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 at 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 of time is constant:

(2)

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

(3)

A vector that 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 the case of a 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 the 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 a speed of 8 m/s, and the last two-thirds of the way – with a speed of 4 m/s. Calculate the average speed of a cyclist.

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