Position, velocity, acceleration

Before a force ever appears on the page, you can already say a great deal about a machine. Where is the piston? How fast is the vehicle? Is the elevator still speeding up? Kinematics is that language — the geometry of motion, stripped of its causes.

Position x(t) is a function of time. Velocity is the first derivative, acceleration the second. In one dimension that is all. In two and three, the same derivatives apply to vectors, which means direction can change even when speed does not — the reason a car on a roundabout is accelerating while the speedometer sits still.

A position–time graph with a steep slope is a fast object. A velocity–time graph’s slope is acceleration; its area is displacement. This is not a mnemonic. It is the fundamental theorem of calculus applied to a shaft encoder.

Sign is a choice of axis, not a moral quality. Negative velocity means the object is moving toward decreasing x. An engineer who treats “negative” as “slowing down” will mis-time a catcher, a brake, or a landing burn.