When we observe a phenomenon in nature, we can do this from different places and ways. For example, when watching the movement of a car, we can observe it by standing on a sidewalk or we can observe inside another moving car. Be that as it may, it is important that the analyses of the movement be consistent, regardless of whose referential we adopt.

The correct way to observe a phenomenon is to adopt a referential and a coordinate system.  For example, suppose that the same phenomenon has been observed from two references, identified by  \(M\) and \(N\), where the system \(N\) moves in a straight line relative to the system \(M\), with relative speed \(v_{ M, N}\). With \( v_{A,N}\)being the speed of a car \(A\), known in the reference \(N\), we may want to know what is the corresponding velocity value \(v_ {A,M}\) of the car \(A\) on referential \(M\). The concept of relative movement allows us to answer this type of question.

Relative Motion

The equation for transforming from one reference to another is known as Galileo's transformation and has the following form: $$\vec{v}_{A,M} = \vec{v}_{A,N} + \vec{v}_{N,M},$$ where \(\vec{v}_{A,M}\) is the speed \(A\) in the reference \(M\), \(\vec{v}_{A,N}\) is the speed \(A\) in the reference \(N\) and \(\vec{v}_{N,M}\) is the speed of reference \(N\) in relation to \(M\).

Principle of movement independence (Galileo's principle)

When a body is under the simultaneous action of several movements, each of them is processed independently of the others. The figure below illustrates a typical case of relative motion.

Airplanes travel over the air, so the speed of the wind influences the movement that can be observed from the Earth. In the figure, all airplanes travel with the same speed in the air, but in different directions. The blue vectors represent the wind velocity measured relative to the Earth, the green vectors represent the velocity of each airplane measured relative to the air, and the red vectors represent the resulting velocity of the airplane relative to the Earth. That is, from the Earth we observe the airplane moving according to the red vector. However, observing the movement from a balloon that moves only carried by the wind, we will observe that the movement of the airplane is according to the green vector.