Overview
This lecture covers the motion of a particle under constant acceleration and emphasizes the derivation, use, and conditions for four key kinematic equations, with a focus on their application to solving homework problems.
Motion Under Constant Acceleration
- A particle is modeled as a point moving in one dimension with constant acceleration.
- Average acceleration equals the change in velocity over the change in time when acceleration is constant.
- Equation 1: ( v_f = v_i + a t ), where ( v_f ) is final velocity, ( v_i ) is initial velocity, ( a ) is acceleration, and ( t ) is time.
Kinematic Equations and Derivations
- Velocity-time graphs: Displacement equals the area under the curve.
- Area under a linear velocity-time graph forms a trapezoid, equivalent to a rectangle using the average velocity.
- Equation 2: ( x_f = x_i + \frac{1}{2}(v_i + v_f)t ), giving displacement from initial and final velocities.
- By substituting for ( v_f ), derive Equation 3: ( x_f = x_i + v_i t + \frac{1}{2} a t^2 ), which does not require the final velocity.
- If time is unknown, rearrange to derive Equation 4: ( v_f^2 = v_i^2 + 2a(x_f - x_i) ), which eliminates time.
Conditions and Usage
- The four kinematic equations only apply when acceleration is constant.
- Use each equation based on given quantities (which variables are known and unknown).
- The "average velocity" formula (( v_{avg} = \frac{v_i + v_f}{2} )) only equals displacement over time if acceleration is constant.
Practical Application in Homework
- Always identify knowns and unknowns to choose the appropriate kinematic equation.
- For displacement problems: use Equation 2 if final velocity is known, Equation 3 if only time and acceleration are known.
- For problems where time is not given: use Equation 4.
- When dealing with direction changes (turning points), total distance traveled is the sum of the absolute values of displacements before and after the turning point.
- Homework question codes reference textbook section and edition; complete questions for covered sections as soon as possible.
Key Terms & Definitions
- Displacement (( \Delta x )) — The change in position (( x_f - x_i )).
- Velocity (( v )) — The rate of change of position.
- Acceleration (( a )) — The rate of change of velocity.
- v_* or v_star — The arithmetic mean of initial and final velocities, used in trapezoid area calculation.
- Average Velocity (( v_{avg} )) — Displacement divided by time interval; equals ( v_* ) only if acceleration is constant.
Action Items / Next Steps
- Read corresponding textbook sections on kinematic equations and derivations.
- Complete all homework questions for sections already covered, especially those using the four kinematic equations.
- For homework with direction change, carefully compute total travel distance as described.
- Review and practice problems involving each of the four kinematic equations.