Basic concepts of kinematics

To apply the theory of kinematics, it is necessary to understand the basic definitions. The main ones will be described below terms used in kinematics.

Material point

A material point (MT) is an idealized object whose volume is considered zero under the conditions the problem in question, for example:

1. The car goes from Moscow to St. Petersburg - we designate the car a material point, since its dimensions do not matter for solving kinematics problems
2. The moon revolves around the earth — let's designate the moon and the earth as material points, since their dimensions do not matter for solving kinematics problems

Under a material point, you can perceive any object within the framework of solving a kinematics problem.

Movement

If the coordinate system in which the position of the MT is considered is really stationary, then such a motion of the MT is called absolute. When the coordinate system is mobile, the motion of the MT is called relative.

In reality, there are no fixed points in the world, therefore, any movement is relative, therefore, when describing movement, it is always necessary to specify the coordinate system in which the calculation is carried out.

Why is it so important to specify a reference point? Imagine that a cat is coming to meet you: if you choose a cat as a reference point — you go towards him, while the cat is standing still.

You can choose absolutely any reference point, but the selected reference point depends on how complex there will be equations of motion and a solution to the problem.

Radius vector

Since we have an established coordinate system, the position of a material point in space can be uniquely described by the vector r. A radius vector is a vector plotted from the origin. In order to describe the movement of a material point, it is necessary to establish a relationship between the position the material point and the time for which the situation has changed. If we are talking about speed, then we I am interested in the dependence of the position on time, which means it will be a function. Mathematically, the position of the point in any moment of time can be described by the radius vector function:

r = r (t)

If we calculate using a general equation, then we will have to work with complex equations involving a large the number of variables in high degrees, in order to simplify the calculations, we can decompose the radius vector into components, so we get parametric equations that depend on time. The essence of the equations is the projection of the radius vector on the coordinate axes, depending on the coordinate system we will have different number of equations:

For R¹:
r_i = r_i(t)

For R²:
r_i = r_i(t)
r_j = r_j(t)

For R³:
r_i = r_i(t)
r_j = r_j(t)
r_k = r_k(t)

The number of equations does not necessarily denote the familiar one-, two-, three-dimensional Cartesian systems coordinates, it can be cylindrical, spherical and others coordinate systems.

Trajectory

A trajectory is a curve representing a set of points along which a material point moves. Any geometric the shape can be a trajectory.

Analytical expression

Analytically, the geometric location of the points (G.M.T.) of the trajectory in three-dimensional space can be given by by the intersection of two planes, the equations of the planes, respectively, can be expressed in parametric dependencies:

f₁ (i, j, k) = 0 plane 1
f₂ (i, j, k) = 0 plane 2
f₁ = f₂ trajectory equation

The trajectory is indicated by the letter s. Considering the movements, we set the position of the material point on the trajectory through time dependence:

s = s (t)

This trajectory function allows us to find the length of the traversed trajectory at any given time in a given direction. The curve length formula is selected based on the selected coordinate system and the problem being solved. In general in the case where the problem is solved through the integral:

l = _t0^t1∫ s(t) dt

Moving

Consider the movement of a material point from position r₀ to position r₁ in time Δt, denote by Δi, Δj, Δk the corresponding displacements along the coordinates i, j k, then the change in the radius vector will be:

Δr = Δi•i + Δj•j + Δk•k

Given that r₁ = r₀ + Δr, the new coordinates will be respectively:

i₁ = i₀ + Δi
j₁ = j₀ + Δj
k₁ = k₀ + Δk

Thus, the displacement vector can be expressed in terms of the initial and final coordinates of motion, or in terms of the initial coordinates and movements along the axes of the reference system.

Speed

Velocity is a vector that shows the direction of movement of the point under study, the magnitude of this vector is equal to the derivative of the radius-vector in time.

∨ = dr/dt

Average velocity vector

The average velocity vector is defined as the ratio of the displacement vector to time

v = Δr / Δt

Instantaneous velocity vector

Using the knowledge of the average speed, the instantaneous speed can be obtained when the time interval tends to zero:

v = lim (Δt → 0)(Δr/Δt) = dr/dt = r'

Unit vector in the direction of the instantaneous velocity vector:

Δ = lim (Δt → 0)(Δr/Δs)

Then the instantaneous velocity vector is expressed as:

v = v · dr = ds/dt = r'

Acceleration

The change of the velocity vector over an infinitesimal time interval is called an instantaneous vector acceleration, or just the acceleration vector:

a = lim (Δt → 0)(Δv/Δt) = dv/dt = v' = d/dt · dr/dt = d²r/dt² = r''

Average acceleration vector

The average acceleration vector is the ratio of the change in the velocity vector to the time during which it occurred change:

a = [v(t + Δt) – v(t)]/Δt = Δv/Δt

unlike the velocity vector, which is always directed tangentially to the trajectory, the acceleration vector can have any direction.