Total momentum of two interacted isolated material points is always constant

Let’s consider a system of n particles, each with its own mass, velocity, and linear momentum. The particles may interact with each other, and external forces may act on them as well. The system as a whole has a total linear momentum , which is defined to be the vector sum of the individual particles' linear momenta. Thus

Let’s suppose that the sum of the external forces acting on a system of particles is zero (the system is isolated) and that no particles leave or enter the system (the system is closed). Putting then we get , or

This important result, called the LAW OF CONSERVATION OF LINEAR MOMENTUM,can also be written as

where the subscripts refer to the values of at initial time i and later time f. Lawof conservation of linear momentum tells us if no external force acts on a system of particles, the total linear momentum of the system remains constant.

The law of conservation of linear momentum is a more general law than Newtonian mechanics itself. It holds in the subatomic realm, where Newton's laws fail. It holds for the highest particle speeds, where Einstein's relativity theory prevails.

Equations of the law of conservation of linear momentum are vector equations and are equivalent to three equations corresponding to the conservation of linear momentum in three mutually perpendicular directions. Depending on the forces acting on a system, linear momentum might be conserved in one or two directions but not in all directions:

If a component of the net external force acting on a closed system is zero along an axis,
the component of the linear momentum of the system along that axis cannot be changed

3.1.1. The centre of mass

The total momentum of the system, as well as momentum of each particle, depends on the chosen reference frame. Theoretically it is possible to find such reference frame, in relation to which the total momentum of the system will equal zero.

Let total momentum of the system in relation to the system K', that moves in relation to the system K with velocity, equals zero:

.

Momentum in relation to the system K is not equal to zero:

.

As velocity of particle in the system K:

then

But total momentum in the system K' equals zero. Then velocity of the system K' in relation to the system K is:

Velocity of reference frame K' in relation to the frame of reference K represents velocity of centre of mass of a system of particles .

The centre of mass of a particles system is the point where its total momentum equals zero.

Taking into account that:

, .

We can easily find the coordinate of the centre of mass:

Fig. 3.1

Therefore the position-vector of the centre of mass in relation to the reference frame K is

Example. Two particles with mass m₁ and m₂, are placed on the axis x in distances x₁ and x₂ from the origin (Fig. 3.1). We’ll find the coordinate of the centre of mass:

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