Motion Graphs: Position vs Time & Velocity vs Time

Describing Motion with Graphs

Position vs Time Graph

• A car moving with a constant, rightward (+) velocity (e.g., +10 m/s) results in a position-time graph with a constant and positive slope.
• A car moving with a rightward (+), changing velocity (accelerating) results in a position-time graph with a changing and positive slope.

Position vs. Time Graphs Summary

• Constant velocity: straight line.
• Changing velocity (acceleration): curved line.

Slope and Velocity Relationship

• "As the slope goes, so goes the velocity."
• Constant velocity corresponds to a constant slope (straight line).
• Changing velocity corresponds to a changing slope (curved line).
• Positive velocity corresponds to a positive slope (moving upwards and to the right).

Calculating Slope

• Pick two points on the line and determine their coordinates.
• Determine the difference in y-coordinates (rise).
• Determine the difference in x-coordinates (run).
• Divide the difference in y-coordinates by the difference in x-coordinates: $slope = rise/run$.

Velocity vs Time Graphs

• A motion described as a constant, positive velocity results in a line of zero slope (a horizontal line has zero slope) when plotted as a velocity-time graph.
• A motion described as a changing, positive velocity results in a sloped line when plotted as a velocity-time graph. The slope of the line is positive, corresponding to the positive acceleration.

Velocity vs. Time Graphs Summary

• Constant positive velocity: Horizontal line above the x-axis representing zero acceleration.
• Changing positive velocity (positive acceleration): Line with a positive slope above the x-axis.

Importance of Slope in Velocity-Time Graphs

• The slope of a velocity-time graph reveals information about the acceleration of the object.
• Lines above x-axis: positive velocity.
• Lines below x-axis: negative velocity.

Speeding Up and Slowing Down

• Speeding up: The magnitude of the velocity is increasing; the line is getting further away from the x-axis.
• Slowing down: The magnitude of the velocity is decreasing; the line is approaching the x-axis.

Check Your Understanding

• An object's motion can be determined by analyzing the graph.

Analyzing a Constant Velocity Motion

• Example: Object moving at a constant velocity of +10 m/s.
• Position increases uniformly over time (0 m, 10 m, 20 m, 30 m, 40 m, 50 m at t=0, 1, 2, 3, 4, 5 seconds).
• Velocity-time graph shows a horizontal line at +10 m/s.
• Acceleration is 0 m/s² (slope of the velocity-time graph).

Analyzing a Changing Velocity Motion

• Example: Object accelerating at a constant rate.
• Velocity-time data shows increasing velocity over time (0, 10, 20, 30, 40, 50 m/s at t=0, 1, 2, 3, 4, 5 seconds).
• Acceleration is 10 m/s² (slope of the velocity-time graph).

Analyzing a Two-Stage Motion

• Velocity-time graph with different slopes in different time intervals.

Two-Stage Rocket Example

• Time intervals: t = 0 - 1 second, t = 1 - 4 second, t = 4 - 12 second
• Accelerations: +40 m/s², +20 m/s², -20 m/s²

Relating the Shape of Velocity-Time Graph to Motion

• Constant, Rightward (+) Velocity:
  • Positive velocity values are plotted.
  • Zero slope (a=0 m/s²).
• Constant, Leftward (-) Velocity:
  • Negative velocity values are plotted.
  • Zero slope (a=0 m/s²).
• Rightward (+) Velocity with a Rightward (+) Acceleration:
  • Positive velocity values are plotted.
  • A positive slope.
• Rightward (+) Velocity with a Leftward (-) Acceleration:
  • Positive velocity values are plotted.
  • A negative slope.
    *Leftward (-) Velocity with a
    Leftward (-) Acceleration
  • Negative velocity values are plotted.
  • A negative slope.

Determining the Slope on a v-t Graph

• Given two points (t1, v1) and (t2, v2) on a velocity-time graph, the slope (acceleration) is calculated as: $Slope = (v2 - v1) / (t2 - t1)$

• Example: Points (0 s, 5 m/s) and (5 s, 25 m/s).

• $Slope = (25 m/s - 5 m/s) / (5 s - 0 s) = 20 m/s / 5 s = 4 m/s^2$

Displacement from Velocity-Time Graph

• The area under the velocity-time graph represents the displacement.

Example 1: Rectangle Area

• Area of a rectangle: $Displacement = length * width$
• Displacement = + 180 m

Example 2: Triangle Area

• Area of a triangle: $Displacement = 0.5 * base * height$
• Displacement = + 80 m

Trapezoid Area

• Area of a trapezoid: $Area = 0.5 * (b) * (h1 + h2)$
• The trapezoid can also be broken into a triangle and a rectangle, and their areas can be computed individually and then summed to find the total displacement.
• Displacement = + 105 m