Velocity from Position

Calculus 3 · Axiom Academy

A GPS log gives you only where something is . Differentiate that position once and you get how it moves, twice and you get how that motion changes. An object traces the plane curve . From that position alone, calculus recovers its velocity , its speed , and its acceleration — three moves on one path. Drag time forward and watch the object walk the path. At each instant its position is the vector — the pair of coordinates that says exactly where it is. Take the derivative of position component-by-component and you get velocity, . It always points along the path — tangent to it — and its length is the speed, . Differentiate again — acceleration Differentiate a second time for acceleration, . Unlike velocity it need not lie along the path — the part of it that points toward the inside of the curve is exactly what bends the motion. One position function, differentiated: , with speed . It's how a phone turns raw GPS fixes into your speed and heading, how flight computers read motion from position, and how any tracker recovers velocity and acceleration from nothing but where something was .

This is the written version of the interactive lesson above. See the full Calculus 3 course.