class="den">2</span></span><span class="var">m</span><span class="var">v</span><span class="sub">f</span><span class="sup">2</span>
        <span class="op">&minus;</span>
        <span class="frac"><span class="num">1</span><span class="den">2</span></span><span class="var">m</span><span class="var">v</span><span class="sub">i</span><span class="sup">2</span>
        <span class="op">=</span>
        <span class="delta-ke">&Delta;<span class="var">KE</span></span>
      </div>
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Calculating Work Done and Mechanical Energy Transfers in Multistep Systems

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      height: auto;
      display: block;
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      justify-content: center;
      min-width: 0;
    }

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      font-size: 11px;
      font-weight: 700;
      color: #64748b;
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      letter-spacing: 0.4px;
      margin-bottom: 2px;
      white-space: normal;
      line-height: 1.2;
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      font-size: 14px;
      font-weight: 800;
      white-space: nowrap;
      overflow: hidden;
      text-overflow: ellipsis;
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</head>
<body>

  <div class="header-bar">
    <div class="step-badge" id="stepLabel">Stage 1: Work on Flat Track</div>
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      <button class="btn active" id="btn0" onclick="goToStep(0)">1. Start</button>
      <button class="btn" id="btn1" onclick="goToStep(1)">2. Ramp Base</button>
      <button class="btn" id="btn2" onclick="goToStep(2)">3. Peak Height</button>
      <button class="btn" id="playBtn" onclick="togglePlay()">▶ Play</button>
    </div>
  </div>

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      <line x1="538" y1="48" x2="570" y2="48" stroke="#cbd5e1" stroke-width="1.5" stroke-dasharray="3,3" />
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        <text x="17" y="22" text-anchor="middle" font-size="13" font-weight="800" fill="#ffffff">4 kg</text>

        
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  <div class="metrics-grid">
    <div class="metric-card">
      <div class="metric-title">Work Input (W_pull)</div>
      <div class="metric-val text-blue" id="wPullVal">+150 J</div>
    </div>
    <div class="metric-card">
      <div class="metric-title">Friction Loss (W_fric)</div>
      <div class="metric-val text-amber" id="wFricVal">-50 J</div>
    </div>
    <div class="metric-card">
      <div class="metric-title" id="energyTitle">Net Work (F_net · d)</div>
      <div class="metric-val text-emerald" id="energyVal">+100 J</div>
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  </div>

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      <text x="335" y="258" fill="#1e2945" font-size="14" font-weight="700" text-anchor="middle">Displacement d = 5 m</text>
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    <text x="224" y="172" fill="#1e2945" font-size="14" font-weight="600">θ = 37°</text>

    
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    <text x="162" y="105" fill="#64748b" font-size="13" font-weight="600" text-anchor="end">F sin(37°) = 12 N</text>
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    <text x="250" y="200" fill="#0284c7" font-size="14" font-weight="700" text-anchor="middle">F cos(37°) = 16 N</text>
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      W = F · d · cos(θ) = (20 N)(5 m) cos(37°) = <span class="highlight-blue">80 J</span>
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Gravitational and Elastic Potential Energy Transformations

Power and Efficiency in Physical Systems

Conservation of Energy in Closed vs. Open Systems

Isolated Systems: When Total Energy Stays Constant

Thermal Dissipation and Efficiency in Mechanical Systems

Calculating Thermal Energy Changes Using Specific Heat Capacity

Calorimetry: Thermal Equilibrium in Isolated Mixtures

Calculating Energy for State Changes Using Specific Latent Heat

Multi-Phase Heating Curves and Total Energy Summation

Synthesizing Gas Behavior: Predicting P-V-T Changes in Closed Systems

science

Work, Power, Thermal Transfers, and Thermodynamics

Estados físicos da matéria e mudanças de estado

Composição da matéria: átomos e moléculas

Evolução dos modelos atômicos: de Dalton a Bohr

Estrutura do átomo: prótons, nêutrons e elétrons

Número atômico, número de massa e isótopos

Distribuição eletrônica e camadas de energia

Estrutura da Matéria e Átomos

水浴加热装置的组装与实验规范

海波（或冰）熔化过程的温度采集与状态记录

绘制并分析晶体熔化的温度—时间图象

石蜡熔化对比实验与晶体/非晶体特征辨析

熔化吸热与凝固放热规律的实际应用

晶体与非晶体熔化规律实验探究

Tabela Periódica: organização e grupos

Propriedades periódicas: raio atômico e eletronegatividade

Metais, não metais e semimetais: propriedades e usos

Ligações iônicas: transferência de elétrons

Ligações covalentes: compartilhamento de elétrons

Ligações metálicas e propriedades dos metais

Tabela Periódica e Ligações Químicas

O que são reações químicas e como identificá-las

Conservação da massa em reações químicas

Balanceamento de equações químicas

Tipos de reações: síntese, decomposição, troca e combustão

Energia nas reações: exotérmicas e endotérmicas

Reações Químicas

Reatividade e toxicidade de materiais no cotidiano

Avaliando riscos de produtos químicos domésticos

Polímeros e plásticos: propriedades e impactos ambientais

Materiais recicláveis e descarte responsável

Corrosão e métodos de proteção de metais

Soluções e concentração no cotidiano

pH e indicadores ácido-base

Propriedades dos Materiais e Química Cotidiana

Formas de energia: cinética, potencial, térmica e luminosa

Conservação de energia em sistemas mecânicos

Trabalho e energia: transformações

Calor e temperatura: conceitos distintos

Transferência de calor: condução, convecção e irradiação

Calor específico e capacidade térmica

Equilíbrio térmico e mudanças de estado

Máquinas térmicas e eficiência energética

Energia e Calor: Transformações e Sistemas Térmicos

Radiações: tipos e origens

Radiação eletromagnética no cotidiano

Radiações ionizantes na saúde e na indústria

Geração de energia nuclear: benefícios e riscos

Protótipos de sistemas térmicos sustentáveis

Radiações, Energia Nuclear e Sistemas Sustentáveis

定滑轮的结构、特点与受力分析

动滑轮的省力特点与距离关系探究

滑轮组的绕线方式与承重绳段数判断

滑轮组拉力大小与移动距离的理论计算

有用功、额外功与总功的概念辨析

机械效率的定义与数学表达式解读

滑轮力学原理与机械效率基础

测量滑轮组机械效率的实验器材组装与调试

匀速竖直拉动弹簧测力计的读数要领

测量定滑轮与动滑轮的机械效率实验

探究物重对滑轮组机械效率的影响

探究动滑轮自重对滑轮组机械效率的影响

探究绳重与摩擦对滑轮组机械效率的影响

滑轮组机械效率实验与影响因素探究

斜面的省力原理与功的原理分析

测量斜面机械效率的实验设计与操作流程

探究斜面倾斜程度对机械效率和拉力的影响

探究斜面粗糙程度对斜面机械效率的影响

轮轴与斜面衍生机械的力学分析

斜面及变形简单机械的原理与效率

机械效率小于100%的物理本质探究

提高滑轮组与斜面机械效率的有效途径

机械效率与功率的联系与本质区别

滑轮组机械效率测量实验的误差来源与反思

水平拉动滑轮组时的机械效率与摩擦力计算

斜面在盘山公路与引桥工程中的实际应用

起重机与缆索吊装系统的机械效率综合分析

机械效率综合计算与拓展探究

机械效率深化认知与综合工程应用

Variables de estado: presión, volumen, temperatura y cantidad de gas

Presión atmosférica y manométrica: instrumentos de medición

Ley de Boyle-Mariotte: procesos isotérmicos

Ley de Charles: procesos isobáricos

Ley de Gay-Lussac: procesos isocóricos

Ley combinada de los gases y condiciones estándar (STP)

Ley de Avogadro y el volumen molar en gases ideales

Gases ideales y ecuación de estado PV=nRT

Leyes empíricas de los gases y ecuación de estado ideal

Ley de Dalton: presiones parciales y fracciones molares

Difusión y efusión de gases: ley de Graham

Gases reales: desviaciones por altas presiones y bajas temperaturas

La ecuación de Van der Waals para gases reales

Licuefacción de gases y condiciones críticas

Modelado comparativo entre el comportamiento de gases ideales y reales

Mezclas gaseosas, difusión y desviaciones del comportamiento real

Sistemas termodinámicos y primera ley

Variables de estado y funciones de trayectoria

Equilibrio térmico y la ley cero de la termodinámica

Energía interna de un sistema (U) y su componente microscópico

Trabajo mecánico por expansión y compresión de fluidos

Trabajo termodinámico en diagramas Presión-Volumen

Termodinámica: sistemas, variables de estado y trabajo mecánico

Temperatura, calor y equilibrio térmico: ley cero

Primera ley de la termodinámica: conservación de la energía

Procesos termodinámicos: isotérmicos, isobáricos, isocóricos y adiabáticos

Fundamentos de termodinámica y procesos de gases

Convención de signos para transferencia de calor y trabajo

Procesos isobáricos: trabajo y transferencia de calor a presión constante

Procesos isocóricos: variación de energía interna sin trabajo mecánico

Procesos isotérmicos: trabajo y calor con energía interna constante

Procesos adiabáticos: transformaciones energéticas con Q = 0

Ciclos termodinámicos cerrados y principios de motores térmicos

Resolución de problemas cuantitativos de la primera ley

Primera ley de la termodinámica y procesos termodinámicos

Calor, temperatura y transferencia térmica

Conducción térmica: transferencia de energía por contacto molecular

Convección térmica: corrientes ascendentes y descendentes de fluidos

Radiación térmica: propagación de calor sin medio material

Capacidad calorífica y calor específico de las sustancias

El calor específico del agua y su rol como termorregulador planetario

Mecanismos de transferencia de calor y sus efectos térmicos

El motor de combustión interna de cuatro tiempos: ciclo Otto

Motores diésel y combustión por alta compresión

La máquina de vapor y el ciclo de Rankine en termoeléctricas

Motores térmicos y ciclos de combustión

Radiación solar y calentamiento diferencial de la superficie terrestre

Gradientes de presión barométrica y la génesis del viento

Brisas de mar y tierra: ciclos diurnos de convección costera

Brisas de montaña y valle: convección térmica orográfica

Circulación global: células convectivas de Hadley, Ferrel y Polar

Efecto Coriolis y los vientos alisios y del oeste

Impacto del calentamiento global en la intensidad de vientos y tormentas

Dinámica de vientos y termodinámica atmosférica

Segunda ley de la termodinámica y entropía

El ciclo de Carnot y el límite teórico de eficiencia

Refrigeradores y bombas de calor: ciclos inversos

Eficiencia energética y etiquetas de consumo en electrodomésticos

Pérdidas por fricción y sistemas de refrigeración en vehículos

Transición hacia motores eléctricos e híbridos

Segunda ley, entropía, refrigeración y eficiencia energética

Tecnologías de control de emisiones: convertidores catalíticos y filtros

Transición energética y desarrollo sostenible en Colombia

Hidrógeno verde como vector energético del futuro

Modelos climáticos y predicción del tiempo atmosférico

Estrategias de mitigación ambiental y transición energética

OGS.10.PS.2

OGS.10.PS.2

Aligned learning resources

Related diagrams