Motion Analysis: Distance-Time and Velocity-Time Graphs

Date: August 12
Learning Target: The primary objective is to analyze the motion of an object using both Distance-Time (D-T) and Velocity-Time (V-T) graphs.
AI Connection: The study of motion graphs utilizes Artificial Intelligence concepts of modeling and visualization to represent physical phenomena.
Mini Lesson: An overview focusing specifically on the interpretation and construction of Distance-Time graphs.
Student Assignment: Practical application through Gizmo - Distance-Time and Velocity-Time Graphs.
Summary and Assessment: Completion of Quiz #1 (accessible via the eCLASS agenda) and the associated Gizmo exercises.

The Concept of Slope: In the context of motion modeling, the slope of a line on a coordinate plane is a fundamental measure used to determine rates of change.
The Slope Formula: Slope is defined as the ratio of the vertical change to the horizontal change between two points on a line. $\text{Slope} = \frac{\text{Rise (Y-Axis)}}{\text{Run (X-Axis)}}$
Step-by-Step Calculation Procedure:
- Identify Points: Select two distinct points on the plotted line.
- Determine Rise: Measure the vertical change (the difference in Y-values) between these points.
- Determine Run: Measure the horizontal change (the difference in X-values) between these points.
- Calculate: Divide the rise by the run.
Numerical Example Calculation:
- In a scenario where the graph shows a vertical rise of $3$ units and a horizontal run of $6$ units: $\text{Slope} = \frac{3}{6}$ $\text{Slope} = 0.5$

Graphing Variables: When creating a Position-Time graph (also referred to as a Distance-Time graph), the X-axis always represents Time, and the Y-axis always represents Distance or Position.
Defining Velocity via Slope: On a Position-Time graph, the slope represents the velocity of the object.
Calculation Example:
- If an object moves forward $3.0\,m$ (Rise) in a duration of $6.0\,s$ (Run): $\text{Velocity} = \frac{3.0\,m}{6.0\,s} = 0.5\,m/s$
Physical Interpretation: A slope of $0.5\,m/s$ indicates that the object is moving forward at a rate of half a meter every second.

Graphing Variables: On a Velocity-Time graph, the X-axis represents Time and the Y-axis represents Velocity ( $m/s$ ).
Defining Acceleration via Slope: On a Velocity-Time graph, the slope represents the acceleration of the object.
Calculation Example:
- If an object speeds up forward by $3.0\,m/s$ (Rise) in a duration of $6.0\,s$ (Run): $\text{Acceleration} = \frac{3.0\,m/s}{6.0\,s} = 0.5\,m/s^2$
Physical Interpretation: A slope of $0.5\,m/s^2$ indicates the object is accelerating, increasing its velocity by $0.5\,m/s$ every second.

Representing an Object At Rest:
- A horizontal line on a D-T graph indicates that the object is at rest.
- In this state, time continues to elapse (advancing along the X-axis), but the distance/position remains constant (no change on the Y-axis).
Representing Constant Speed:
- A straight, diagonal line moving upward indicates an object traveling at a constant speed.
- Speed Comparison: When comparing two lines on the same D-T graph, the line with the greater (steeper) slope represents the object moving at a higher velocity.
Representing Return to Home:
- A line with a negative slope (moving downward toward the X-axis) indicates the object is returning to the starting point.
- Distance = 0: This is defined as "home."
- Distance > 0: Any value away from zero represents a position away from home.
- In this profile, the object is specifically getting closer to the zero-distance mark.
Representing Displacement over Time:
- A curved line on a Displacement-Time graph indicates a change in velocity over time, reflecting either acceleration or deceleration rather than a constant speed.