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<title>Adventure 14 – Activity 1 (Solution)</title>
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<h1>🎯 Adventure 14 — Activity 1 (Solution)</h1>
<div class="subtitle">Completed derivation and numerical results for the Moon’s orbit</div>
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<h2>2) Key Equations and Derivation</h2>
<p>The Moon stays in orbit because gravity provides exactly the centripetal force required for circular motion.</p>
<p class="eq"><b>Newton’s gravitational force:</b> F = GMm / R²</p>
<p class="eq"><b>Circular motion requirement:</b> F = mv² / R</p>
<p>Set them equal:</p>
<p class="eq"><b>GMm / R² = mv² / R</b></p>
<p>Cancel <b>m</b> and multiply by <b>R</b>:</p>
<p class="eq"><b>GM / R = v²</b></p>
<p>Therefore:</p>
<p class="eq"><b>v = √(GM / R)</b></p>
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<h2>3) Compute the Moon’s Orbital Speed</h2>
<p>Using <b>G = 6.67 × 10<sup>−11</sup></b>, <b>M = 5.97 × 10<sup>24</sup> kg</b>, and <b>R = 3.84 × 10<sup>8</sup> m</b>:</p>
<p><b>v ≈ 1018 m/s</b></p>
<p><b>v ≈ 1.02 km/s</b></p>
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<h2>4) Compute the Orbital Period</h2>
<p>Circumference:</p>
<p class="eq"><b>2πR ≈ 2.41 × 10<sup>9</sup> m</b></p>
<p>Time = distance / speed:</p>
<p><b>T ≈ 2.37 × 10<sup>6</sup> s</b></p>
<p>Convert to days:</p>
<p><b>T ≈ 27.4 days</b></p>
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<h2>5) Interpretation</h2>
<p><b>Why does the Moon not crash into Earth?</b><br>The Moon is constantly falling toward Earth, but its sideways velocity keeps missing Earth.</p>
<p><b>What if the Moon moved more slowly?</b><br>Its orbit would shrink and it could spiral inward.</p>
<p><b>What if the Moon moved more quickly?</b><br>Its orbit would grow larger, and if it became fast enough it could escape Earth completely.</p>
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<h2>1) The Big Idea</h2>
<p><b>The Moon stays in orbit because gravity provides exactly the centripetal force needed for circular motion.</b></p>
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