A Classical Approach of Newtonian Mechanics

6 CONSERVATION OF MOMENTUM 6.2 Two-component systems

This is particularly the case if the internal forces, f 12 and f 21 , are complicated in
nature.

Suppose that there are no external forces acting on the system (i.e., F 1 = F 2 =

0 ), or, equivalently, suppose that the sum of all the external forces is zero (i.e.,

F = F 1 + F 2 = 0). In this case, according to Eq. (6.6), the motion of the centre
of mass is governed by Newton’s first law of motion: i.e., it consists of uniform

motion in a straight-line. Hence, in the absence of a net external force, the motion

of the centre of mass is almost certainly far simpler than that of the component

masses.

Now, the velocity of the centre of mass is written

m 1 x ̇ 1 + m 2 x ̇ 2

vcm = x ̇cm =

m 1 + m 2

. (6.7)

We have seen that in the absence of external forces vcm is a constant of the motion
(i.e., the centre of mass does not accelerate). It follows that, in this case,

m 1 x ̇ 1 + m 2 x ̇ 2 (6.8)

is also a constant of the motion. Recall, however, from Sect. 4.3, that momentum

is defined as the product of mass and velocity. Hence, the momentum of the first

mass is written p 1 = m 1 x ̇ 1 , whereas the momentum of the second mass takes the

form p 2 = m 2 x ̇ 2. It follows that the above expression corresponds to the total

momentum of the system:

P = p 1 + p 2. (6.9)

Thus, the total momentum is a conserved quantity—provided there is no net

external force acting on the system. This is true irrespective of the nature of the

internal forces. More generally, Eq. (6.6) can be written

dP

= F. (6.10)

dt

In other words, the time derivative of the total momentum is equal to the net

external force acting on the system—this is just Newton’s second law of motion

applied to the system as a whole.