The signs marking kilometer 135 and 13 indicate a certain position on a road
Let’s suppose that we have a material point, remembering that a material point is a body whose dimensions are not taken into account, which moves on a certain rectilinear trajectory s as the reference system (figure below). On this trajectory we will indicate a direction and we will set the origin 0 (zero) for the measurement of the segments. The position of a mobile, at each instant, is the algebraic measure of the oriented segment that connects the origin 0 to the point where the mobile is located.
To note the positions we will adopt s, t to represent the time interval and the subscript “0” symbol ( t 0 , S 0 , V 0 etc.) designating “initial”. Therefore, t 0 represents the initial count of time, V 0 is the initial velocity and S 0 represents the initial position.
The value assigned to the position of the movable does not necessarily coincide with that of the space traversed by the movable up to the instant considered: it simply indicates the coordinates of the point in relation to the origin of the adopted reference system.
We can say that on highways, the main form of location is the signs corresponding to the kilometric landmarks. For example, a sign with the indication “km 135” informs that that point of the highway is located 135 km from a point taken as the origin. This point is commonly called ground zero .
Schematically, we make use of an oriented line, representing the trajectory, in which we indicate the positions of interest, as shown in the figure below. Note that, unlike what happens on highways, it may be in our interest to include negative positions. Therefore, we will adopt the arrowhead to indicate the direction in which the position values increase.
In the time interval Δt = t – t 0 , the material point passes from the initial position S 0 to the S position. This variation of positions of the material point in this time interval is called displacement of the mobile. The displacement measure (ΔS) in a given time interval is obtained by the algebraic difference between the final (S) and initial (S 0 ) positions occupied by the mobile in that time interval.
Let’s see how all this is used: let’s suppose that a small straight stretch of an athlete’s run is represented as shown in the figure below. We associate an oriented line s to the trajectory, marking, on it, the extremes S 0 and S, occupied by the material point that represents the athlete, at the instants t 0 and t , respectively.
We can calculate the athlete’s displacement from the measurement of his position in relation to the starting point, that is, in relation to the origin. For each instant, we have a value for the position occupied by the mobile. Between any two instants, the displacement of the mobile is given by: