Describing Speed and Distance Using Vectors

When discussing projectile motion, we have considered the formula for the position $\mathbf{x}(t)$ and the velocity $\mathbf{v}(t)$ of the object on its trajectory, which have both been vectors. However, we have not yet discussed the speed or the distance of the object in precise mathematical terms.

For both of these concepts, there is a numerical representation but no direction is attached to them. When we talk about speed we communicate how fast an object is moving, but not which direction it is moving in. When we discuss distance, we are talking about how far away something is, but not the direction in which it is located. In both of these, we are in fact discussing the magnitude of a corresponding vector quantity:

Distance $D(t)$ is the magnitude of the position vector $\mathbf{x}(t)$ (relative to some other reference point $\mathbf{x_{ref}}$ )
Speed $S(t)$ is the magnitude of the velocity vector $\mathbf{v}(t)$

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