We are again considering a system in which there is zero net external force (the forces associated
with the collision are internal in nature). It follows that the total momentum of
the system is a conserved quantity. However, unlike before, we must now treat the total
momentum as a *vector* quantity, since we are no longer dealing with 1-dimensional
motion. Note that if the collision takes place wholly within the - plane, as indicated in Fig. 55,
then it is sufficient to equate the - and - components of the total momentum before and after the collision.

Consider the -component of the system's total momentum. Before the collision, the
total -momentum is simply , since the second object is initially
stationary, and the first object is initially moving along the -axis with
speed . After the collision, the -momentum of the first object is
: *i.e.*, times the -component of the
first object's final velocity. Likewise, the final -momentum of the second
object is
. Hence, momentum conservation in the -direction
yields

Consider the -component of the system's total momentum. Before the collision, the
total -momentum is zero, since there is initially no motion along the -axis.
After the collision, the -momentum of the first object is
: *i.e.*, times the -component of the
first object's final velocity. Likewise, the final -momentum of the second
object is
. Hence, momentum conservation in the -direction
yields

For the special case of an *elastic* collision, we can equate the
total kinetic energies of the two objects before and after the collision. Hence,
we obtain

Figure 56 shows a 2-dimensional totally inelastic collision. In this
case, the first object, mass , initially moves along the -axis
with speed . On the other hand, the second object, mass , initially moves at an
angle to the -axis with speed . After the collision, the two
objects stick together and move off at an angle to the -axis with
speed . Momentum conservation along the -axis yields

Given the initial conditions (