Displacement

Consider a body moving in 1 dimension: e.g., a train traveling down a straight railroad track, or a truck driving down an interstate in Kansas. Suppose that we have a team of observers who continually report the location of this body to us as time progresses. To be more exact, our observers report the distance $x$ of the body from some arbitrarily chosen reference point located on the track on which it is constrained to move. This point is known as the origin of our coordinate system. A positive $x$ value implies that the body is located $x$ meters to the right of the origin, whereas a negative $x$ value implies that the body is located $\vert x\vert$ meters to the left of the origin. Here, $x$ is termed the displacement of the body from the origin. See Fig. 2. Of course, if the body is extended then our observers will have to report the displacement $x$ of some conveniently chosen reference point on the body (e.g., its centre of mass) from the origin.

Our information regarding the body's motion consists of a set of data points, each specifying the displacement $x$ of the body at some time $t$. It is usually illuminating to graph these points. Figure 3 shows an example of such a graph. As is often the case, it is possible to fit the data points appearing in this graph using a relatively simple analytic curve. Indeed, the curve associated with Fig. 3 is

$$x = 1 + t + \frac{t^2}{2} - \frac{t^4}{4}.$$ (11)

Figure 2: Motion in 1 dimension

Figure 3: Graph of displacement versus time