Kinematics in 1 D – Comprehensive Notes
Introduction to Kinematics in 1 D
- 1 D (one dimension): Motion confined to a single straight line, chosen to be the x–axis.
- Kinematics: Pure description of motion (no forces/causes), so focus is on how position, velocity, and acceleration vary with time for a particle restricted to the x–axis.
- Particle can move forwards ( +x ) or backwards ( –x ) only.
Position & Displacement
- Position variables
- Initial position: x0 at time t0.
- Later position: x at time t.
- Displacement (vector)
- Symbol: \vec{s} (often written simply as s in 1 D).
- Definition: s = x - x_0.
- Direction is encoded in its sign:
- s > 0 → displacement to the right ( +x ).
- s < 0 → displacement to the left ( –x ).
- Magnitude of displacement
- Denoted |s| (absolute‐value bars).
- |s| is always non-negative regardless of sign of s.
- Notation caution
- In 1 D, writing s without an arrow does NOT mean magnitude; it still means x - x_0. Use |s| for magnitude explicitly.
Average Velocity
- In any interval [t_0, t], the particle may speed up, slow down, or reverse, but we can still compute an average.
- Definition
- \bar{v} = \dfrac{\text{displacement}}{\text{time interval}} = \dfrac{s}{t - t_0}.
- Alternative notation: \bar{v} = \dfrac{\Delta x}{\Delta t} where \Delta x = s and \Delta t = t - t_0.
- Units: length / time (e.g.
\text{m·s}^{-1}).
Instantaneous Velocity
- Everyday idea: the speedometer reading of a car at a single instant, even while accelerating.
- Mathematical formulation
- Compute \bar{v} over a time interval \Delta t that ends (or centers) at the instant of interest t.
- Shrink \Delta t → 0 so the interval collapses to that single instant.
- Limit definition
- v = \lim_{\Delta t \to 0} \dfrac{\Delta x}{\Delta t}.
- In calculus language: v = \dfrac{dx}{dt}.
- Direction/sign conventions identical to displacement.
Average Acceleration
- Consider instantaneous velocities at t0 (value v0) and t (value v).
- Definition
- \bar{a} = \dfrac{\Delta v}{\Delta t} = \dfrac{v - v0}{t - t0}.
- Units: length / time² (e.g.
\text{m·s}^{-2}).
Instantaneous Acceleration
- Analogy with velocity: shrink the time interval over which average acceleration is measured.
- Limit definition
- a = \lim_{\Delta t \to 0} \dfrac{\Delta v}{\Delta t}.
- In calculus terms: a = \dfrac{dv}{dt}.
- Physically: rate at which velocity changes at a single moment.
Direction & Vector Representation in 1 D
- Even though only one spatial dimension exists, quantities like displacement, velocity, and acceleration are still vectors because they can point either way along the line.
- Graphically, arrows can be drawn on the x-axis to show direction; in formulas, sign does the job.
- Positive sign means pointing toward +x (right); negative sign means toward –x (left).
Summary Connections & Practical Notes
- Hierarchy
- Position x(t) → first derivative → velocity v(t) → second derivative → acceleration a(t).
- These foundational definitions allow later derivation of constant-acceleration equations, graphical area interpretations, and numerical modeling.
- Real-world relevance: tracking cars on a straight highway, falling objects along a vertical line (if upward chosen +x), motion of elevators, etc.
- Ethical/engineering implication: precise definitions ensure consistent communication and safety in system designs (e.g.
braking distances require understanding of acceleration).
