## The equation of rectilinear uniform motion

January 5th, 2012

Given that the trajectory of a material point (MP) motion is a straight line, the motion is called “a rectilinear motion”. The rectilinear motion can be described by the one-dimensional coordinate system. Let us suppose that at time $0$ s MP was in the beginning of motion, at time $t_0$ s MP was at the point with coordinate $x_0$, and at time $t$ MP was at the point with coordinate $x$.

Then the position of the MP at time $t_0$ is characterized by radius – vector $\vec r_0$ , MP position at time $t$ is characterized by the radius – vector $\vec r$ (Fig. 1).

Fig.1. The displacement and coordinates of MP during the rectilinear motion.

We see from Fig. 1 that in this case the vectors $\Delta \vec r$ and $\Delta \vec s$ coincide, and the path traversed by the MP during the $(t-t_0)$ is equal to:

$\mid \Delta \vec r\mid = x-x_0$                (1)

Let us consider the linear motion of the MP, where the path traversed by MP per unit time is constant:

$\frac {\Delta x}{\Delta t} = \frac {\mid \Delta \vec s \mid}{\Delta t} = \frac {\mid \Delta \vec r \mid}{\Delta t} = const$                (2)

Let $t_0 = 0$. Then equation (2) can be rewritten as:

$\frac {\Delta s}{t} = const$                  (3)

A vector which has a module determined by the equation (3), with the direction that coincides with the direction of movement of the MP is called the speed of uniform rectilinear motion

$\vec v =$  $\frac {\vec s-\vec s_0}{t}$                 (4)

From equation (4) we obtain the kinematic equation of uniform motion in a vector form

$\vec s(t) = \vec s_0 + \vec v \cdot t$                 (5)

The projection of the vector equation (5) on the coordinate axis is one of four scalar equations listed below together with their plots:

1. $x(t) = x_0 + v_x \cdot t$

Fig.2.  The plot of the MP path motion with a positive velocity from the initial positive coordinate. 2. $x(t) = -x_0+v_x \cdot t$

Fig.3. The plot of the MP path motion with a positive velocity from the initial negative coordinate.

3.  $x(t) = x_0-v_x \cdot t$

Fig. 4. The plot of the MP path motion with a negative velocity from the initial positive coordinate.

4. $x(t) = -x_0-v_x \cdot t$

Fig. 5. The plot of the MP path motion with a negative velocity from the initial negative coordinate. In case of spatial coordinate system the vector equation (5) can be represented by a system of scalar equations projected on each of axes

$OX$, $OY$, $OZ$:

(The information has been taken from e-book «Fundamentals of classical physics” (author V.D. Shvets), which was published in 2007 by “1C-Multimedia Ukraine” publishing company. Copyright is reserved by certificate Number 55955 from 06.08.2014. More information can be found at https://ipood-kiev.academia.edu/ValentynaShvets/Books ).