3.4 Motion in space

We have now seen how to describe curves in the plane and in space, and how to determine their properties, such as arc length and curvature. All of this leads to the main goal of this chapter, which is the description of motion along plane curves and space curves. We now have all the tools we need; in this section, we put these ideas together and look at how to use them.

Motion vectors in the plane and in space

Our starting point is using vector-valued functions to represent the position of an object as a function of time. All of the following material can be applied either to curves in the plane or to space curves. For example, when we look at the orbit of the planets, the curves defining these orbits all lie in a plane because they are elliptical. However, a particle traveling along a helix moves on a curve in three dimensions.

Since $\text{r}\left(t\right)$ can be in either two or three dimensions, these vector-valued functions can have either two or three components. In two dimensions, we define $\text{r}\left(t\right)=x\left(t\right)\text{i}+y\left(t\right)\text{j}$ and in three dimensions $\text{r}\left(t\right)=x\left(t\right)\text{i}+y\left(t\right)\text{j}+z\left(t\right)\text{k}.$ Then the velocity, acceleration, and speed can be written as shown in the following table.