9.6 Solving Motion Problems Using Parametric and Vector-Valued Functions

Parametric and vector-valued functions are especially useful for motion because they describe position in terms of time. If ${\mathbf{r}(t)=\langle x(t),y(t)\rangle}$ then ${\mathbf{r}'(t)}$ is velocity and ${\mathbf{r}''(t)}$ is acceleration. In parametric form, velocity components are ${\frac{dx}{dt}}$ and ${\frac{dy}{dt}}$, and speed is ${\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}}$ This lets you analyze direction, speed, and changing motion all from the same representation.

You should distinguish between velocity and speed. Velocity is a vector and can have negative components, while speed is always nonnegative because it is the magnitude of the velocity vector. Motion problems may ask for position, velocity, acceleration, slope of the path, or speed at a given time, so you need to be comfortable moving among these related quantities.