Overview
Kinematics links displacement, velocity and acceleration using calculus.
The relationships
- Velocity = rate of change of displacement: v = ds/dt
- Acceleration = rate of change of velocity: a = dv/dt
- Going back: s = ∫ v dt and v = ∫ a dt
Key points
- At rest / instantaneously stationary → v = 0.
- Maximum velocity → a = 0 (dv/dt = 0).
- Distance travelled comes from integrating velocity (mind direction changes).
Worked example
s = t³ − 6t² + 9t. Then v = 3t² − 12t + 9. The particle is at rest when v = 0: 3(t − 1)(t − 3) = 0 → t = 1 s and t = 3 s.
Common mistakes
- Confusing displacement (a vector, can be negative) with distance (always positive).
Exam tips
- Differentiate to go down (s → v → a); integrate to go back up.