A Classical Approach of Newtonian Mechanics

6 CONSERVATION OF MOMENTUM 6.6 Collisions in 1 - dimension

the colliding objects. In a totally inelastic collision, the two objects stick together

after the collision, so that vf1 = vf2.

Let us, now, consider elastic collisions in more detail. Suppose that we trans-

form to a frame of reference which co-moves with the centre of mass of the

system. The motion of a multi-component system often looks particularly simple

when viewed in such a frame. Since the system is subject to zero net external
force, the velocity of the centre of mass is invariant, and is given by

vcm =^

m 1 vi1 + m 2 vi2 m 1 + m 2

=

m 1 vf1 + m 2 vf2

. (6.40)
m 1 + m 2

An object which possesses a velocity v in our original frame of reference—henceforth,

termed the laboratory frame—possesses a velocity vJ = v − vcm in the centre of
mass frame. It is easily demonstrated that

viJ 1

viJ 2

m 2 = − (vi2 m 1 + m 2 m 1 = + (vi2 m 1 + m 2

— vi1

), (6.41)

), (6.42)

vfJ 1

vfJ 2

The above equations yield

m 2 = − (vf2 m 1 + m 2 m 1 = + (vf2 m 1 + m 2

— vf1

), (6.43)

). (6.44)

— piJ 1 = piJ 2 = μ (vi2 − vi1), (6.45) −pfJ 1 = pfJ 2 = μ (vf2 − vf1), (6.46)

where μ = m 1 m 2 /(m 1 + m 2 ) is the so-called reduced mass, and piJ 1 = m 1 viJ 1

is the initial momentum of the first object in the centre of mass frame, etc. In

other words, when viewed in the centre of mass frame, the two objects approach

one another with equal and opposite momenta before the collision, and diverge

from one another with equal and opposite momenta after the collision. Thus, the

centre of mass momentum conservation equation,

piJ 1 + piJ 2 = pfJ 1 + pfJ 2 , (6.47)