Two Graphing Stories — Modeling Physical Motion
Can a picture tell two different stories? Translate physical events into piecewise mathematical graphs and discover how slope reveals velocity, rest intervals, and elevation changes.
Student & Teacher Overview: Lesson 5
Motion & Graphical TranslationIn this lesson, you will analyze two classic motion scenarios: a hiker descending and ascending a canyon (Elevation vs. Time) and a skateboarder accelerating down a ramp and coasting across flat ground (Speed vs. Time). Use the animated simulator below to link physical motion directly with piecewise coordinate graphs.
Core Student Outcomes
- Match physical motion scenarios with piecewise continuous functions and coordinate graphs.
- Distinguish between an Elevation vs. Time graph (where $y$ represents physical vertical altitude) and a Speed vs. Time graph (where $y$ represents rate of motion).
- Interpret the physical meaning of negative, positive, and zero slopes in different physical contexts.
Teacher Insight
Beware the "Graph as Picture" trap: students frequently mistake a graph's shape for the physical road or hill. Remind students that on a Speed vs. Time graph, a downward slope does not mean the skateboarder is going downhill—it means they are slowing down!
Animated Graphing Story Studio
Select a motion scenario below. Play the animation or drag the timeline scrubber to observe how the visual motion directly traces the coordinate graph in real time.
Critical Comparison: Position Graphs vs. Velocity Graphs
Understanding the physical variable on the vertical axis ($y$) changes how we interpret every single slope and feature of the graph:
- Positive Slope: Moving upward in space (climbing altitude).
- Negative Slope: Moving downward in space (descending altitude).
- Zero Slope ($m = 0$): Standing completely still at a constant altitude.
- Positive Slope: Accelerating (moving faster and faster).
- Negative Slope: Decelerating / Braking (slowing down).
- Zero Slope ($m = 0$): Moving at a constant cruising speed (NOT standing still!).
Key Vocabulary: Lesson 5
Piecewise Function
A function defined by multiple sub-functions, each applying to a distinct interval of the independent variable (domain).
Dependent vs. Independent
Time ($t$) is almost universally independent (horizontal axis), while physical quantities like altitude or velocity depend on time.
Constant Speed
Motion without acceleration; on a speed graph, it produces a flat horizontal line ($a = 0$), while on a position graph, it produces a linear sloped line.
Acceleration
The rate of change of velocity with respect to time ($a = \frac{\Delta v}{\Delta t}$), represented by the slope of a velocity-time graph.
Exit Ticket: Quick Mastery Check
Demonstrate your understanding of Lesson 5 concepts to log mastery to your profile.