Where does the trajectory lead?

The motion of any body can be described if there is a way to determine its position in space at each moment. To do this, we need to have a reference body, (to know what kind of object we will consider its movement), and also to establish for ourselves the way in which we will describe this movement.

Since the bodies have dimensions (that is, some extent in space), we need to decide in which cases we can neglect them and not trace the motion of each point. This is possible in two cases: if the body moves in such a way that all the lines drawn in it move in parallel to themselves (this motion is called translational motion), and also if the body dimensions are allowed to be neglected under the conditions of the problem (consider the body as a material point). This happens if the path traveled by the body is many times greater than its physical dimensions.

In mechanics, by default, the body is taken for a material point, unless otherwise specified.

The line of motion of a point in space is a trajectory of motion. What it is? The concept of "trajectory", the definition of which is given by classical mechanics, implies the totality of all positions sequentially occupied by a material point in space.

To determine the position that a material point occupies in space at any particular time, use the concepts of radius vector or coordinate system. The values of the coordinates x, y, z characterize the linear arrangement of the point relative to the corresponding axes. The formula for changing these coordinates (or the position of the radius vector) is the formula by which its trajectory is determined.

Since the movement occurs not only in space, but also in time, the third component of the frame of reference is a device for measuring time (a clock or a stopwatch). Together with the coordinate system and the starting point (the reference body) they form the necessary "set" for describing the motion of our material point.

Let the trajectory of the motion be an arc with the origin at the point M1, the coordinates of which are X1, Y1 and Z1, and the ending at the point M2 (coordinates X2, Y2, Z2). The distance the material point travels along its trajectory (the length of the arc | M1M2 |) will be called the length of its path. This is a scalar quantity.

If we draw a directed segment (vector) r from the point M1 to the point M2, then it will be called the displacement of the material point. This concept is not identical to the concept of the path. The path and the displacement of the point coincide only in the case of a straight line motion.

The kinematic law of motion (or the method of determining its coordinates at any moment) is a function of time and can take the analytical form of the coordinate function or the radius vector of the variable t, which denotes the time of motion. It can be expressed by a formula, in the form of a table or as a graph.

With uniform motion, there is such a thing as the velocity of a material point. The speed is the quotient of dividing the displacement by the time of travel. If the trajectory is straight, but the body moves unevenly, that is, at different speeds in different parts of the path, then we can talk about the average speed.

In mechanics, motion of various kinds is considered - uniform rectilinear, uniformly accelerated rectilinear and uniform along the circumference.

The characteristics of the mechanical motion are relative, the motion can be considered immediately in two or more coordinate systems, some of them are fixed, others are mobile. For example, a car moves along the road relative to a pedestrian walking along it (a mobile point), which itself moves relative to the growing tree near the road (fixed reference point). In this case, the speed of the body (car) will be composed of two speeds - its speed relative to the first - mobile - the system (pedestrian) and the speed of the pedestrian relative to the fixed (tree).