Points to Ponder

The location of the origin and which direction is called positive along an axis are completely up to you. Before you assign signs to quantities like displacement, velocity, or acceleration, you must first clearly state these choices.

When a particle is speeding up, its acceleration points in the same direction as its velocity. When it is slowing down, its acceleration points opposite to its velocity. This rule holds true no matter how you set up your coordinate axes.

The sign of acceleration alone does not tell you whether speed is increasing or decreasing. That sign depends on which direction you chose as positive. For instance, if upward is positive, gravity's acceleration is $a = -g$. A falling object has this negative acceleration, yet its speed increases. A ball thrown upward has the same negative acceleration, but its speed decreases.

A particle can have zero velocity at an instant without having zero acceleration. For example, a ball thrown straight up reaches its highest point where $v = 0$, but the acceleration due to gravity $g$ still acts on it at that exact moment.

In the kinematic equations for constant acceleration (like $v = u + at$, $s = ut + \frac12 at^2$, $v^2 = u^2 + 2as$), every quantity is algebraic — it carries a positive or negative sign. These equations work for any one-dimensional motion with constant acceleration, provided you substitute all values with their correct signs. …