Question
Physics 211 - This is a review question on the material covered in the last three questions, starting with the June 13, 1997 question. (Signs. Terminology.)
These questions dealt with the concepts of position, displacement, and distance travelled. They also dealt with signs and their dependence on the choice of the direction of an axis.
The following two diagrams show two points and an x-axis. The positions of the two points are measured by their x-coordinates. In both diagrams, the origin x = 0 lies between the two points. The origin is not indicated. The only difference between the two diagrams is that in diagram (a) the x-axis points to the right while in diagram (b) it points to the left.
Today's question refers to a motion from point P2 to point P1. Which of the following quantities are positive, which ones are negative: the position coordinates of the two points, the displacement, and the distance travelled? Indicate your answers in the chart below by either a plus or a minus sign. The two rows refer to the choices of axes illustrated in diagrams (a) and (b). The symbol s heading the last column denotes the distance travelled. Two entries in the table have already been made as examples.
Answer
The two diagrams (a) and (b) are drawn below. This time, however, the origin of the x-axis is indicated and specific values for the coordinates x1 and x2 have been assumed, to make things more concrete. Also the direction of the displacement from P1 to P2 has been indicated by a bold arrow.
The following table lists the correct signs.
Explanation. The important thing to realize is that signs depend on the choice of axis. An x-axis can be chosen to point to the right or to the left and the signs of certain quantities (not all quantities) change accordingly.
It should be clear that the signs of the position coordinates x1 and x2 get reversed when the direction of the x-axis is reversed while the origin remains the same.
In case (a), the displacement is opposite to the direction of the x-axis. Therefore, the x-values decrease in the direction of the displacement so that the displacement Dx = x1 - x2 is negative. In the given example, Dx = x1 - x2 = -4 - 10 = -14 m. The distance travelled is, by definition, always positive (or zero). In the given example, it is equal to 14 m.
In case (b), the displacement is in the same direction as the x-axis. Therefore, the x-values increase in the direction of the displacement so that the displacement Dx = x1 - x2 is positive. In the given example, Dx = x1 - x2 = 4 - (-10) = 14 m. The distance travelled is, by definition, always positive (or zero). It is again equal to 14 m.