Units 1 & 2 (Kinematics and Motion Modelling)
Introduce kinematics in a calculus context by modelling linear motion using displacement, velocity, and acceleration functions. Explore how differentiation and integration describe motion, interpret motion graphs, and solve problems involving time, distance, and change in velocity.

Unit 2: Calculus in Motion (Kinematics)

2.1 Modelling Motion with Functions

• Displacement Function: s(t), position of an object over time

• Velocity: first derivative of displacement, v(t) = s′(t)

• Acceleration: second derivative, a(t) = v′(t) = s″(t)

• Units: e.g. m, m/s, m/s² (depending on context)

2.2 Interpreting Motion Graphs

• Displacement-Time Graphs: shape and slope indicate velocity

• Velocity-Time Graphs: slope gives acceleration; area under curve gives displacement

• Acceleration-Time Graphs: area gives change in velocity

• Turning Points: when velocity = 0 (object at rest or changing direction)

2.3 Solving Motion Problems

• Given s(t): find v(t), a(t) using differentiation

• Given a(t) or v(t): find v(t), s(t) using integration (+ initial conditions)

• Total Displacement: ∫ₐᵇ v(t) dt

• Total Distance: integrate |v(t)| over given time intervals

2.4 Real-World Applications

• Vertical motion under gravity: s(t) = –½gt² + v₀t + s₀

• Projectile motion in one dimension

• Determining maximum height, time of flight, and impact velocity

• Use of initial conditions to solve for constants in integration

2.5 Technology and Graphical Analysis

• CAS for solving and checking derivative/integral expressions

• Graphing motion functions to visualise movement over time

• Interpreting motion behaviour through graphical intersections and extrema