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:

my_site_kinematics_1

(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 ).

TASKS

Task 1. A cyclist is riding the first third of the way with the speed of 8 m/s, and the last two-thirds of the way – with the speed of 4 m/s. Calculate the average speed of a cyclist.

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(The information has been taken from book «Mechanics” (author V.D. Shvets. G.P. Polovina), which was published in 2007 by “University Ukraine” publishing house. More information can be found at https://ipood-kiev.academia.edu/ValentynaShvets/Books ).

4 Responses to “The equation of rectilinear uniform motion”

  1. Francis says:

    That’s a smart answer to a difficult question.

  2. Blayke says:

    Your posts really straightened me out. Thanks!

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