Lesson 2: 4-2 Straight-Line Motion: Connecting Posting, Velocity and Accleration
Duration of Days: 2
Lesson Objective
Establish the hierarchical relationship between position, velocity, and acceleration using derivatives.
Determine when a particle is at rest, moving right (or up), or moving left (or down) based on the sign of the velocity.
Calculate displacement vs. total distance traveled over a specified time interval.
Use the relationship between the signs of velocity and acceleration to determine if an object’s speed is increasing or decreasing.
Calculate average velocity and average acceleration using the Difference Quotient.
If an object's velocity is negative but its acceleration is positive, what is happening to its speed?
How can a particle have a total distance of 10 meters but a displacement of 0 meters?
Why is velocity considered a vector quantity while speed is a scalar quantity?
At what point in a particle's path must the velocity be zero if it is changing direction?
Position s(t) or x(t)
Velocity v(t)
Acceleration a(t)
Speed |v(t)|
Initial Position s(0)
Instantaneous Velocity
Average Velocity
Displacement
Total Distance "At Rest"
Speeding Up / Slowing Down
Topic 4.2: Straight-Line Motion: Connecting Position, Velocity, and Acceleration.
Standard CHA-3.B: Determine the values of the derivative of a position function to solve problems involving motion.
Mathematical Practice 1.D: Identify an appropriate mathematical rule or procedure based on the classification of a given problem.
Description:
Students learn to differentiate position to find velocity and differentiate velocity to find acceleration. The lesson emphasizes "sign analysis"—using a number line to track when v(t) and a(t) are positive or negative. Students often use the FlippedMath "Sign Chart" method to visualize these changes over time.
Purpose: Motion is the most "tangible" application of calculus. Mastering this section allows students to handle both Multiple Choice and Free Response questions that ask for justifications of motion.
DOK 1 (Recall): Finding v(t) and a(t) given a polynomial position function.
DOK 2 (Skill/Concept): Identifying intervals where a particle moves left vs. right; calculating displacement.
DOK 3 (Strategic Thinking): Justifying if speed is increasing or decreasing at a specific time t by comparing the signs of v(t) and a(t).
DOK 4 (Extended Thinking): Given a velocity-time graph, sketching the general shape of the position and acceleration functions.
Scaffolding (Support):The Physics Ladder: Provide a visual "ladder" showing Position (Deriv) to Velocity (Deriv) to Acceleration.
Sign Chart Checklist: Give students a checklist for "Speeding Up/ slowing Down.
Extension (Challenge):Vertical Motion with Gravity: Have students model a projectile using an equation and determine the maximum height.
Multiple Particles: Challenge students with a "collision" problem: Given two different position functions, at what time do the particles have the same velocity?
College Board AP Classroom Assessments