Both before and after the collision, the two objects move with *constant velocity*.
Let and be the velocities of the first and second objects, respectively,
before the collision. Likewise, let and be the velocities of
the first and second objects, respectively,
after the collision. During the collision itself, the first object exerts a large
transitory force on the second, whereas the second object exerts an
equal and opposite force
on the first. In fact, we can model the collision
as equal and opposite *impulses* given to the two objects at the instant in time
when they come together.

We are clearly considering a system in which there is zero net external force (the forces
associated with the collision are internal in nature). Hence, the total momentum of the
system is a conserved quantity. Equating the total momenta before and after the
collision, we obtain

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

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 (217) remains valid for inelastic collisions--however, Eq. (218)
is invalid. Thus, generally speaking, 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 the colliding objects.
In a totally inelastic collision, the two objects *stick together* after the collision, so
that .

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

(219) |

(220) | |||

(221) | |||

(222) | |||

(223) |

The above equations yield

where is the so-called

(226) |

The centre of mass kinetic energy conservation equation takes the form

In other words, the

Equations (217) and (228) can be combined to give the following
pair of equations which fully specify the final velocities (in the laboratory frame) of two objects
which collide elastically, given their initial velocities:

Let us, now, consider some special cases. Suppose that two *equal mass* objects collide elastically.
If then Eqs. (229) and (230) yield

(231) | |||

(232) |

In other words, the two objects simply

Suppose that the second object is much more massive than the first (*i.e.*, )
and is initially at rest (*i.e.*, ). In this case, Eqs. (229) and (230) yield

(233) | |||

(234) |

In other words, the velocity of the light object is effectively

Suppose, finally, that the second object is much lighter than the first (*i.e.*, )
and is initially at rest (*i.e.*, ). In this case, Eqs. (229) and (230) yield

(235) | |||

(236) |

In other words, the motion of the massive object is essentially unaffected by the collision, whereas the light object ends up going

Let us, now, consider totally inelastic collisions in more detail. In a totally inelastic
collision the two objects stick together after colliding, so they
end up moving with the same final velocity
. In this case, Eq. (217) reduces to

(237) |

Suppose that the second object is initially at rest (*i.e.*, ). In this
special case, the common final velocity of the two objects is

(238) |

(239) |