Graphical Kinematics

<h1 id="lecture-21-the-language-of-motion">Lecture 2.1: The Language of Motion</h1>
<h3 id="unit-2-the-art-of-observation">Unit 2: The Art of Observation</h3>

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<h2 id="todays-mission">Today’s Mission</h2>

<ul>
 <li>
 Bridge Back: In the rocket lab, the changing slope of the velocity graph told a story about the engine firing and cutting off. Today, we’ll formalize the rules for reading that story.
 </li>
 <li>
 Thinking Lens: We will look for Patterns in graphs to understand the story of an object’s motion.
 </li>
 <li>Essential Questions:</li>
 <li>
 <ol>
 <li>What is the difference between distance and displacement?</li>
 </ol>
 </li>
 <li>
 <ol>
 <li>How can we find velocity from a position-time graph?</li>
 </ol>
 </li>
 <li>
 <ol>
 <li>What do the slope and area of a velocity-time graph represent?</li>
 </ol>
 </li>
</ul>

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<h2 id="where-are-you-position-distance--displacement">Where Are You? Position, Distance, & Displacement</h2>

<ul>
 <li>Position ($x$): An object’s location relative to a defined origin.</li>
 <li>Distance: The total path length traveled. It is a scalar.</li>
 <li>
 Displacement ($\Delta\vec{x}$): The change in position ($\vec{x}{final} - \vec{x}{initial}$). It is a vector (magnitude and direction).
 </li>
 <li>Example: You walk $7.0 \text{ m}$ East, then turn around and walk $3.0 \text{ m}$ West.</li>
 <li>
 <ul>
 <li>Distance traveled: $7.0 \text{ m} + 3.0 \text{ m} = 10.0 \text{ m}$</li>
 </ul>
 </li>
 <li>
 <ul>
 <li>Displacement: $4.0 \text{ m, East}$</li>
 </ul>
 </li>
</ul>

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<h2 id="position-vs-time-graphs">Position vs. Time Graphs</h2>

A position vs. time graph shows an object’s position at every moment.

The most important rule:
The slope of a position vs. time graph represents the object’s velocity.

[\text{slope} = \frac{\text{rise}}{\text{run}} = \frac{\Delta x}{\Delta t} = \text{velocity}]

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<h2 id="position-vs-time-accelerated-motion">Position vs. Time: Accelerated Motion</h2>

When an object accelerates, its position vs. time graph becomes curved.

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<h2 id="reading-the-story-of-a-p-t-graph">Reading the Story of a P-T Graph</h2>

<ul>
 <li>Positive Slope: Object is moving in the positive direction (away from the origin).</li>
 <li>Negative Slope: Object is moving in the negative direction (toward the origin).</li>
 <li>Zero Slope (Horizontal Line): Object is at rest (velocity is zero).</li>
 <li>Steeper Slope: Higher speed.</li>
 <li>Curved Line: The velocity is changing (acceleration is occurring).</li>
</ul>

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<h2 id="velocity-vs-time-graphs">Velocity vs. Time Graphs</h2>

<ol>
 <li>The slope of the graph represents acceleration.
$\text{slope} = \frac{\Delta v}{\Delta t} = \text{acceleration}$</li>
 <li>The area under the graph represents displacement.
$\text{area} = \text{velocity} \times \text{time} = \frac{\text{m}}{\text{s}} \cdot \text{s} = \text{m}$</li>
</ol>

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<h2 id="velocity-vs-time-accelerated-motion">Velocity vs. Time: Accelerated Motion</h2>

Key observations:
<ul>
 <li>Constant slope = constant acceleration</li>
 <li>Area under the curve = total displacement</li>
 <li>Steeper slope = greater acceleration</li>
</ul>

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<h2 id="example-analyzing-the-rocket-launch">Example: Analyzing the Rocket Launch</h2>

Let’s analyze the first 10 seconds of the rocket’s motion from our lab data.

<ul>
 <li>
 Acceleration ($0-4s$):
slope = $\frac{40.0 \text{ m/s} - 0.0 \text{ m/s}}{4.0 \text{ s} - 0.0 \text{ s}} = 10.0 \text{ m/s}^2$
 </li>
 <li>
 Displacement ($0-4s$):
area = $\frac{1}{2} \cdot \text{base} \cdot \text{height}$
area = $\frac{1}{2} (4.0 \text{ s})(40.0 \text{ m/s}) = 80.0 \text{ m}$
 </li>
</ul>

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<h2 id="check-for-understanding">Check for Understanding</h2>
Describe the rocket’s motion from t = 4.0 s to t = 8.2 s.

<ol>
 <li>What is its acceleration?
 <ul>
 <li>It’s the slope. It’s constant and negative ($\approx -9.8 \text{ m/s}^2$).</li>
 </ul>
 </li>
 <li>Is it speeding up or slowing down?
 <ul>
 <li>It’s slowing down. The velocity is positive but decreasing.</li>
 </ul>
 </li>
 <li>How could we find its displacement during this time?
 <ul>
 <li>Find the area of the trapezoid between t=4s and t=8.2s.</li>
 </ul>
 </li>
</ol>

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<h2 id="answering-our-essential-questions">Answering Our Essential Questions</h2>

1. What is the difference between distance and displacement?
<ul>
 <li>Distance is the total path length traveled (scalar)</li>
 <li>Displacement is the change in position with direction (vector)</li>
</ul>

2. How can we find velocity from a position-time graph?
<ul>
 <li>Velocity = slope of the position-time graph</li>
 <li>Steeper slope = higher speed; negative slope = motion toward origin</li>
</ul>

3. What do the slope and area of a velocity-time graph represent?
<ul>
 <li>Slope = acceleration (rate of velocity change)</li>
 <li>Area = displacement (total change in position)</li>
</ul>

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<h2 id="summary-and-next-steps">Summary and Next Steps</h2>

Key Rules:
<ul>
 <li>Position-Time:
 <ul>
 <li>Slope = Velocity</li>
 </ul>
 </li>
 <li>Velocity-Time:
 <ul>
 <li>Slope = Acceleration</li>
 <li>Area = Displacement</li>
 </ul>
 </li>
 <li>
 Bookend Statement: A graph of motion is a story waiting to be read.
 </li>
 <li>Bridge Forward: The skills of interpreting slope and area on these motion graphs are exactly what you’ll need to master Problem Set 2.1.</li>
</ul>