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 s MP was in the beginning of motion, at time s MP was at the point with coordinate , and at time MP was at the point with coordinate .

Then the position of the MP at time is characterized by radius – vector , MP position at time is characterized by the radius – vector (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 and coincide, and the path traversed by the MP during the is equal to:

(1)

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

(2)

Let . Then equation (2) can be rewritten as:

(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*

(4)

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

(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.

Fig.2. The plot of the MP path motion with a positive velocity from the initial positive coordinate. 2.

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

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

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

, , :

(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 ).