|
Server : Apache/2.4.62 System : FreeBSD fbsdweb2.web.rcn.net 14.1-RELEASE FreeBSD 14.1-RELEASE releng/14.1-n267679-10e31f0946d8 GENERIC amd64 User : www ( 80) PHP Version : 8.3.8 Disable Function : NONE Directory : /domains/drsuper/CalculusMechanics/ |
Upload File : |
<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8" />
<title>Adventure 4 — Solutions</title>
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
<style>
body {
font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", sans-serif;
margin: 0;
padding: 0;
background: #f5f5fb;
color: #222;
}
.page {
width: 100%;
box-sizing: border-box;
padding: 1.8rem 5vw 3rem;
background: #ffffff;
max-width: 980px;
margin: 0 auto;
}
h1 { margin: 0.2rem 0 0.25rem; font-size: 2rem; }
.subtitle { color: #555; margin: 0 0 1.2rem; line-height: 1.45; }
h2 { margin-top: 1.4rem; margin-bottom: 0.4rem; font-size: 1.25rem; }
p { margin: 0.35rem 0 0.7rem; line-height: 1.55; }
ul { margin-top: 0.3rem; padding-left: 1.2rem; }
li { margin: 0.25rem 0; }
a { color: #1f4ed8; text-decoration: none; }
a:hover { text-decoration: underline; }
.toplinks { font-size: 0.95rem; margin-bottom: 0.8rem; }
.card {
border: 1px solid #dde3ff;
background: #f6f7ff;
border-radius: 14px;
padding: 1rem 1rem 0.95rem;
margin-top: 0.7rem;
}
.callout {
background: #f3f6ff;
border-radius: 10px;
border: 1px solid #dde3ff;
padding: 0.9rem 1rem;
font-size: 0.98rem;
margin: 1rem 0;
}
.questions {
background: #fff9e5;
border-radius: 10px;
border: 1px solid #f0e0a0;
padding: 0.9rem 1rem;
font-size: 0.98rem;
margin: 1rem 0;
}
.blank {
display:inline-block;
border-bottom: 2px solid #999;
min-width: 220px;
height: 1.05em;
vertical-align: baseline;
}
.blank.sm { min-width: 120px; }
.blank.lg { min-width: 360px; }
code { background:#fff; border:1px solid #d9def5; padding:2px 6px; border-radius:8px; }
.footer {
margin-top: 2.2rem;
font-size: 0.9rem;
color: #777;
text-align: left;
border-top: 1px solid #eee;
padding-top: 0.8rem;
}
/* --- Adventure 4 two-column layout (questions left, chart right) --- */
.page { max-width: none !important; } /* full width */
.two-col {
display: grid;
grid-template-columns: 1.15fr 0.85fr;
gap: 2rem;
align-items: start;
clear: both;
}
.right-col { text-align: center; }
.right-col img {
width: 115%; /* expanded another 20% */
height: auto;
border-radius: 12px;
border: 1px solid #dde3ff;
display: inline-block;
}
.chart-sticky {
position: sticky;
top: 1rem;
}
@media (max-width: 900px) {
.two-col { grid-template-columns: 1fr; }
.chart-sticky { position: static; }
.right-col img { width: 100%; }
}
.top-buttons{
display: flex;
gap: 10px;
margin-left: 12px; /* ← move right */
}
button {
background: #e9f0ff;
border: 2px solid #6b8cff;
border-radius: 0;
padding: 10px 18px;
font-size: 1rem;
cursor: pointer;
box-shadow: 0 3px 0 #4f6fe6;
transition: all 0.12s ease;
}
/* Hover */
button:hover {
background: #f3f7ff;
}
/* Pressed */
button:active {
background: #cddcff;
box-shadow: 0 1px 0 #4f6fe6;
transform: translateY(2px);
}
/* Optional: "checked" feeling after click */
button.checked {
background: #dff5e1;
border-color: #3a9b3a;
box-shadow: inset 0 2px 4px rgba(0,0,0,0.15);
}
button.sampled {
background: #fff3d9;
border-color: #d9a441;
box-shadow: inset 0 2px 4px rgba(0,0,0,0.15);
}
</style>
<style>
.wide-page { max-width: none !important; }
.two-col {
display: grid;
grid-template-columns: 1.1fr 0.9fr;
gap: 2rem;
align-items: start;
}
.chart-sticky {
position: sticky;
top: 1rem;
}
</style>
</head>
<body>
<div class="page wide-page">
<div style="display:flex; flex-wrap:wrap; gap:12px;">
<a href="index.html"><button style="font-size:16px; padding:12px 16px;">🧭 Hub</button></a>
<a href="adventure_4_home.html"><button style="font-size:16px; padding:12px 16px;">🏠 Home</button></a>
<a href="adventure_4.html"><button style="font-size:16px; padding:12px 16px;">✏️ Activity</button></a>
</div>
<h1>✅ Adventure 4 — Solutions</h1>
<div class="subtitle">
These show how to read the graphs, compute areas, and connect derivatives/integrals to motion.
</div>
<div class="card">
</div>
<div class="two-col">
<div class="left-col">
<h2>1) Height vs Time</h2>
<ul>
<li><code>y(4)=980−5·16=900</code> m</li>
<li><code>y(10)=980−5·100=480</code> m</li>
<li>Distance fallen from 4→10: <code>900−480=420</code> m</li>
</ul>
<h2>2) Velocity and Distance</h2>
<ul>
<li>Speed magnitude: <code>|v(t)|=gt=10t</code> (m/s)</li>
<li>Area 0→4: triangle <code>½·4·40=80</code> m</li>
<li>Area 0→10: triangle <code>½·10·100=500</code> m</li>
<li>Check with height: <code>980−y(4)=80</code> and <code>980−y(10)=500</code> ✓</li>
</ul>
<h2>3) Hitting the Ground</h2>
<ul>
<li><code>980−5t²=0 → t²=196 → t=14</code> s</li>
<li>Impact speed magnitude: <code>|v(14)|=10·14=140</code> m/s</li>
<li><code>140×3.6=504</code> km/h</li>
</ul>
<h2>4) Acceleration Check</h2>
<ul>
<li>Rectangle area from 4→10: <code>10·6=60</code> m/s</li>
<li>Meaning: area under acceleration = change in velocity. <code>v(10)−v(4)=100−40=60</code> ✓</li>
</ul>
<h2>★ Calculus (Derivatives + Integrals)</h2>
<ul>
<li><code>y(t)=980−5t² → v(t)=−10t</code> (negative means y decreases). Speed = <code>10t</code>.</li>
<li><code>a(t)=v′(t)=−10</code> (downward). Magnitude is 10.</li>
<li>Integrate back (using initial conditions): <code>v(t)=∫(−10)dt=−10t</code>, <code>y(t)=∫(−10t)dt=980−5t²</code>.</li>
</ul>
</div>
<div class="right-col chart-sticky">
<img src="images/adventure_4_chart.png" alt="Adventure 4 DiVA Charts">
</div>
</div>
<div class="footer">Dr. Super & Spark — Powered by ChatGPT</div>
</div>
</body>
</html>