Physics Lecture: Intuitive Understanding of Velocity and Acceleration

Introduction

• Explore traditional physics concepts and formulas.
• Emphasize understanding as common sense ideas.

Basic Assumptions

• Positive number: velocity to the right.
• Negative number (not covered): velocity to the left.
• Operate in one dimension.

Example Scenario

• Initial Velocity:
  • 5 meters per second (m/s) to the right.
• Constant Acceleration:
  • 2 meters per second squared (m/s²) to the right.
• Duration:
  • Change in time (Δt) = 4 seconds (s).

Key Questions

• How fast are we going after 4 seconds?
• How far have we traveled over these 4 seconds?

Calculating Final Velocity

• Initial Velocity (vᵢ): 5 m/s
• Acceleration (a): 2 m/s²
• Time (t): 4 s
• Formula: [ v_f = v_i + (a \times t) ]
  • Calculate: [ v_f = 5 \text{ m/s} + (2 \text{ m/s²} \times 4 \text{ s}) = 13 \text{ m/s} ]
• Intuitive Understanding:
  • Each second, velocity increases by 2 m/s.
  • Calculation breakdown: 5, 7, 9, 11, 13 m/s sequentially.

Calculating Distance Traveled

• Distance = Area under velocity-time graph.
• Graph Breakdown:
  • Two shapes: Rectangle + Triangle

Rectangle Area

• Base (Time): 4 s
• Height (Initial Velocity): 5 m/s
• Area (Distance without Acceleration): 5 m/s × 4 s = 20 meters

Triangle Area

• Base (Time): 4 s
• Height (Change in Velocity): 13 m/s - 5 m/s = 8 m/s
• Area (Additional Distance due to Acceleration):
  • Formula: [ \frac{1}{2} \times \text{Base} \times \text{Height} ]
  • Calculate: [ \frac{1}{2} \times 4 \text{ s} \times 8 \text{ m/s} = 16 \text{ meters} ]

Total Distance

• Total = Rectangle Area + Triangle Area
• Total Distance = 20 m + 16 m = 36 meters (to the right)

General Formula Derivation

• Distance (d) = Δt × (vᵢ + ( \frac{1}{2} \times (v_f - v_i) ))
• Simplifies to: [ d = \Delta t \times \left( v_i + \frac{1}{2}(v_f + v_i) \right) ]
• This gives us the average velocity times time formula for distance.

Average Velocity Concept

• Average of initial and final velocity: [ v_{avg} = \frac{v_i + v_f}{2} ]
• Valid for constant acceleration.
• Allows for simplified distance calculation: [ d = v_{avg} \times \Delta t ]

Conclusion

• Emphasize reasoning rather than memorization of formulas.
• Understand and derive formulas logically for intuitive comprehension.