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

Position s(t) tells you where the particle is. Its derivative, velocity v(t), tells you how fast and which way it's moving: positive means moving in the positive direction, negative means moving in the negative direction. The derivative of velocity, acceleration a(t), tells you how velocity is changing. Speed is just |v(t)|, so speed increases when velocity is moving away from zero and decreases when velocity is moving toward zero. That happens exactly when v(t) and a(t) share a sign (speeding up) or have opposite signs (slowing down) — same logic as a function increasing when its derivative is positive.

Stack the three graphs on one time axis and they read like a story: wherever the position graph has a horizontal tangent, velocity is zero, and that's a candidate for a direction change. The particle actually changes direction only if velocity crosses zero and switches sign there, not just touches it. Wherever velocity is zero, acceleration tells you concavity of position — a local max or min on the position graph lines up with a sign change in v. Where position is rising, v > 0; where position is falling, v < 0; where position is flat, v = 0.

This topic is the bridge between pure derivative rules and the motion problems that follow, like finding total distance by integrating speed, or relating acceleration to force. Direction changes matter because distance traveled adds up the absolute value of displacement on each interval where velocity keeps one sign — you cannot just plug endpoints into position and subtract, because that only gives net displacement.