Time–Speed–Distance (Kinematics) – Advanced Study Notes
(MIT‑level treatment for Karnataka TET – Quantitative Aptitude)
Objective – Build a rigorous, complete understanding of the time–speed–distance relationship from first principles, extend it to vectorial and calculus‑based contexts, handle edge cases, and solve a challenging problem that blends theory with application.
1. Foundations (First Principles)
| Symbol | Meaning | Units (SI) |
|---|---|---|
| (t) | Time | seconds (s) |
| (s) | Distance (arc length) | meters (m) |
| (v) | Speed (scalar) | m s(^{-1}) |
| (\mathbf{v}) | Velocity (vector) | m s(^{-1}) |
| (a) | Acceleration (scalar) | m s(^{-2}) |
| (\mathbf{a}) | Acceleration (vector) | m s(^{-2}) |
| (u) | Initial speed | m s(^{-1}) |
| (x) | Displacement | m |
1.1 Definition of Speed and Velocity
- Speed is the scalar rate of change of distance: [ v = \frac{\Delta s}{\Delta t};. ]
- Velocity is the vector rate of change of displacement: [ \mathbf{v} = \frac{\Delta \mathbf{x}}{\Delta t};. ]
Remark – Speed is always non‑negative; velocity can be positive or negative depending on the chosen reference direction.
1.2 Average vs. Instantaneous
- Average speed