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<h1>✅ Adventure 5 — Solutions</h1>
<div class="subtitle">Drag Racing — 1/4 mile (400 m), constant acceleration <code>a = 8 m/s²</code>.</div>
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<h2>1) How far, how fast?</h2>
<ul>
<li>Distance at 5 s: <code>x(5)=½·8·5² = 100 m</code></li>
<li>Distance at 10 s: <code>x(10)=½·8·10² = 400 m</code></li>
<li>Velocity at 5 s: <code>v(5)=8·5 = 40 m/s</code></li>
<li>Velocity at 10 s: <code>v(10)=8·10 = 80 m/s</code></li>
</ul>
<h2>2) Area means distance</h2>
<ul>
<li>Area 0→5: triangle <code>½·5·40 = 100 m</code></li>
<li>Area 0→10: triangle <code>½·10·80 = 400 m</code></li>
<li>These match the distances above ✓</li>
</ul>
<h2>3) Slope means velocity</h2>
<ul>
<li>Slope (tangent) at t=5 is about <code>40 m/s</code>, which matches <code>v(5)</code>.</li>
</ul>
<h2>4) Finish line (¼ mile = 400 m)</h2>
<ul>
<li>Solve <code>½·8·t² = 400</code> → <code>4t²=400</code> → <code>t=10 s</code></li>
<li>Velocity at finish: <code>v=8·10=80 m/s</code></li>
<li>In km/h: <code>80×3.6 = 288 km/h</code></li>
</ul>
<h2>5) Acceleration check</h2>
<ul>
<li>Δv from 0→5: <code>a·t = 8·5 = 40 m/s</code></li>
<li>Δv from 0→10: <code>8·10 = 80 m/s</code></li>
<li>Meaning: area under <code>a(t)</code> gives change in velocity.</li>
</ul>
<h2>6) (***) Calculus</h2>
<ul>
<li><code>x(t)=½·a·t²</code> → <code>dx/dt = v(t)=a·t</code></li>
<li><code>v(t)=a·t</code> → <code>dv/dt = a</code> (constant)</li>
<li>Integrals: <code>∫a dt = a·t = v(t)</code>, and <code>∫v(t) dt = ∫a·t dt = ½·a·t² = x(t)</code></li>
</ul>
<h2>7) (***) Kinetic Energy and Work — Table</h2>
<table>
<thead>
<tr>
<th>Time (s)</th>
<th>Velocity (m/s)</th>
<th>Distance (m)</th>
<th>KE (kJ)</th>
<th>Work (kJ)</th>
</tr>
</thead>
<tbody>
<tr><td>2</td><td>16</td><td>16</td><td>128</td><td>128</td></tr>
<tr><td>5</td><td>40</td><td>100</td><td>800</td><td>800</td></tr>
<tr><td>10</td><td>80</td><td>400</td><td>3200</td><td>3200</td></tr>
</tbody>
</table>
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