1.3 Position

The above exercises point to an interesting feature of the frame of reference: that the specification of position of the object (values of coordinates) depends on the choice of origin of the given frame of reference. We have already seen that description of motion depends on the state of observer i.e. the attached system of reference. This additional dependence on the choice of origin of the reference would have further complicated the issue, but for the linear distance between any two points in a given system of reference, is found to be independent of the choice of the origin. For example, the linear distance between the markings 6 and 12 is ‘2r’, irrespective of the choice of the origin.

Plotting motion

Position of a point in the volumetric space is a three dimensional description. A plot showing positions of an object during a motion is an actual description of the motion in so far as the curve shows the path of the motion and its length gives the distance covered. A typical three dimensional motion is depicted as in the figure below :

In the figure, the point like object is deliberately shown not as a point, but with finite dimensions. This has been done in order to emphasize that an object of finite dimensions can be treated as point when the motion is purely translational.

The three dimensional description of positions of an object during motion is reduced to be two or one dimensional description for the planar and linear motions respectively. In two or one dimensional motion, the remaining coordinates are constant. In all cases, however, the plot of the positions is meaningful in following two respects :

• The length of the curve (i.e. plot) is equal to the distance.
• A tangent in forward direction at a point on the curve gives the direction of motion at that point

Description of motion

Position is the basic element used to describe motion. Scalar properties of motion like distance and speed are expressed in terms of position as a function of time. As the time passes, the positions of the motion follow a path, known as the trajectory of the motion. It must be emphasized here that the path of motion (trajectory) is unique to a frame of reference and so is the description of the motion.

To illustrate the point, let us consider that a person is traveling on a train, which is moving with the velocity v along a straight track. At a particular moment, the person releases a small pebble. The pebble drops to the ground along the vertical direction as seen by the person.

The same incident, however, is seen by an observer on the ground as if the pebble followed a parabolic path (See Figure blow). It emerges then that the path or the trajectory of the motion is also a relative attribute, like other attributes of the motion (speed and velocity). The coordinate system of the passenger in the train is moving with the velocity of train ( v ) with respect to the earth and the path of the pebble is a straight line. For the person on the ground, however, the coordinate system is stationary with respect to earth. In this frame, the pebble has a horizontal velocity, which results in a parabolic trajectory.