A Classical Approach of Newtonian Mechanics

6 CONSERVATION OF MOMENTUM 6.6 Collisions in 1 - dimension

momentum of the system is a conserved quantity. Equating the total momenta

before and after the collision, we obtain

m 1 vi1 + m 2 vi2 = m 1 vf1 + m 2 vf2. (6.38)

This equation is valid for any 1 - dimensional collision, irrespective its nature. Note

that, assuming we know the masses of the colliding objects, the above equation

only fully describes the collision if we are given the initial velocities of both ob-

jects, and the final velocity of at least one of the objects. (Alternatively, we could

be given both final velocities and only one of the initial velocities.)

There are many different types of collision. An elastic collision is one in which

the total kinetic energy of the two colliding objects is the same before and after

the collision. Thus, for an elastic collision we can write

1

m v 2 +

1
m^ v^2 =^1 m^ v^2 +^1 m^ v^2 ,^ (6.39)^
2

(^1) i1
2
(^2) i2
2
(^1) f1
2
(^2) f2
in addition to Eq. (6.38). Hence, in this case, the collision is fully specified once
we are given the two initial velocities of the colliding objects. (Alternatively, we
could be given the two final velocities.)
The majority of collisions occurring in real life are not elastic in nature. Some
fraction of the initial kinetic energy of the colliding objects is usually converted
into some other form of energy—generally heat energy, or energy associated with
the mechanical deformation of the objects—during the collision. Such collisions
are termed inelastic. For instance, a large fraction of the initial kinetic energy of
a typical automobile accident is converted into mechanical energy of deformation
of the two vehicles. Inelastic collisions also occur during squash/racquetball/handball
games: in each case, the ball becomes warm to the touch after a long game,
because some fraction of the ball’s kinetic energy of collision with the walls of
the court has been converted into heat energy. Equation (6.38) remains valid
for inelastic collisions—however, Eq. (6.39) is invalid. Thus, generally speak-
ing, an inelastic collision is only fully characterized when we are given the initial
velocities of both objects, and the final velocity of at least one of the objects.
There is, however, a special case of an inelastic collision—called a totally inelastic
collision—which is fully characterized once we are given the initial velocities of