Lecture Notes on Equations of Motion and Graphical Analysis

Introduction

• Discussion on different graphs in equations of motion:
  • Displacement-Time Graph
  • Velocity-Time Graph
  • Acceleration-Time Graph
• Importance of understanding the meaning of graphs and their slopes.

Displacement-Time Graph (S-T Graph)

• Understanding the Graph:

  • The angle made by the graph with the positive x-axis represents the slope of the graph.
  • Slope formula:
    [ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} ]
  • For S-T graph:
    • y-axis = Displacement
    • x-axis = Time
    • Slope = Change in Displacement/Change in Time = Velocity.

• Key Points:

  • A steeper angle corresponds to higher velocity.
  • Slope = 0 implies object is at rest.
  • If the slope value is negative, it implies the object is moving in the negative direction (backward).

Numerical Example: S-T Graph Analysis

1. Given Points:
   • A: (0, 0)
   • B: (10, 10)
   • C: (15, 10)
   • D: (20, 15)
2. Regions:
   • Region O-A:
     • Velocity = [ \frac{10 - 0}{10 - 0} = 1 \text{m/s} ]
   • Region A-B:
     • Slope = 0
     • Velocity = 0
   • Region B-C:
     • Velocity = [ \frac{15 - 10}{20 - 15} = 1 \text{m/s} ]

Important Observations

• **Velocity Relation:
  • Positive slope = Positive velocity
  • Zero slope = At rest
  • Negative slope = Moving backward**
• The angle of slope (theta) determines velocity:
  • [ \tan(\theta) = \text{slope} ]

Velocity-Time Graph (V-T Graph)

• The slope of the V-T graph represents acceleration:
  • Slope formula again applies:
  • Velocity change over time corresponds to acceleration.
• Area under the V-T graph represents displacement.
  • Positive area = body moving forward
  • Negative area = body moving backward.

Key Concepts:

• Acceleration:
  • Positive slope = Positive acceleration.
  • Negative slope = Negative acceleration.
• Jerk: (rate of change of acceleration):
  • Not commonly tested, but understanding change in velocity through area of acceleration-time graph can be beneficial.

Summary

1. Understand the slope and areas of different graphs:
   • Slope = Measure of rate of change (velocity/acceleration).
   • Area = Measure of cumulative effect (displacement).
2. Analyze graphs to predict motion:
   • Positive/Negative/Zero values signify forward/backward motion or rest state.

Practice Problems

• Draw corresponding V-T graphs for given S-T graphs.
• Determine the areas and slopes for various given V-T graphs to find displacement and acceleration values.

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