Overview
This lecture covers the core rules and examples of kinematics in mechanics, focusing on differentiation, integration, graph interpretation, and key equations of motion.
Differentiation & Integration in Kinematics
- Differentiation is used to find velocity from displacement and acceleration from velocity.
- Integration is used to find velocity from acceleration and displacement from velocity.
- Use radians for trigonometry in mechanics-related problems.
- Displacement (s) is commonly used; velocity (v) = ds/dt; acceleration (a) = dv/dt or d²s/dt².
- Integration: v = ∫a dt, s = ∫v dt.
- Average speed = total distance / total time.
Interpreting Kinematics Graphs
- Displacement-time graph: gradient = velocity; straight line = constant speed; curved line = acceleration; horizontal line = stationary.
- The maximum point on a displacement-time graph is furthest from the start.
- Velocity-time graph: gradient = acceleration; area under the graph = displacement; maximum value = fastest speed.
- Acceleration-time graph: area under the graph = velocity; gradient (jerk) often not required.
SUVAT Equations (Constant Acceleration)
- SUVAT equations apply only with constant acceleration:
- v = u + at
- s = [(u + v)/2] × t
- v² = u² + 2as
- s = ut + ½at²
- s = displacement, u = initial velocity, v = final velocity, a = acceleration, t = time.
Example Problems
- If s = 3t² – 12, when t = 0, displacement = –12 m, so distance = 12 m (scalar).
- To find when displacement is 0, solve 0 = 3t² – 12 → t = 2 s.
- Velocity at t = 2: ds/dt = 6t → 6 × 2 = 12 m/s.
- Average speed in 3 s: final displacement = 15 m, initial displacement = –12 m, total distance = 27 m; average speed = 27/3 = 9 m/s.
- Stone projected upward from 13 m with 8 m/s: greatest height = 13 + 3.2 = 16.2 m.
- Velocity when hitting ground: use v² = u² + 2as with downward acceleration, v = 18 m/s.
Key Terms & Definitions
- Displacement (s) — distance moved in a specific direction.
- Velocity (v) — rate of change of displacement.
- Acceleration (a) — rate of change of velocity.
- SUVAT equations — set of equations relating s, u, v, a, and t under constant acceleration.
- Jerk — rate of change of acceleration.
Action Items / Next Steps
- Practice more kinematics questions, especially using SUVAT equations.
- Review trigonometry in radians for mechanics.
- Prepare for next class on further kinematics problem-solving.