1.15 Rectilinear motion

A motion along straight line is called rectilinear motion. In general, it need not be one – dimensional; it can take place in a two dimensional plane or in three dimensional space. But, it is always possible that rectilinear motion be treated as one dimensional motion, by suitably orienting axes of the coordinate system. This fact is illustrated here for motion along an inclined plane. The figure below depicts a rectilinear motion of the block as it slides down the incline. In this particular case, the description of motion in the coordinate system, as shown, involves two coordinates (x and y).

The reorientation of the coordinate system renders two dimensional description (requiring x and y values) of the motion to one dimensional (requiring only x value). A proper selection, most of the time, results in simplification of measurement associated with motion. In the case of the motion of the block, we may choose the orientation such that the progress of motion is along the positive x – direction as shown in the figure. A proper orientation of the coordinates results in positive values of quantities like displacement and velocity. It must be emphasized here that we have complete freedom in choosing the orientation of the coordinate system. The description of the rectilinear motion in independent of the orientation of axes.

In rectilinear motion, we are confined to the measurement of movement of body in only one direction. This simplifies expressions of quantities used to describe motion. In the following section, we discuss (also recollect from earlier discussion) the simplification resulting from motion in one dimension (say in x –direction).

Position vector in rectilinear motion

Position still requires three coordinates for specification. But, only one of them changes during the motion; remaining two coordinates remain constant. In practice, we choose one dimensional reference line to coincide with the path of the motion. It follows then that position of the particle under motion is equal to the value of x – coordinate (others being zero).

Corresponding position vector also remains a three dimensional quantity. However, if the path of motion coincides with the reference direction and origin of the reference coincides with origin, then position vector is simply equal to component vector in x – direction i.e

The position vectors corresponding to points A, B and C as shown in the figure are 2 i , 4 i and 6 i . units respectively.

As displacement is equal to change in position vector, the displacement for the indicated positions are given as :

$$\begin{array}{l}\mathbf{AB}=(x_2-x_1)\mathbf{i}=\Delta x\mathbf{i}=(4-2)\mathbf{i}=2\mathbf{i}\\ \mathbf{BC}=(x_2-x_1)\mathbf{i}=\Delta x\mathbf{i}=(6-4)\mathbf{i}=2\mathbf{i}\\ \mathbf{AC}=(x_2-x_1)\mathbf{i}=\Delta x\mathbf{i}=(6-2)\mathbf{i}=4\mathbf{i}\end{array}$$

Vector interpretation and equivalent system of scalars

Rectilinear motion involves motion along straight line and thus is described usually in one dimension. Further, rectilinear motion involves only one way of changing direction i.e. the particle under motion can only reverse motion. The particle can move either in positive x-direction or in negative x-direction. There is no other possible direction valid in rectilinear motion.