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<h1>✅ Adventure 7 — Solutions</h1>
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Filled tables and sample explanations. Use these to check your work.
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<h2>1) Velocity estimates and chart values</h2>
<ul>
<li><b>t=5:</b> slope ≈ 3.3 km/s, chart 3.9 km/s → 3900 m/s</li>
<li><b>t=30:</b> slope ≈ 7.9 km/s, chart 7.8 km/s → 7800 m/s</li>
<li><b>t=58:</b> slope ≈ 10 km/s, chart 11 km/s → 11000 m/s</li>
</ul>
<h2>2) Acceleration (slope of velocity)</h2>
<ul>
<li><b>t=5:</b> 13 m/s² (matches chart)</li>
<li><b>t=30:</b> 0 m/s² (matches chart)</li>
<li><b>t=58:</b> 5.66 m/s² (matches chart)</li>
</ul>
<h2>3) Mass from m(t)</h2>
<ul>
<li><b>m(5)=</b> 50,000 − 2,000·5 = 40,000 kg</li>
<li><b>m(30)=</b> 30,000 kg</li>
<li><b>m(58)=</b> 30,000 − 600(58−55) = 27,000 kg</li>
</ul>
<h2>4) Mass change m′(t)</h2>
<ul>
<li><b>t=5:</b> m′ = −2000 kg/min = 33.3 kg/s</li>
<li><b>t=30:</b> m′ = 0</li>
<li><b>t=58:</b> m′ = −600 kg/min = 10 kg/s</li>
</ul>
<h2>5) Force</h2>
<ul>
<li><b>t=5:</b> m·a = 520,000 N; v·m′ = 129,870 N; F ≈ 390,130 N</li>
<li><b>t=30:</b> F ≈ 0</li>
<li><b>t=58:</b> m·a = 152,820 N; v·m′ = 110,000 N; F ≈ 42,820 N</li>
</ul>
<h2>Interpretation (sample answers)</h2>
<ul>
<li><b>Force is largest at launch</b> because acceleration and fuel burn are both large.</li>
<li><b>Force is near zero while coasting</b> because a≈0 and m′=0.</li>
<li><b>Force increases near 58 min</b> because the rocket starts another burn (escape burn).</li>
</ul>
<h2>(***) Why acceleration can be exact but velocity slope is not</h2>
<p>
The velocity curve is piecewise linear, so its slope (acceleration) is constant on each stage, making rise/run exact.
The distance curve is curved, so rise/run gives an average slope over an interval, not the instantaneous slope.
</p>
<h2>What makes the rocket go forward?</h2>
<p>Fuel is ejected backward; the reaction pushes the rocket forward (and the rocket becomes lighter as it burns fuel).</p>
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<img src="images/adventure_7_chart_1.png" alt="Rocket Distance–Velocity–Acceleration charts">
<img src="images/adventure_7_chart_2.png" alt="Rocket Mass–Momentum–Force charts">
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