Kinematics in 1-D

"1D" means motion is confined to a single straight line (chosen as the x-axis).
We track a single particle that can move only left or right along this line.
Kinematics = purely describing motion of a particle along the x-axis (no forces or causes considered).

Choose an origin and positive direction on the x-axis.
Notation:
- t₀ = initial time
- x₀ = initial position of the particle
- t = a later time
- x = position of the particle at time ‘t’
The particle may move (speed up, slow down, reverse); We are free to sample it at any two instants at t₀ and t

s = x - x₀
Displacement = final position minus the initial position, where x is the final position and x₀is the initial position of the particle.
Magnitude: |\vec{s}| (absolute value: drop the sign, keep the size).
Direction encoded by the sign:
- \vec{s}>0 ⇒ motion toward +x (right).
- \vec{s}<0 ⇒ motion toward –x (left).
Caution with notation:
- In 1D, we do not switch to plain s for magnitude; s without an arrow still means x-x_0 (contains a sign).
- Must wrap with vertical bars |\,| to talk about the purely positive magnitude.

Captures “overall rate of change of position” during the interval [t_0,t].
Formula:
- \Delta x \equiv x-x_0 (identical to \vec{s}).
- \Delta t \equiv t-t_0 (always positive).
Unit intuition: metres per second (m/s).
Conceptual picture: even if the particle sped up, slowed down, or reversed, \bar{v} is the single constant velocity that would carry it from x_0 to x in the same total time.

Everyday sense: the speedometer reading of a car at one instant (direction included by sign).
Constructed mathematically by shrinking the averaging window:
- Consider average velocity over a smaller and smaller interval \Delta t around the chosen instant t.
- In the limit \Delta t \to 0, average velocity approaches the true instantaneous value.
Definition (calculus form):
v=\lim_{\Delta t\to0}\frac{\Delta x}{\Delta t}=\frac{dx}{dt}.
Still a vector in 1D (sign = direction).

Over [t0,t]: \bar{a}=\frac{\Delta v}{\Delta t}=\frac{v-v0}{t-t_0}.
- v0: instantaneous velocity at t0.
- v : instantaneous velocity at t.

Repeat the limiting idea used for velocity, but now with velocity changes:
- Shrink \Delta t so the average acceleration increasingly approximates the value right at time t.
Definition:
a=\lim_{\Delta t\to0}\frac{\Delta v}{\Delta t}=\frac{dv}{dt}=\frac{d^2x}{dt^2}.
Units: m/s$^2$.

Diagram: particle starting left of x_0, weaving back and forth, finally to the right of x.
Car analogy: Even during acceleration, we intuitively talk about its instantaneous speedometer reading; mathematics backs this up with the limit definition.

Displacement, velocity, and acceleration in 1D are signed quantities; the sign contains directional information, eliminating the need for arrows in written form once the axis is fixed.
Average quantities smooth over complicated motion; instantaneous quantities capture the exact state at a single instant via calculus limits.
These definitions form the foundational language for later dynamics (where we introduce forces) and for solving motion problems analytically.