# 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 perpendicular 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 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.

**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 carâ€™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. 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 Q. Then the distance moved by the car is OP = +240 m. **This distance is called the path length **traversed by the car**. ** Again the car moves from Q to O and then moves back from P to Q. During this course of motion, the path length traversed is OQ + QO = + 240 m + (+240 m) = + 480 m. Path length is a **Scalar quantity **â€” a quantity that has a magnitude only and no direction.

**Displacement**

It is useful to define another quantity called displacement. It is nothing but the change in position within the given time period. Let x_{1} and x_{2} be the positions of an object at time t_{1} and t_{2}. Then its displacement, denoted by Î” x, in time Î” t = (t_{2} â€“ t_{1} ), is given by the difference between the final and initial positions.

Î”x = x_{2} â€“ x_{1}

_{ }

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

If x_{2} > x_{1} , Î”x is positive; and if x_{2} < x_{1} , Î”x is negative.

Displacement has both magnitude and direction. Such quantities are represented by **Vectors**. Presently, we are dealing with motion along a straight line (also called **rectilinear motion**) only.

**The magnitude of displacement may or may not be equal to the path length traversed by an object.**

**Displacement - Time Graph**

When the displacement of a body is plotted on Y- axis and time on the X-axis then, the graph so obtained is called as displacement - time graph.

Following visuals shows the characteristics of displacement - time graph.