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  <title>Adventure 6 β€” Student Activity</title>
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      <a href="adventure_6_home.html"><button style="font-size:16px; padding:12px 16px;">🏠 Home</button></a>
      <a href="adventure_6_activity_2_solution.html"><button style="font-size:16px; padding:12px 16px;">βœ… Solutions</button></a>
    </div>

    <h1>✏️ Adventure 6 β€” Student Activity</h1>
    <div class="subtitle">
      Use the chart to connect <b>area</b> to <b>distance</b> and <b>velocity</b>.
      Motion: speed up, then slow down to a stop (no friction).
    </div>

    <div class="two-col">
      <div class="left-col">

        <h2>✳ Equations of Motion (given)</h2>
        <ul>
          <li><code>a(t)=8-2t</code></li>
          <li><code>v(t)=t(8-t)</code></li>
          <li><code>s(t)=4tΒ²-(1/3)tΒ³</code></li>
        </ul>

        <h2>1) How far, how fast? (Read the distance graph)</h2>
        <div class="row-grid">
          <div>
            <label for="s2"><b>Distance traveled in 2 seconds (m):</b></label>
            <input id="s2" type="number" step="0.1" value="">
          </div>
          <div>
            <label for="s27"><b>Distance traveled from 2 to 7 seconds (m):</b></label>
            <input id="s27" type="number" step="0.1" value="">
          </div>
        </div>

        <h2>2) Area means distance (use the velocity curve area in blue font)</h2>
        <div class="row-grid">
          <div>
            <label for="int02"><b>Area under v(t) from 0β†’2 (m):</b></label>
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          </div>
        </div>
        
        <h2>3) Velocity as area under acceleration</h2>
        <div class="row-grid">
          <div>
            <label for="a_area"><b>Signed area under a(t) from 0β†’8 (m/s):</b></label>
            <input id="a_area" type="number" step="0.1" value="">
          </div>
          <div>
            <label for="v8"><b>Velocity at 8 seconds from the velocity graph (m/s):</b></label>
            <input id="v8" type="number" step="0.1" value="">
          </div>
          <div>
            <label for="close2"><b>Are these two numbers equal?</b> (Yes/No)</label>
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          </div>
        </div>

        <h2>4) Acceleration check (0β†’5)</h2>
        <div class="row-grid">
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            <label for="dv05"><b>Change in velocity from 0β†’5 (m/s):</b></label>
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          </div>
          <div>
            <label for="explain"><b>1 sentence: what does area under a(t) mean?</b></label>
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          </div>
        </div>

        <h2>β˜… Calculus (optional)</h2>
        <ul>
          <li><code>s'(t)=v(t)</code> and <code>v'(t)=a(t)</code></li>
          <li><code>∫v(t)dt</code> gives distance (area idea).</li>
        </ul>

      </div>

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    <div id="hintBox" class="hint">
      Hint: <b>Area under v(t)</b> is <b>distance</b>. <b>Signed area under a(t)</b> is <b>change in velocity</b>.
      Use the labels on the curves (values at each second) to estimate.
    </div>

    <div id="solutionBox" class="solution">
      <strong>Reference values</strong>
      <ul>
        <li>Rectangle totals: Ξ”t=1 β†’ 84.00 m, Ξ”t=0.5 β†’ 85.00 m, Ξ”t=0.25 β†’ 85.25 m</li>
        <li>Exact integral (true distance): 85.33 m</li>
        <li>Velocity at 8 s: 0 m/s</li>
        <li>Change in velocity 0β†’5: 15 m/s</li>
      </ul>
    </div>

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Anon7 - 2021