physics
posted by James .
Two balls have masses 44 kg and 86 kg. The
44 kg ball has an initial velocity of 80 m/s (to
the right) along a line joining the two balls
and the 86 kg ball is at rest. The 86 kg
ball has in initial velocity of −24 m/s. The
two balls make a headon elastic collision with
each other.
44 kg
80 m/s
86 kg
−24 m/s
What is the final velocity of the 44 kg ball?

You know the momentum is the same before and after
and
you know the kinetic energy is the same before and after
u = final velocity of 44 kg ball
v = final velocity of 86 kg ball
44*80 + 86* 24 = 44 u + 86 v
(1/2)44(80^2)+(1/2)86(24^2) =(1/2)44u^2+(1/2)86v^2
solve for u 
Since they have not told you the final velocity of at least one ball, you are going to have to use conservation of momentum AND kinetic energy to come up with answers. (Conservation of kinetic energy may not be a good assumption for some kinds of balls that do not bounce well, such as squash balls).
The math gets rather tedious, and I will leave it up to you. You can probably find the solution formula online.
A convenient way to solve this type of problem is to use a coordinate system that moves with the center of mass. In that system, each ball will simply reverse direction and keep the same speed. Then transform speeds back to lab coordinates.