Solving Motion Problems Using Parametric and Vector-Valued Functions

A particle moving in the plane has a position vector (x(t), y(t)). Its velocity vector is (x'(t), y'(t)) — two separate rates, one for horizontal motion and one for vertical. Speed collapses these two numbers into one: it's the length of the velocity vector, √(x'(t)² + y'(t)²). No matter how x' and y' individually behave, speed is always a single nonnegative number because it measures only how fast the particle covers ground, not which way it's heading.

To get speed, direction, and distance out of one velocity vector, start by differentiating x(t) and y(t) to get the components. Speed is the magnitude of that vector, as above. Direction comes from the vector itself — the angle it makes, or simply its ratio of components, tells you which way the particle is instantaneously moving. Total distance traveled over an interval [a, b] is the integral of speed with respect to t: integral from a to b of √(x'(t)² + y'(t)²) dt. This works because distance is accumulated speed, the same way position is accumulated velocity in one dimension — except now you can't just subtract endpoint values, since the particle might double back or curve, so you must integrate the speed directly.

This ties directly to acceleration, which is the derivative of the velocity vector, (x''(t), y''(t)). Acceleration can point in a totally different direction from velocity, and that's exactly what lets speed stay constant while the path curves — the acceleration only redirects the vector, it doesn't have to stretch or shrink it. Vector-valued motion problems in this unit all come back to this same pattern: differentiate for velocity and acceleration, take magnitudes for speed, integrate magnitudes for distance.