Straight-Line Motion: Connecting Position, Velocity, and Acceleration

Position s(t) tells you where a particle is on a line at time t. Velocity v(t) = s'(t) tells you how fast and in which direction it's moving right now. Acceleration a(t) = v'(t) = s''(t) tells you how velocity itself is changing. Each one is the slope of the graph above it: velocity is the slope of the position graph, acceleration is the slope of the velocity graph. A particle can be at position zero and still be moving fast, because position zero just means it's crossing a reference point. It says nothing about speed. In fact a particle often passes through s = 0 at its maximum speed, the way a pendulum swings fastest through the bottom of its arc.

Reading the three graphs stacked together is the real skill. Where velocity is zero, the position graph has a horizontal tangent, a turning point. Where velocity is positive, position is increasing; where negative, position is decreasing. Where acceleration and velocity share the same sign, the particle speeds up; where they have opposite signs, it slows down. Acceleration zero with velocity nonzero means constant speed at that instant, often a point where velocity is at a local max or min.

This topic is the direct application of derivatives and second derivatives to a physical quantity with units, setting up the rest of the unit: related rates reuse the same chain of derivatives, and later topics ask you to interpret these rates in applied settings rather than just compute them.