Constant Acceleration Equations Derivation

Introduction

• Purpose: To derive the two constant acceleration equations graphically.
• Importance: Understanding the origin of these equations is useful and comforting, although memorization of the proofs is not required for tests.

Derivation of First Equation

• Starting Point: Constant acceleration.
• Acceleration as a Function of Time:
  • Acceleration is constant.
  • Using area calculation to transition from an acceleration graph to a velocity graph.
• Velocity Change (Δv):
  • Calculated as base * height from the graph.
  • Set initial time t_initial = 0, thus Δv = v_final - v_initial.
  • Rearranging gives: v_final = v_initial + acceleration * time.
  • Note: Use positive or negative values for velocities (v's) and acceleration (a). Time (t) is always positive.

Derivation of Second Equation

• From Velocity to Position Graph:
  • Area under the velocity-time graph is considered.
  • Split the area into a rectangle and a triangle.
  • Total area = ΔxA (triangle) + ΔxB (rectangle).
  • ΔxA = 0.5 * base(t) * height(v_final - v_initial).
  • ΔxB = base(t) * height(v_initial).
  • Algebraic manipulation yields: Δx = 0.5 * a * t^2 + v_initial * t.
• Position Function: Quadratic in nature.
  • Rearranged to conventional form: x_final = x_initial + v_initial * t + 0.5 * a * t^2.
  • Importance of Sign: Correct numerical answers depend on appropriate sign orientation for x's, v's, and a's.

Conclusion

• Equations to Remember:
  1. v_final = v_initial + acceleration * time
  2. x_final = x_initial + v_initial * time + 0.5 * a * t^2
• These equations are crucial and will be used frequently throughout the course. Understanding their derivation helps in grasping the logic behind their application.