Position, Path Length And Displacement

Earlier you learnt that motion is change in position of an object with time. In order to specify position, we need to use a reference point and a set of axes. It is convenient to choose a rectangular coordinate system consisting of three mutually perpenducular axes, labelled X-, Y-, and Z- axes. The point of intersection of these three axes is called origin (O) and serves as the reference point. The coordinates (x, y. z) of an object describe the position of the object with respect to this coordinate system. To measure time, we position a clock in this system. This coordinate system along with a clock constitutes a frame of reference.

If one or more coordinates of an object change with time, we say that the object is in motion. Otherwise, the object is said to be at rest with respect to this frame of reference. 

The choice of a set of axes in a frame of reference depends upon the situation. For example, for describing motion in one dimension, we need only one axis. To describe motion in two/three dimensions, we need a set of two/ three axes. 

Description of an event depends on the frame of reference chosen for the description. For example, when you say that a car is moving on a road, you are describing the car with respect to a frame of reference attached to you or to the ground. But with respect to a frame of reference attached with a person sitting in the car, the car is at rest. 

To describe motion along a straight line, we can choose an axis, say X-axis, so that it coincides with the path of the object. We then measure the position of the object with reference to a conveniently chosen origin, say O, as shown in Fig. 3.1. Positions to the right of O are taken as positive and to the left of O, as negative. Following this convention, the position coordinates of point P and Q in Fig. 3.1 are +360 m and +240 m. Similarly, the position coordinate of point R is –120 m.

Path length 

Consider the motion of a car along a straight line. We choose the x-axis such that it coincides with the path of the car’s motion and origin of the axis as the point from where the car started moving, i.e. the car was at x = 0 at t = 0 (Fig. 1). Let P, Q and R represent the positions of the car at different instants of time. Consider two cases of motion. In the first case, the car moves from O to P. Then the distance moved by the car is OP = +360 m. 

Fig 1: x-axis, origin and positions of a car at different times.

This distance is called the path length traversed by the car. In the second case, the car moves from O to P and then moves back from P to Q. During this course of motion, the path length traversed is OP + PQ = + 360 m + (+120 m) = + 480 m. Path length is a scalar quantity — a quantity that has a magnitude only and no direction (see Chapter 4).

Displacement 

It is useful to define another quantity displacement as the change in position. Let $x_1$ and $x_2$ be the positions of an object at time $t_1$ and $t_2$. Then its displacement, denoted by ∆x, in time ∆t = ($t_2$ - $t_1$), is given by the difference between the final and initial positions: 

∆x = $x_2-x_1$ 

(We use the Greek letter delta (∆) to denote a change in a quantity.) 

If $x_2$ > $x_1$ , ∆x is positive; and if $x_2$ < $x_1$ , ∆x is negative. 

Displacement has both magnitude and direction. Such quantities are represented by vectors. You will read about vectors in the next chapter. Presently, we are dealing with motion along a straight line (also called rectilinear motion) only. In one-dimensional motion, there are only two directions (backward and forward, upward and downward) in which an object can move, and these two directions can easily be specified by + and – signs. For example, displacement of the car in moving from O to P is: 

∆x = $x_2-x_1$  = (+360 m) – 0 m = +360 m 

The displacement has a magnitude of 360 m and is directed in the positive x direction as indicated by the + sign. Similarly, the displacement of the car from P to Q is 

240 m – 360 m = – 120 m. 

The negative sign indicates the direction of displacement. Thus, it is not necessary to use vector notation for discussing motion of objects in one-dimension. 

The magnitude of displacement may or may not be equal to the path length traversed by an object. For example, for motion of the car from O to P, the path length is +360 m and the displacement is +360 m. In this case, the magnitude of displacement (360 m) is equal to the path length (360 m). But consider the motion of the car from O to P and back to Q. In this case, 

the path length = (+360 m) + (+120 m) = + 480 m. 

However, 

the displacement = (+240 m) – (0 m) = + 240 m. 

Thus, the magnitude of displacement (240 m) is not equal to the path length (480 m). 

The magnitude of the displacement for a course of motion may be zero but the corresponding path length is not zero. For example, if the car starts from O, goes to P and then returns to O, the final position coincides with the initial position and the displacement is zero. However, the path length of this journey is 

OP + PO = 360 m + 360 m = 720 m. 

Motion of an object can be represented by a position-time graph as you have already learnt about it. Such a graph is a powerful tool to represent and analyse different aspects of motion of an object. For motion along a straight line, say X-axis, only x-coordinate varies with time and we have an x-t graph. Let us first consider the simple case in which an object is stationary, e.g. a car standing still at x = 40 m. The position-time graph is a straight line parallel to the time axis, as shown in Fig. 2(a). 

 Fig. 2: Position-time graph of (a) stationary object, and (b) an object in uniform motion.

If an object moving along the straight line covers equal distances in equal intervals of time, it is said to be in uniform motion along a straight line. Fig. 2(b) shows the position-time graph of such a motion.

Fig 3: Position-time graph of a car

Now, let us consider the motion of a car that starts from rest at time t = 0 s from the origin O and picks up speed till t = 10 s and thereafter moves with uniform speed till t = 18 s. Then the brakes are applied and the car stops at t = 20 s and x = 296 m. The position-time graph for this case is shown in Fig. 3. We shall refer to this graph in our discussion in the following sections.