Overview
This lecture explains how to derive the four main kinematic equations used in constant acceleration motion, and summarizes their forms and variable meanings.
Deriving the Kinematic Equations
- The slope of a velocity vs. time graph represents acceleration: ( a = \frac{\Delta v}{\Delta t} ).
- If initial time is zero, ( a = \frac{v_f - v_i}{t} ).
- Rearranging gives the first kinematic equation: ( v_f = v_i + a t ).
Displacement from Area Under Curve
- The area under a velocity-time graph shows displacement (( \Delta x )).
- The total area is a rectangle (( v_i t )) plus a triangle (( \frac{1}{2}(v_f - v_i)t )).
- Substitute for ( v_f - v_i ) to get the second equation: ( \Delta x = v_i t + \frac{1}{2} a t^2 ).
Average Velocity Approach
- Average velocity (( v_{avg} )) equals displacement divided by time: ( v_{avg} = \frac{\Delta x}{t} ).
- For constant acceleration, ( v_{avg} = \frac{v_i + v_f}{2} ).
- Substituting gives the third kinematic equation: ( \Delta x = \frac{v_i + v_f}{2} t ).
Eliminating Time for Velocity-Displacement Relation
- Substitute ( t = \frac{v_f - v_i}{a} ) into the third equation.
- Simplifying gives: ( v_f^2 = v_i^2 + 2a\Delta x ) (the fourth kinematic equation).
Summary of the Four Kinematic Equations
- ( v_f = v_i + a t )
- ( \Delta x = v_i t + \frac{1}{2} a t^2 )
- ( \Delta x = \frac{v_i + v_f}{2} t )
- ( v_f^2 = v_i^2 + 2 a \Delta x )
Key Terms & Definitions
- Displacement (( \Delta x )) — The change in position.
- Velocity (( v )) — The speed of an object in a given direction.
- Initial velocity (( v_i )) — Starting velocity.
- Final velocity (( v_f )) — Velocity after a time period.
- Acceleration (( a )) — The rate of change of velocity.
- Time (( t )) — The duration of motion.
Action Items / Next Steps
- Review and memorize all four kinematic equations.
- Watch the next video for tips on choosing the correct equation for specific problems.