Kinematics: displacement, velocity and acceleration

The three quantities of motion

• Kinematics is the study of motion using calculus. Three quantities describe a moving object, all depending on time $t$:
  • Displacement ($s$) — the object's position relative to its start. It can be negative (behind the start).
  • Velocity ($v$) — how fast the displacement changes, with direction (negative velocity = moving backwards). Speed is velocity without the direction.
  • Acceleration ($a$) — how fast the velocity changes.

The calculus links

• Velocity is the rate of change of displacement: $v = \dfrac{ds}{dt}$.
• Acceleration is the rate of change of velocity: $a = \dfrac{dv}{dt}$.
• So differentiate to go down the chain: displacement $\to$ velocity $\to$ acceleration (differentiate once for velocity, twice for acceleration).

Going the other way

• Anti-differentiate to go up the chain: acceleration $\to$ velocity $\to$ displacement (remembering $+c$ each time).
• Use a given condition (the starting position or starting velocity) to find each constant of integration.

Key moments

• At rest / stationary: velocity $v = 0$.
• Maximum or minimum velocity: acceleration $a = 0$.
• Back at the starting point: displacement $s = 0$.