Cosmos in the Classroom

The Language of Motion

SPHYSUnit The Geometry of Impact

Thinking Lens

patterns

Lecture 3.1: The Language of Motion

A graph of motion is a story waiting to be read.

Today’s Essential Questions

  • What is the difference between distance and displacement?
  • How can we find velocity from a position-time graph?
  • What does the area under a velocity-time graph represent?

Connecting to Our Last Investigation

In the traffic collision data, you saw patterns of braking, reacting, and crashing. Those events are all stories told with motion. Today, we’ll learn the formal language and graphical tools of kinematics—the study of motion—to describe that story with scientific precision.

Position, Distance, & Displacement

These three terms seem similar, but have precise meanings in physics.

  • Position (x): An object’s specific location relative to an origin point.
  • Distance: A scalar quantity. It is the total path length traveled.
  • Displacement ($\Delta x$): A vector quantity. It is the straight-line change in position from the start point to the end point, including direction.

Example: If you walk 5 meters east and then 3 meters west, you have traveled a distance of 8 meters, but your displacement is 2 meters east.

Speed vs. Velocity

This is another critical scalar-vector pair.

  • Speed: A scalar. It tells you how fast you are going (distance / time).
  • Velocity (v): A vector. It tells you how fast you are going and in what direction (displacement / time).
\[v_{avg} = \frac{\Delta x}{\Delta t} = \frac{x_{final} - x_{initial}}{t_{final} - t_{initial}}\]

Reading the Story: Position vs. Time Graphs

A graph of an object’s position versus time tells a complete story. The most important feature of this graph is its slope.

The slope of a position-time graph is velocity.

  • Steep slope: High velocity
  • Shallow slope: Low velocity
  • Zero slope (flat line): Zero velocity (stopped)
  • Negative slope: Negative velocity (moving in the reverse direction)
Image concept "position_vs_time_graph_slopes" not found

The Next Chapter: Velocity vs. Time Graphs

A velocity-time graph also tells a story.

  • The slope of this graph tells you the acceleration.
  • The area under the graph tells you the displacement.
Image concept "velocity_vs_time_graph_area" not found

Acceleration

Acceleration (a) is the rate at which an object’s velocity changes. Because velocity is a vector, you are accelerating if you are:

  1. Speeding up
  2. Slowing down
  3. Changing direction
\[a_{avg} = \frac{\Delta v}{\Delta t} = \frac{v_{final} - v_{initial}}{t_{final} - t_{initial}}\]

Thinking Lens: Patterns

Graphs allow us to see patterns in motion that would otherwise be invisible. A simple line can represent a complex journey.

Question: How can the shape of a line on a position-time graph (straight, curved upwards, curved downwards) tell us the entire story of an object’s acceleration without ever looking at a velocity graph?

Preparing for Our Next Task

The skills you have learned today—interpreting the slope and area on motion graphs—are exactly what you will need to successfully complete Problem Set 1 (Kinematic Graphs).

Summary: Answering Our Questions

  • What is the difference between distance and displacement? Distance is a scalar (total path traveled), while displacement is a vector (the change from start to end position).

  • How can we find velocity from a position-time graph? Velocity is the slope of the position-time graph.

  • What does the area under a velocity-time graph represent? The area under a velocity-time graph represents the displacement.

Prompt: In 2-3 sentences, explain how graphs can tell a complete story about an object’s motion.